QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Quandle Homology: Computations andApplications

J. Scott Carter and Masahico Saito

Knots in Washington, Dec. 4 – Nov. 6, 2009, GWU

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Outline

1 Organization

2 Quandle tables

3 Quandle homology

4 Cocycle invariants

5 Classical and virtual knots

6 Knotted surfaces

7 Algebras, categories and others

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Organization

A selfish overview –– I want to have a list of what are known and what areavailable (I forget)

Computationsby computers, and algebraic methods for:

QuandlesHomology groupsCocycle invariants

Applications

Classical and virtual knotsKnotted surfacesAlgebras and categories

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Organization

A selfish overview –– I want to have a list of what are known and what areavailable (I forget)

Computationsby computers, and algebraic methods for:

QuandlesHomology groupsCocycle invariants

Applications

Classical and virtual knotsKnotted surfacesAlgebras and categories

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Organization

A selfish overview –– I want to have a list of what are known and what areavailable (I forget)

Computationsby computers, and algebraic methods for:

QuandlesHomology groupsCocycle invariants

Applications

Classical and virtual knotsKnotted surfacesAlgebras and categories

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Organization

A selfish overview –– I want to have a list of what are known and what areavailable (I forget)

Computationsby computers, and algebraic methods for:

QuandlesHomology groupsCocycle invariants

Applications

Classical and virtual knotsKnotted surfacesAlgebras and categories

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Outline

1 Organization

2 Quandle tables

3 Quandle homology

4 Cocycle invariants

5 Classical and virtual knots

6 Knotted surfaces

7 Algebras, categories and others

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle tables

Computers

[CKS] Appendix, up to 6 elements[Nelson & Co.] Including biquandles, semi-quandles, etc.[Grana-Vendramin] GAP programs for racks, quandles,

and their homology

Algebraic methods

[Joyce, Matveev] Group cosets[Nelson] Classification of finite Alexander quandles[Grana] Order p2 (known for p for SYBE)[CENS, Andruskiewitsch-Grana] Cocycle extensions[Niebrzydowski-Przytycki] Burnside keis

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle tables

Computers

[CKS] Appendix, up to 6 elements[Nelson & Co.] Including biquandles, semi-quandles, etc.[Grana-Vendramin] GAP programs for racks, quandles,

and their homology

Algebraic methods

[Joyce, Matveev] Group cosets[Nelson] Classification of finite Alexander quandles[Grana] Order p2 (known for p for SYBE)[CENS, Andruskiewitsch-Grana] Cocycle extensions[Niebrzydowski-Przytycki] Burnside keis

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle tables

Computers

[CKS] Appendix, up to 6 elements[Nelson & Co.] Including biquandles, semi-quandles, etc.[Grana-Vendramin] GAP programs for racks, quandles,

and their homology

Algebraic methods

[Joyce, Matveev] Group cosets[Nelson] Classification of finite Alexander quandles[Grana] Order p2 (known for p for SYBE)[CENS, Andruskiewitsch-Grana] Cocycle extensions[Niebrzydowski-Przytycki] Burnside keis

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle tables

Computers

[CKS] Appendix, up to 6 elements[Nelson & Co.] Including biquandles, semi-quandles, etc.[Grana-Vendramin] GAP programs for racks, quandles,

and their homology

Algebraic methods

[Joyce, Matveev] Group cosets[Nelson] Classification of finite Alexander quandles[Grana] Order p2 (known for p for SYBE)[CENS, Andruskiewitsch-Grana] Cocycle extensions[Niebrzydowski-Przytycki] Burnside keis

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle tables

Computers

[CKS] Appendix, up to 6 elements[Nelson & Co.] Including biquandles, semi-quandles, etc.[Grana-Vendramin] GAP programs for racks, quandles,

and their homology

Algebraic methods

[Joyce, Matveev] Group cosets[Nelson] Classification of finite Alexander quandles[Grana] Order p2 (known for p for SYBE)[CENS, Andruskiewitsch-Grana] Cocycle extensions[Niebrzydowski-Przytycki] Burnside keis

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle tables

Computers

[CKS] Appendix, up to 6 elements[Nelson & Co.] Including biquandles, semi-quandles, etc.[Grana-Vendramin] GAP programs for racks, quandles,

and their homology

Algebraic methods

[Joyce, Matveev] Group cosets[Nelson] Classification of finite Alexander quandles[Grana] Order p2 (known for p for SYBE)[CENS, Andruskiewitsch-Grana] Cocycle extensions[Niebrzydowski-Przytycki] Burnside keis

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle tables

Computers

[CKS] Appendix, up to 6 elements[Nelson & Co.] Including biquandles, semi-quandles, etc.[Grana-Vendramin] GAP programs for racks, quandles,

and their homology

Algebraic methods

[Joyce, Matveev] Group cosets[Nelson] Classification of finite Alexander quandles[Grana] Order p2 (known for p for SYBE)[CENS, Andruskiewitsch-Grana] Cocycle extensions[Niebrzydowski-Przytycki] Burnside keis

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle tables

Computers

[CKS] Appendix, up to 6 elements[Nelson & Co.] Including biquandles, semi-quandles, etc.[Grana-Vendramin] GAP programs for racks, quandles,

and their homology

Algebraic methods

[Joyce, Matveev] Group cosets[Nelson] Classification of finite Alexander quandles[Grana] Order p2 (known for p for SYBE)[CENS, Andruskiewitsch-Grana] Cocycle extensions[Niebrzydowski-Przytycki] Burnside keis

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle tables

Computers

[CKS] Appendix, up to 6 elements[Nelson & Co.] Including biquandles, semi-quandles, etc.[Grana-Vendramin] GAP programs for racks, quandles,

and their homology

Algebraic methods

[Joyce, Matveev] Group cosets[Nelson] Classification of finite Alexander quandles[Grana] Order p2 (known for p for SYBE)[CENS, Andruskiewitsch-Grana] Cocycle extensions[Niebrzydowski-Przytycki] Burnside keis

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle tables

Computers

[CKS] Appendix, up to 6 elements[Nelson & Co.] Including biquandles, semi-quandles, etc.[Grana-Vendramin] GAP programs for racks, quandles,

and their homology

Algebraic methods

[Joyce, Matveev] Group cosets[Nelson] Classification of finite Alexander quandles[Grana] Order p2 (known for p for SYBE)[CENS, Andruskiewitsch-Grana] Cocycle extensions[Niebrzydowski-Przytycki] Burnside keis

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle tables

Computers

[CKS] Appendix, up to 6 elements[Nelson & Co.] Including biquandles, semi-quandles, etc.[Grana-Vendramin] GAP programs for racks, quandles,

and their homology

Algebraic methods

[Joyce, Matveev] Group cosets[Nelson] Classification of finite Alexander quandles[Grana] Order p2 (known for p for SYBE)[CENS, Andruskiewitsch-Grana] Cocycle extensions[Niebrzydowski-Przytycki] Burnside keis

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Outline

1 Organization

2 Quandle tables

3 Quandle homology

4 Cocycle invariants

5 Classical and virtual knots

6 Knotted surfaces

7 Algebras, categories and others

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology

Computers

[CJKLS] For a small number of quandles[CKS] Up to 6 element quandles, errors corrected by [NP][CEGS] For a generalized theory of [AG], for R3

(Still very limited knowledge for generalized theories)[Grana-Vendramin] GAP program[Niebrzydowski-Przytycki] Dihedral quandles, up to H12

for R3. The “delayed Fibonacci conjecture” for Rp

fn = fn−1 + fn−3, and f (1) = f (2) = 0, f (3) = 1.(Spy’s report: Maybe solved this year by Nosaka)

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology

Computers

[CJKLS] For a small number of quandles[CKS] Up to 6 element quandles, errors corrected by [NP][CEGS] For a generalized theory of [AG], for R3

(Still very limited knowledge for generalized theories)[Grana-Vendramin] GAP program[Niebrzydowski-Przytycki] Dihedral quandles, up to H12

for R3. The “delayed Fibonacci conjecture” for Rp

fn = fn−1 + fn−3, and f (1) = f (2) = 0, f (3) = 1.(Spy’s report: Maybe solved this year by Nosaka)

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology

Computers

[CJKLS] For a small number of quandles[CKS] Up to 6 element quandles, errors corrected by [NP][CEGS] For a generalized theory of [AG], for R3

(Still very limited knowledge for generalized theories)[Grana-Vendramin] GAP program[Niebrzydowski-Przytycki] Dihedral quandles, up to H12

for R3. The “delayed Fibonacci conjecture” for Rp

fn = fn−1 + fn−3, and f (1) = f (2) = 0, f (3) = 1.(Spy’s report: Maybe solved this year by Nosaka)

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology

Computers

[CJKLS] For a small number of quandles[CKS] Up to 6 element quandles, errors corrected by [NP][CEGS] For a generalized theory of [AG], for R3

(Still very limited knowledge for generalized theories)[Grana-Vendramin] GAP program[Niebrzydowski-Przytycki] Dihedral quandles, up to H12

for R3. The “delayed Fibonacci conjecture” for Rp

fn = fn−1 + fn−3, and f (1) = f (2) = 0, f (3) = 1.(Spy’s report: Maybe solved this year by Nosaka)

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology

Computers

[CJKLS] For a small number of quandles[CKS] Up to 6 element quandles, errors corrected by [NP][CEGS] For a generalized theory of [AG], for R3

(Still very limited knowledge for generalized theories)[Grana-Vendramin] GAP program[Niebrzydowski-Przytycki] Dihedral quandles, up to H12

for R3. The “delayed Fibonacci conjecture” for Rp

fn = fn−1 + fn−3, and f (1) = f (2) = 0, f (3) = 1.(Spy’s report: Maybe solved this year by Nosaka)

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology

Computers

[CJKLS] For a small number of quandles[CKS] Up to 6 element quandles, errors corrected by [NP][CEGS] For a generalized theory of [AG], for R3

(Still very limited knowledge for generalized theories)[Grana-Vendramin] GAP program[Niebrzydowski-Przytycki] Dihedral quandles, up to H12

for R3. The “delayed Fibonacci conjecture” for Rp

fn = fn−1 + fn−3, and f (1) = f (2) = 0, f (3) = 1.(Spy’s report: Maybe solved this year by Nosaka)

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology

Computers

[CJKLS] For a small number of quandles[CKS] Up to 6 element quandles, errors corrected by [NP][CEGS] For a generalized theory of [AG], for R3

(Still very limited knowledge for generalized theories)[Grana-Vendramin] GAP program[Niebrzydowski-Przytycki] Dihedral quandles, up to H12

for R3. The “delayed Fibonacci conjecture” for Rp

fn = fn−1 + fn−3, and f (1) = f (2) = 0, f (3) = 1.(Spy’s report: Maybe solved this year by Nosaka)

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology

Computers

[CJKLS] For a small number of quandles[CKS] Up to 6 element quandles, errors corrected by [NP][CEGS] For a generalized theory of [AG], for R3

(Still very limited knowledge for generalized theories)[Grana-Vendramin] GAP program[Niebrzydowski-Przytycki] Dihedral quandles, up to H12

for R3. The “delayed Fibonacci conjecture” for Rp

fn = fn−1 + fn−3, and f (1) = f (2) = 0, f (3) = 1.(Spy’s report: Maybe solved this year by Nosaka)

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology

Computers

[CJKLS] For a small number of quandles[CKS] Up to 6 element quandles, errors corrected by [NP][CEGS] For a generalized theory of [AG], for R3

(Still very limited knowledge for generalized theories)[Grana-Vendramin] GAP program[Niebrzydowski-Przytycki] Dihedral quandles, up to H12

for R3. The “delayed Fibonacci conjecture” for Rp

fn = fn−1 + fn−3, and f (1) = f (2) = 0, f (3) = 1.(Spy’s report: Maybe solved this year by Nosaka)

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology

Computers

[CJKLS] For a small number of quandles[CKS] Up to 6 element quandles, errors corrected by [NP][CEGS] For a generalized theory of [AG], for R3

(Still very limited knowledge for generalized theories)[Grana-Vendramin] GAP program[Niebrzydowski-Przytycki] Dihedral quandles, up to H12

for R3. The “delayed Fibonacci conjecture” for Rp

fn = fn−1 + fn−3, and f (1) = f (2) = 0, f (3) = 1.(Spy’s report: Maybe solved this year by Nosaka)

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology (cont.)

Algebraic methods

[Litherland-Nelson] Rank[Mochizuki] H2, H3 for order p dihedral quandles, with

explicit cocycle constructions[AS] Polynomial cocycles of Alexander quandles[Niebrzydowski-Przytycki] Homology operations∗a(c) = (c ∗ a), ha(c) = (c , a)“Algebraic twist spinning”

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology (cont.)

Algebraic methods

[Litherland-Nelson] Rank[Mochizuki] H2, H3 for order p dihedral quandles, with

explicit cocycle constructions[AS] Polynomial cocycles of Alexander quandles[Niebrzydowski-Przytycki] Homology operations∗a(c) = (c ∗ a), ha(c) = (c , a)“Algebraic twist spinning”

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology (cont.)

Algebraic methods

[Litherland-Nelson] Rank[Mochizuki] H2, H3 for order p dihedral quandles, with

explicit cocycle constructions[AS] Polynomial cocycles of Alexander quandles[Niebrzydowski-Przytycki] Homology operations∗a(c) = (c ∗ a), ha(c) = (c , a)“Algebraic twist spinning”

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology (cont.)

Algebraic methods

[Litherland-Nelson] Rank[Mochizuki] H2, H3 for order p dihedral quandles, with

explicit cocycle constructions[AS] Polynomial cocycles of Alexander quandles[Niebrzydowski-Przytycki] Homology operations∗a(c) = (c ∗ a), ha(c) = (c , a)“Algebraic twist spinning”

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology (cont.)

Algebraic methods

[Litherland-Nelson] Rank[Mochizuki] H2, H3 for order p dihedral quandles, with

explicit cocycle constructions[AS] Polynomial cocycles of Alexander quandles[Niebrzydowski-Przytycki] Homology operations∗a(c) = (c ∗ a), ha(c) = (c , a)“Algebraic twist spinning”

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology (cont.)

Algebraic methods

[Litherland-Nelson] Rank[Mochizuki] H2, H3 for order p dihedral quandles, with

explicit cocycle constructions[AS] Polynomial cocycles of Alexander quandles[Niebrzydowski-Przytycki] Homology operations∗a(c) = (c ∗ a), ha(c) = (c , a)“Algebraic twist spinning”

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology (cont.)

Algebraic methods

[Litherland-Nelson] Rank[Mochizuki] H2, H3 for order p dihedral quandles, with

explicit cocycle constructions[AS] Polynomial cocycles of Alexander quandles[Niebrzydowski-Przytycki] Homology operations∗a(c) = (c ∗ a), ha(c) = (c , a)“Algebraic twist spinning”

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology (cont.)

[Niebrzydowski-Przytycki] Homology operations∗a(c) = (c ∗ a), ha(c) = (c , a)

Pictures for “Algebraic twist spinning”

K KK KK

(1) (2) (3) (4) (5)

1

1aa

0a

a

*aj

j

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology (cont.)

Algebraic methods

[Litherland-Nelson] Rank[Mochizuki] H2, H3 for order p dihedral quandles, with

explicit cocycle constructions[AS] Polynomial cocycles of Alexander quandles[Niebrzydowski-Przytycki] Homology operations∗a(c) = (c ∗ a), ha(c) = (c , a)“Algebraic twist spinning”(Spies report: Partial derivatives)[Nosaka] Alexander quandles of order p

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology (cont.)

Algebraic methods

[Litherland-Nelson] Rank[Mochizuki] H2, H3 for order p dihedral quandles, with

explicit cocycle constructions[AS] Polynomial cocycles of Alexander quandles[Niebrzydowski-Przytycki] Homology operations∗a(c) = (c ∗ a), ha(c) = (c , a)“Algebraic twist spinning”(Spies report: Partial derivatives)[Nosaka] Alexander quandles of order p

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – quandle homology (cont.)

Algebraic methods

[Litherland-Nelson] Rank[Mochizuki] H2, H3 for order p dihedral quandles, with

explicit cocycle constructions[AS] Polynomial cocycles of Alexander quandles[Niebrzydowski-Przytycki] Homology operations∗a(c) = (c ∗ a), ha(c) = (c , a)“Algebraic twist spinning”(Spies report: Partial derivatives)[Nosaka] Alexander quandles of order p

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Outline

1 Organization

2 Quandle tables

3 Quandle homology

4 Cocycle invariants

5 Classical and virtual knots

6 Knotted surfaces

7 Algebras, categories and others

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – cocycle invariants

Computers

[CJKS] Early computer results[CEGS] For AG theory, for twist-spuns[Nelson & Co.] Enhanced invariants, quandles,

biquandles, etc.[AS] For polynomial cocycles[Smudde] Data base for knot table

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – cocycle invariants

Computers

[CJKS] Early computer results[CEGS] For AG theory, for twist-spuns[Nelson & Co.] Enhanced invariants, quandles,

biquandles, etc.[AS] For polynomial cocycles[Smudde] Data base for knot table

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – cocycle invariants

Computers

[CJKS] Early computer results[CEGS] For AG theory, for twist-spuns[Nelson & Co.] Enhanced invariants, quandles,

biquandles, etc.[AS] For polynomial cocycles[Smudde] Data base for knot table

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – cocycle invariants

Computers

[CJKS] Early computer results[CEGS] For AG theory, for twist-spuns[Nelson & Co.] Enhanced invariants, quandles,

biquandles, etc.[AS] For polynomial cocycles[Smudde] Data base for knot table

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – cocycle invariants

Computers

[CJKS] Early computer results[CEGS] For AG theory, for twist-spuns[Nelson & Co.] Enhanced invariants, quandles,

biquandles, etc.[AS] For polynomial cocycles[Smudde] Data base for knot table

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – cocycle invariants

Computers

[CJKS] Early computer results[CEGS] For AG theory, for twist-spuns[Nelson & Co.] Enhanced invariants, quandles,

biquandles, etc.[AS] For polynomial cocycles[Smudde] Data base for knot table

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – cocycle invariants

Computers

[CJKS] Early computer results[CEGS] For AG theory, for twist-spuns[Nelson & Co.] Enhanced invariants, quandles,

biquandles, etc.[AS] For polynomial cocycles[Smudde] Data base for knot table

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – cocycle invariants

12/1/09 5:08 PMDatabase: Quandle Cocycle Knot Invariants

Page 1 of 5http://shell.cas.usf.edu/quandle/Invariants/database/database.php

Database of Quandle Cocycle Knot Invariants

This page is the front end to our database. By checking the desired boxes you can create a table of values ofdifferent quandle cocycle knot invariants.

See the NOTES page for comments about the contents and use of the quandle cocycle knot invariantdatabase.

Crossing Numbers of Knots

Select the crossing numbers of the prime knots that you want in your table.

8 and fewer 9 10 11 12

Knot Information

Check the boxes of the knot information that you want in your table.

NumberedKnot

Name

Braid

Word

Crossing

Number

Braid

Length

Number of

Strands

Alexander

Polynomial

Quandle Cocycle Invariants with Alexander Quandles and Mochizuki Cocycles

Dihedral Quandles with Mochizuki 3-Cocycles f(x,y,z)=(x-y)[(2zp-yp)-(2z-y)p]/p mod p

R2 R3 R5 R7 R11

R13 R17 R19 R23 R29

R31 R37 R41 R43 R47

Cocycle values

Values of Mochizuki 3-cocycle formula f(x,y,z)=(x-y)[(2zp-yp)-(2z-y)p]/p mod p

Alexander Quandles with Mochizuki 2-cocycles f(x,y)=(x-y)p

2[t,t-1] / 2[t,t-1] / 2[t,t-1] / 2[t,t-1] /

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – cocycle invariants

Computers

[CJKS] Early computer results[CEGS] For AG theory, for twist-spuns[Nelson & Co.] Enhanced invariants, quandles,

biquandles, etc.[AS] For polynomial cocycles[Smudde] Data base for knot table

Algebraic methods

[CENS,AG] ExtensionsX ×φ A, (x , a) ∗ (y , b) = (x / y , a + φ(x , y))[Asami-Kuga,Iwakiri,Satoh,Shima] For twist-spuns using

Mochizuki cocycles[Satoh] For spatial graphs using Mochizuki cocycles

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – cocycle invariants

Computers

[CJKS] Early computer results[CEGS] For AG theory, for twist-spuns[Nelson & Co.] Enhanced invariants, quandles,

biquandles, etc.[AS] For polynomial cocycles[Smudde] Data base for knot table

Algebraic methods

[CENS,AG] ExtensionsX ×φ A, (x , a) ∗ (y , b) = (x / y , a + φ(x , y))[Asami-Kuga,Iwakiri,Satoh,Shima] For twist-spuns using

Mochizuki cocycles[Satoh] For spatial graphs using Mochizuki cocycles

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – cocycle invariants

Computers

[CJKS] Early computer results[CEGS] For AG theory, for twist-spuns[Nelson & Co.] Enhanced invariants, quandles,

biquandles, etc.[AS] For polynomial cocycles[Smudde] Data base for knot table

Algebraic methods

[CENS,AG] ExtensionsX ×φ A, (x , a) ∗ (y , b) = (x / y , a + φ(x , y))[Asami-Kuga,Iwakiri,Satoh,Shima] For twist-spuns using

Mochizuki cocycles[Satoh] For spatial graphs using Mochizuki cocycles

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – cocycle invariants

Computers

[CJKS] Early computer results[CEGS] For AG theory, for twist-spuns[Nelson & Co.] Enhanced invariants, quandles,

biquandles, etc.[AS] For polynomial cocycles[Smudde] Data base for knot table

Algebraic methods

[CENS,AG] ExtensionsX ×φ A, (x , a) ∗ (y , b) = (x / y , a + φ(x , y))[Asami-Kuga,Iwakiri,Satoh,Shima] For twist-spuns using

Mochizuki cocycles[Satoh] For spatial graphs using Mochizuki cocycles

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – cocycle invariants

Computers

[CJKS] Early computer results[CEGS] For AG theory, for twist-spuns[Nelson & Co.] Enhanced invariants, quandles,

biquandles, etc.[AS] For polynomial cocycles[Smudde] Data base for knot table

Algebraic methods

[CENS,AG] ExtensionsX ×φ A, (x , a) ∗ (y , b) = (x / y , a + φ(x , y))[Asami-Kuga,Iwakiri,Satoh,Shima] For twist-spuns using

Mochizuki cocycles[Satoh] For spatial graphs using Mochizuki cocycles

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Computations – cocycle invariants

Computers

[CJKS] Early computer results[CEGS] For AG theory, for twist-spuns[Nelson & Co.] Enhanced invariants, quandles,

biquandles, etc.[AS] For polynomial cocycles[Smudde] Data base for knot table

Algebraic methods

[CENS,AG] ExtensionsX ×φ A, (x , a) ∗ (y , b) = (x / y , a + φ(x , y))[Asami-Kuga,Iwakiri,Satoh,Shima] For twist-spuns using

Mochizuki cocycles[Satoh] For spatial graphs using Mochizuki cocycles

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Outline

1 Organization

2 Quandle tables

3 Quandle homology

4 Cocycle invariants

5 Classical and virtual knots

6 Knotted surfaces

7 Algebras, categories and others

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – classical and virtual knots

[Fenn-Rourke] Chirality of trefoil

[Nelson & Co.] Variety of enhanced invariants for virtualknots

[Satoh] Chirality of spatial graphs

[Eiserman] Characterization of the unknot,knot coloring polynomials (generalizations)

[CESS] Minimal numbers of type III moves

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – classical and virtual knots

[Fenn-Rourke] Chirality of trefoil

[Nelson & Co.] Variety of enhanced invariants for virtualknots

[Satoh] Chirality of spatial graphs

[Eiserman] Characterization of the unknot,knot coloring polynomials (generalizations)

[CESS] Minimal numbers of type III moves

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – classical and virtual knots

[Fenn-Rourke] Chirality of trefoil

[Nelson & Co.] Variety of enhanced invariants for virtualknots

[Satoh] Chirality of spatial graphs

[Eiserman] Characterization of the unknot,knot coloring polynomials (generalizations)

[CESS] Minimal numbers of type III moves

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – classical and virtual knots

[Fenn-Rourke] Chirality of trefoil

[Nelson & Co.] Variety of enhanced invariants for virtualknots

[Satoh] Chirality of spatial graphs

[Eiserman] Characterization of the unknot,knot coloring polynomials (generalizations)

[CESS] Minimal numbers of type III moves

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – classical and virtual knots

[Fenn-Rourke] Chirality of trefoil

[Nelson & Co.] Variety of enhanced invariants for virtualknots

[Satoh] Chirality of spatial graphs

[Eiserman] Characterization of the unknot,knot coloring polynomials (generalizations)

[CESS] Minimal numbers of type III moves

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – classical and virtual knots

[Fenn-Rourke] Chirality of trefoil

[Nelson & Co.] Variety of enhanced invariants for virtualknots

[Satoh] Chirality of spatial graphs

[Eiserman] Characterization of the unknot,knot coloring polynomials (generalizations)

[CESS] Minimal numbers of type III moves

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – classical and virtual knots

Pictures for:

[CESS] Minimal numbers of type III moves

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – classical and virtual knots

[Fenn-Rourke] Chirality of trefoil

[Nelson & Co.] Variety of enhanced invariants for virtualknots

[Satoh] Chirality of spatial graphs

[Eiserman] Characterization of the unknot,knot coloring polynomials (generalizations)

[CESS] Minimal numbers of type III moves

[CESSW] Non-existence of checkerboard coloring ofvirtual knots

[AERSS] Tangle embedding

[ABDHKS] Determinants for random knots

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – classical and virtual knots

[Fenn-Rourke] Chirality of trefoil

[Nelson & Co.] Variety of enhanced invariants for virtualknots

[Satoh] Chirality of spatial graphs

[Eiserman] Characterization of the unknot,knot coloring polynomials (generalizations)

[CESS] Minimal numbers of type III moves

[CESSW] Non-existence of checkerboard coloring ofvirtual knots

[AERSS] Tangle embedding

[ABDHKS] Determinants for random knots

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – classical and virtual knots

[Fenn-Rourke] Chirality of trefoil

[Nelson & Co.] Variety of enhanced invariants for virtualknots

[Satoh] Chirality of spatial graphs

[Eiserman] Characterization of the unknot,knot coloring polynomials (generalizations)

[CESS] Minimal numbers of type III moves

[CESSW] Non-existence of checkerboard coloring ofvirtual knots

[AERSS] Tangle embedding

[ABDHKS] Determinants for random knots

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – classical and virtual knots

[Fenn-Rourke] Chirality of trefoil

[Nelson & Co.] Variety of enhanced invariants for virtualknots

[Satoh] Chirality of spatial graphs

[Eiserman] Characterization of the unknot,knot coloring polynomials (generalizations)

[CESS] Minimal numbers of type III moves

[CESSW] Non-existence of checkerboard coloring ofvirtual knots

[AERSS] Tangle embedding

[ABDHKS] Determinants for random knots

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – classical and virtual knots

Pictures for:

[ABDHKS] Determinants for random knots

40320.0

0.1

36

0.16

4 12

0.2

0.08

0.02

24

0.18

x

0.04

8 2820

0.06

0.12

0.14

16 36

700

28

500

124

1,000

900

40

800

3224

600

20168

400

300

200

100

0

Determinant distributions of random polygonal knots in theunit cube.

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – classical and virtual knots

[Fenn-Rourke] Chirality of trefoil

[Nelson & Co.] Variety of enhanced invariants for virtualknots

[Satoh] Chirality of spatial graphs

[Eiserman] Characterization of the unknot,knot coloring polynomials (generalizations)

[CESS] Minimal numbers of type III moves

[CESSW] Non-existence of checkerboard coloring ofvirtual knots

[AERSS] Tangle embedding

[ABDHKS] Determinants for random knots

[S] Minimum Fox colors

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – classical and virtual knots

[Fenn-Rourke] Chirality of trefoil

[Nelson & Co.] Variety of enhanced invariants for virtualknots

[Satoh] Chirality of spatial graphs

[Eiserman] Characterization of the unknot,knot coloring polynomials (generalizations)

[CESS] Minimal numbers of type III moves

[CESSW] Non-existence of checkerboard coloring ofvirtual knots

[AERSS] Tangle embedding

[ABDHKS] Determinants for random knots

[S] Minimum Fox colors

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – classical and virtual knots

Pictures for:

[S] Minimum Fox colors

1 33 1

2

0 0

21 1

0 0

Links with four colors for any p, have certain properties in thequandle cocycle invariant. Relations to Milnor’s invariant issuspected.

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – classical and virtual knots

[Fenn-Rourke] Chirality of trefoil[Nelson & Co.] Variety of enhanced invariants for virtual

knots[Satoh] Chirality of spatial graphs[Eiserman] Characterization of the unknot,

knot coloring polynomials (generalizations)[CESS] Minimal numbers of type III moves[CESSW] Non-existence of checkerboard coloring of

virtual knots[AERSS] Tangle embedding[ABDHKS] Determinants for random knots[S] Minimum Fox colors[Ishii-Tanaka] Invariants of embedded handle-bodies[Inoue] Volume is a quandle cocycle inv.[Hamaya-Inoue] Chern-Simons inv is a quandle cocycle

inv.

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – classical and virtual knots

[Fenn-Rourke] Chirality of trefoil[Nelson & Co.] Variety of enhanced invariants for virtual

knots[Satoh] Chirality of spatial graphs[Eiserman] Characterization of the unknot,

knot coloring polynomials (generalizations)[CESS] Minimal numbers of type III moves[CESSW] Non-existence of checkerboard coloring of

virtual knots[AERSS] Tangle embedding[ABDHKS] Determinants for random knots[S] Minimum Fox colors[Ishii-Tanaka] Invariants of embedded handle-bodies[Inoue] Volume is a quandle cocycle inv.[Hamaya-Inoue] Chern-Simons inv is a quandle cocycle

inv.

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – classical and virtual knots

[Fenn-Rourke] Chirality of trefoil[Nelson & Co.] Variety of enhanced invariants for virtual

knots[Satoh] Chirality of spatial graphs[Eiserman] Characterization of the unknot,

knot coloring polynomials (generalizations)[CESS] Minimal numbers of type III moves[CESSW] Non-existence of checkerboard coloring of

virtual knots[AERSS] Tangle embedding[ABDHKS] Determinants for random knots[S] Minimum Fox colors[Ishii-Tanaka] Invariants of embedded handle-bodies[Inoue] Volume is a quandle cocycle inv.[Hamaya-Inoue] Chern-Simons inv is a quandle cocycle

inv.

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – classical and virtual knots

[Fenn-Rourke] Chirality of trefoil[Nelson & Co.] Variety of enhanced invariants for virtual

knots[Satoh] Chirality of spatial graphs[Eiserman] Characterization of the unknot,

knot coloring polynomials (generalizations)[CESS] Minimal numbers of type III moves[CESSW] Non-existence of checkerboard coloring of

virtual knots[AERSS] Tangle embedding[ABDHKS] Determinants for random knots[S] Minimum Fox colors[Ishii-Tanaka] Invariants of embedded handle-bodies[Inoue] Volume is a quandle cocycle inv.[Hamaya-Inoue] Chern-Simons inv is a quandle cocycle

inv.

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Outline

1 Organization

2 Quandle tables

3 Quandle homology

4 Cocycle invariants

5 Classical and virtual knots

6 Knotted surfaces

7 Algebras, categories and others

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – knotted surfaces

[CJKLS,CEGS] Invertibility

[Satoh-Shima, Hatakenaka, Iwakiri, Mohamad-Yashiro,Kamada-Oshiro] Minimal triple point numbers

[SS] Minimal broken sheet numbers

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – knotted surfaces

[CJKLS,CEGS] Invertibility

[Satoh-Shima, Hatakenaka, Iwakiri, Mohamad-Yashiro,Kamada-Oshiro] Minimal triple point numbers

[SS] Minimal broken sheet numbers

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – knotted surfaces

[CJKLS,CEGS] Invertibility

[Satoh-Shima, Hatakenaka, Iwakiri, Mohamad-Yashiro,Kamada-Oshiro] Minimal triple point numbers

[SS] Minimal broken sheet numbers

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – knotted surfaces

[CJKLS,CEGS] Invertibility

[Satoh-Shima, Hatakenaka, Iwakiri, Mohamad-Yashiro,Kamada-Oshiro] Minimal triple point numbers

[SS] Minimal broken sheet numbers

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – knotted surfaces

Picture for:

[SS] Minimal broken sheet numbers

The spun trefoil needs 4 sheets

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – knotted surfaces

[CJKLS,CEGS] Invertibility

[Satoh-Shima, Hatakenaka, Iwakiri, Mohamad-Yashiro,Kamada-Oshiro] Minimal triple point numbers

[SS] Minimal broken sheet numbers

[CSS] Ribbon concordance

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – knotted surfaces

[CJKLS,CEGS] Invertibility

[Satoh-Shima, Hatakenaka, Iwakiri, Mohamad-Yashiro,Kamada-Oshiro] Minimal triple point numbers

[SS] Minimal broken sheet numbers

[CSS] Ribbon concordance

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – knotted surfaces

Picture for:

[CSS] Ribbon concordance

0F

Cocycle inv detects which knotted spheres are not ribbonconcordant.

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – knotted surfaces

[CJKLS,CEGS] Non-invertibility

[Satoh-Shima, Hatakenaka, Iwakiri, Mohamad-Yashiro,Kamada-Oshiro] Minimal triple point numbers

[SS] Minimal broken sheet numbers

[CSS] Ribbon concordance

[Iwakiri] Unknotting numbers

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – knotted surfaces

[CJKLS,CEGS] Non-invertibility

[Satoh-Shima, Hatakenaka, Iwakiri, Mohamad-Yashiro,Kamada-Oshiro] Minimal triple point numbers

[SS] Minimal broken sheet numbers

[CSS] Ribbon concordance

[Iwakiri] Unknotting numbers

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Outline

1 Organization

2 Quandle tables

3 Quandle homology

4 Cocycle invariants

5 Classical and virtual knots

6 Knotted surfaces

7 Algebras, categories and others

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – algebras, categories and others

[Andruskiewitsch-Grana] Pointed Hopf algebras

[CCES] Categorical self-distributivity for Lie alg

[CCES] Categorical self-distributivity for the adjoint mapfor Hopf alg

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – algebras, categories and others

[Andruskiewitsch-Grana] Pointed Hopf algebras

[CCES] Categorical self-distributivity for Lie alg

[CCES] Categorical self-distributivity for the adjoint mapfor Hopf alg

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – algebras, categories and others

[Andruskiewitsch-Grana] Pointed Hopf algebras

[CCES] Categorical self-distributivity for Lie alg

[CCES] Categorical self-distributivity for the adjoint mapfor Hopf alg

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – algebras, categories and others

[Andruskiewitsch-Grana] Pointed Hopf algebras

[CCES] Categorical self-distributivity for Lie alg

[CCES] Categorical self-distributivity for the adjoint mapfor Hopf alg

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – algebras, categories and others

Picture for:

[CCES] Categorical self-distributivity for the adjoint mapfor Hopf alg

(2)

!1

(1)ba

b

(2)

b a b

a b

b (1)

bba b

(2)

(3)

(3)

S(b ) a(2) b(3)

(3)

S(b ) S

The adjoint map of Hopf algebras is an analogue ofconjugation quandle.

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – algebras, categories and others

[Andruskiewitsch-Grana] Pointed Hopf algebras

[CCES] Categorical self-distributivity for Lie alg

[CCES] Categorical self-distributivity for the adjoint mapfor Hopf alg

[CCES] 2-Quandles and 2-cocycles (Spy’s report: comingsoon)

[Zablow] Dehn quandle

[Kamada-Matsumoto] Elliptic fibrations and Dehnquandles

[Niebrzydowski-Przytycki] Trefoil quandle as Dehnquandle

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – algebras, categories and others

[Andruskiewitsch-Grana] Pointed Hopf algebras

[CCES] Categorical self-distributivity for Lie alg

[CCES] Categorical self-distributivity for the adjoint mapfor Hopf alg

[CCES] 2-Quandles and 2-cocycles (Spy’s report: comingsoon)

[Zablow] Dehn quandle

[Kamada-Matsumoto] Elliptic fibrations and Dehnquandles

[Niebrzydowski-Przytycki] Trefoil quandle as Dehnquandle

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – algebras, categories and others

[Andruskiewitsch-Grana] Pointed Hopf algebras

[CCES] Categorical self-distributivity for Lie alg

[CCES] Categorical self-distributivity for the adjoint mapfor Hopf alg

[CCES] 2-Quandles and 2-cocycles (Spy’s report: comingsoon)

[Zablow] Dehn quandle

[Kamada-Matsumoto] Elliptic fibrations and Dehnquandles

[Niebrzydowski-Przytycki] Trefoil quandle as Dehnquandle

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – algebras, categories and others

[Andruskiewitsch-Grana] Pointed Hopf algebras

[CCES] Categorical self-distributivity for Lie alg

[CCES] Categorical self-distributivity for the adjoint mapfor Hopf alg

[CCES] 2-Quandles and 2-cocycles (Spy’s report: comingsoon)

[Zablow] Dehn quandle

[Kamada-Matsumoto] Elliptic fibrations and Dehnquandles

[Niebrzydowski-Przytycki] Trefoil quandle as Dehnquandle

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Applications – algebras, categories and others

[Andruskiewitsch-Grana] Pointed Hopf algebras

[CCES] Categorical self-distributivity for Lie alg

[CCES] Categorical self-distributivity for the adjoint mapfor Hopf alg

[CCES] 2-Quandles and 2-cocycles (Spy’s report: comingsoon)

[Zablow] Dehn quandle

[Kamada-Matsumoto] Elliptic fibrations and Dehnquandles

[Niebrzydowski-Przytycki] Trefoil quandle as Dehnquandle

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Final remarks

My motivation was:A selfish overview –– I wanted to have a list of what are known and what areavailable (I forget)But still I hope this will be obsolete next year!

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Final remarks

My motivation was:A selfish overview –– I wanted to have a list of what are known and what areavailable (I forget)But still I hope this will be obsolete next year!

QuandleHomology:

Computationsand

Applications

J.S. Carter &M. Saito

Organization

Quandle tables

Quandlehomology

Cocycleinvariants

Classical andvirtual knots

Knottedsurfaces

Algebras,categories andothers

Final remarks

My motivation was:A selfish overview –– I wanted to have a list of what are known and what areavailable (I forget)But still I hope this will be obsolete next year!