Two generalizations of the Multivariable Alexander Polynomial

The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions

Two generalizations of the MultivariableAlexander Polynomial

Iva Halacheva(University of Toronto)

Knots in Washington XLIDecember 6, 2015


Table of Contents

1 The original MVALong virtual knots

2 The Archibald generalizationCircuit algebrasAlexander Half DensitiesThe invariant

3 The Bar-Natan generalizationMetamonoidsThe invariant

4 Some resultsReducing the Archibald invariantMaking the connection

5 Future directions


Long virtual knots

Setup

We will consider:vMVA=the MVA for regular long virtual knots and links

Ingredients

Given a virtual link L with n components, diagram DL :

Label all the arcs of DL (using a set S). To each componentassociate a variable, {ti}ni=1.

Build the Alexander matrix:

M(DL ) ∈ MS\{a}×S(Z(ti | i = 1, . . . , n)), a = incoming long arc

s3

s1

t2 t1

s2

7→s1 s2 s3

s3 1 − t2 −1 t1


Long virtual knots

Setup


Ingredients





s3

s1

t2 t1

s2

7→s1 s2 s3

s3 1 − t2 −1 t1


Long virtual knots

Setup


Ingredients





s3

s1

t2 t1

s2

7→s1 s2 s3

s3 1 − t2 −1 t1


Long virtual knots

Setup (cont’d)

s2

t1s3

t2

s1

7→s1 s2 s3

s3 t2 − 1 −t1 1

DefinitionThe Multivariable Alexander Polynomial for a regular long virtuallink L is defined as:

vMVA(L) =∏

i

t−µ(i)/2i

det M′(DL )

tl − 1

where,µ(i) = #{i-th component goes over a crossing}

M′(DL ) = M(DL ) with the column a deleted

tl = variable of the long strand


Long virtual knots

Setup (cont’d)

s2

t1s3

t2

s1

7→s1 s2 s3

s3 t2 − 1 −t1 1

DefinitionThe Multivariable Alexander Polynomial for a regular long virtuallink L is defined as:

vMVA(L) =∏

i

t−µ(i)/2i

det M′(DL )

tl − 1

where,µ(i) = #{i-th component goes over a crossing}

M′(DL ) = M(DL ) with the column a deleted

tl = variable of the long strand


Long virtual knots

Additional arcs: We might need to subdivide an arc in laterconstructions.

b−→

b

a7→

a ba −1 1

Lemma

Dividing an arc will not change the value of vMVA(L).


Circuit algebras

Virtual Tangles

Underlying structure: Circuit Algebras.

DefinitionA (oriented) circuit diagram with k inputs is a disk with k internaldisks removed, and (oriented) pairings among the boundary points.

DefinitionA circuit algebra is a collection of spacesV = {Vn}n∈N andmorphisms F = {Fd}d a circuit diagram.

DefinitionAn oriented circuit algebra is a collection of spacesV = {Vn,m}n,m∈N (n incoming and m outgoing components) andmorphisms F = {Fd}d an oriented circuit diagram.


Circuit algebras

Virtual Tangles






Circuit algebras

Virtual Tangles






Circuit algebras

Example: A circuit diagram.

: V1,2 ⊗ V3,2 −→ V1,1

Remarks:

Circuit algebras form a category.

Virtual tangles are a circuit algebra, CA⟨!,"

⟩.

Goal: Invariant of tangles which is a circuit algebra morphism.


Circuit algebras

Example: A circuit diagram.

: V1,2 ⊗ V3,2 −→ V1,1

Remarks:

Circuit algebras form a category.

Virtual tangles are a circuit algebra, CA⟨!,"

⟩.

Goal: Invariant of tangles which is a circuit algebra morphism.


Alexander Half Densities

Given S a finite set, R a ring, we’ll denote:

Λk (S) := k -th exterior power of the formal R-module with basis S

Definition

Let X in and Xout denote the labelling sets of the incoming andoutgoing arcs of a tangle (we require n =

∣∣∣X in∣∣∣ =

∣∣∣Xout∣∣∣). The

corresponding Alexander Half Density is:

AHD(X in,Xout) := Λn(Xout) ⊗ Λn(X in ∪ Xout)

Remark: To guarantee that∣∣∣X in

∣∣∣ =∣∣∣Xout

∣∣∣, we might have to breaksome arcs artificially.



Given S a finite set, R a ring, we’ll denote:

Λk (S) := k -th exterior power of the formal R-module with basis S

Definition

Let X in and Xout denote the labelling sets of the incoming andoutgoing arcs of a tangle (we require n =

∣∣∣X in∣∣∣ =

∣∣∣Xout∣∣∣). The

corresponding Alexander Half Density is:

AHD(X in,Xout) := Λn(Xout) ⊗ Λn(X in ∪ Xout)

Remark: To guarantee that∣∣∣X in

∣∣∣ =∣∣∣Xout

∣∣∣, we might have to breaksome arcs artificially.



The morphisms are defined using interior multiplication:

Definition (The Gluing Map)

Suppose {aj ⊗ bj ∈ AHD(X inj ,X

outj )}j=1,...,m with the elements to be

glued labelled the same. Let G =⋃

j X inj ∩

⋃j Xout

j denote the setof elements to be glued. The gluing map produces:

iG

m∧j=1

aj

⊗ iG

m∧j=1

bj

∈ AHD

⋃j

X inj − G,

⋃j

Xoutj − G

Where iG is interior multiplication.

Proposition

(AHD, gluing) is a circuit algebra.



The morphisms are defined using interior multiplication:

Definition (The Gluing Map)

Suppose {aj ⊗ bj ∈ AHD(X inj ,X

outj )}j=1,...,m with the elements to be

glued labelled the same. Let G =⋃

j X inj ∩

⋃j Xout

j denote the setof elements to be glued. The gluing map produces:

iG

m∧j=1

aj

⊗ iG

m∧j=1

bj

∈ AHD

⋃j

X inj − G,

⋃j

Xoutj − G

Where iG is interior multiplication.

Proposition

(AHD, gluing) is a circuit algebra.


The invariant

Construction of a tangle invariant:

tMVA : {Regular v-tangles} −→ {Alexander Half Densities}

Given an n-tangle T with a labelled diagram DT :

Construct the Alexander matrix M(DT ) as before:

M(DT ) =

internal Xout X in

internal

Xout

Let w ∈ Λn

(Xout

), a choice of ordering of the labels of Xout.

Let M(DT )i1<...<in = the minor of M(DT ) computed using all“internal columns” and those labelled ci1 , . . . , cin in Xout ∪ X in.


The invariant





M(DT ) =

internal Xout X in

internal

Xout

Let w ∈ Λn(Xout




The invariant





M(DT ) =

internal Xout X in

internal

Xout

Let w ∈ Λn

(Xout




The invariant

Theorem (Archibald)

An invariant of regular, virtual n-tangles can be defined by, for anytangle T:

tMVA(T) =n∏

k=1

t−µ(k)/2k w ⊗

∑i1<...<in

M(DT )i1<...<in ci1 ∧ . . . ∧ cin

In particular:

a2

b1

t2a1

t1

b2

7→

a1

b2

t1a2

t2

b1

7→


The invariant

Theorem (Archibald)


tMVA(T) =n∏

k=1

t−µ(k)/2k w ⊗

∑i1<...<in

M(DT )i1<...<in ci1 ∧ . . . ∧ cin

In particular:

a2

b1

t2a1

t1

b2

7→

a1 a2 b1 b2

a1 1 0 −1 0a2 1 − t2 t1 0 −1

a1

b2

t1a2

t2

b1

7→

a1 a2 b1 b2

a1 1 0 −1 0a2 t2 − 1 1 0 −t1


The invariant

Theorem (Archibald)


tMVA(T) =n∏

k=1

t−µ(k)/2k w ⊗

∑i1<...<in

M(DT )i1<...<in ci1 ∧ . . . ∧ cin

In particular:

a2

b1

t2a1

t1

b2

7→t−1/21 a1 ∧ a2 ⊗ [b1 ∧ b2 + (t2 − 1)b1 ∧ a1

−(t1)b1 ∧ a2 + b2 ∧ a1 + (t1)a1 ∧ a2]

a1

b2

t1a2

t2

b1

7→t−1/21 a2 ∧ a1 ⊗ [(t1)b2 ∧ b1 − (t1)b2 ∧ a1

+b1 ∧ a2 + (t2 − 1)b1 ∧ a1 + a2 ∧ a1]


The invariant

Example: Suppose we are considering the following tangle T .

a1 a2

b1 b2

7→

b1 b2 a1 a2

b1 t2 0 −1 1 − t1b2 0 t1 1 − t2 −1

tMVA(T) = t−1/21 t−1/2

2 b1 ∧ b2 ⊗ [(t1t2)b1 ∧ b2 + (t2(1 − t2))b1 ∧ a1

−(t2)b1 ∧ a2 + (t1)b2 ∧ a1 − (t1(1 − t1))b2 ∧ a2 + (t1 + t2 − t1t2)a1 ∧ a2]


The invariant


a1 a2

b1 b2

7→

b1 b2 a1 a2

b1 t2 0 −1 1 − t1

b2 0 t1 1 − t2 −1

tMVA(T) = t−1/21 t−1/2

2 b1 ∧ b2 ⊗ [(t1t2)b1 ∧ b2 + (t2(1 − t2))b1 ∧ a1



The invariant


a1 a2

b1 b2

7→

b1 b2 a1 a2

b1 t2 0 −1 1 − t1b2 0 t1 1 − t2 −1

tMVA(T) = t−1/21 t−1/2

2 b1 ∧ b2 ⊗ [(t1t2)b1 ∧ b2 + (t2(1 − t2))b1 ∧ a1



The invariant


a1 a2

b1 b2

7→

b1 b2 a1 a2

b1 t2 0 −1 1 − t1b2 0 t1 1 − t2 −1

tMVA(T) = t−1/21 t−1/2

2 b1 ∧ b2 ⊗ [(t1t2)b1 ∧ b2 + (t2(1 − t2))b1 ∧ a1



The invariant

Properties of the tMVA:

It is a circuit algebra morphism.

Satisfies the extra “Overcrossings Commute” relation, so is awelded tangle invariant.

Can renormalize for usual tangles to get R1 invariance:

tMVA′ =∏

k

t rot(k)/2k tMVA

Can recover vMVA:

tMVA(

a−→ T

b−→

)ta − 1

= vMVA(

a−→ T

b−→

)(b ⊗ b − a ⊗ a)

Gives easy verification of many local vMVA relations(Conway’s second and third identity, Murakami’s fifth axiom,doubled delta move).


Metamonoids

DefinitionA meta-monoid is a collection of sets {GX }X=a finite set together withmaps between them:

mxyz : G{x,y}∪X −→ G{z}∪X “multiplication”∗ : GX × GY −→ GX∪Y “union”ηx : G{x}∪X −→ GX “deletion”σx

z : G{x}∪X −→ G{z}∪X “renaming”ex : GX −→ G{x}∪X “identity”

satisfying:1 “Monoid axioms”

mxyz ◦ ex = σ

yz = myx

z ◦ ex

mxyz ◦muv

x = muxz ◦mvy

x

2 A list of “set manipulation” axioms.


Metamonoids

Examples:

The collection {GX }X=a finite set where G is a monoid andGX = {f : X → G}, with the standard operations, is ameta-monoid.

Pure, regular virtual tangles form a meta-monoid, withGX = {tangles with strand labels in X} and the standardoperations.

Note: A property for monoids that does not, in general, holdfor meta-monoids.

ηxg ∗ ηyg = g, for g ∈ G{x,y}


Metamonoids

Examples:

The collection {GX }X=a finite set where G is a monoid andGX = {f : X → G}, with the standard operations, is ameta-monoid.

Pure, regular virtual tangles form a meta-monoid, withGX = {tangles with strand labels in X} and the standardoperations.

Note: A property for monoids that does not, in general, holdfor meta-monoids.

ηxg ∗ ηyg = g, for g ∈ G{x,y}


Metamonoids

The target space:

Definition

For a finite set X , let R = Z(Tx | x ∈ X). Take:

BX = R ×MatX×X (R)

And the operations:λ a b Xa α β θ

b γ δ ε

X φ ψ Ξ

mabc

−−−−−−→µB1−β

ta ,tb→tc

λµ c Xc γ + αδ/µ ε + δθ/µ

X φ + αψ/µ Ξ + ψθ/µ

(λ1 X1

X1 A1,λ2 X2

X2 A2

)∗−−→

λ1λ2 X1 X2

X1 A1 0X2 0 A2

Proposition

BX with the above maps has a meta-monoid structure.


Metamonoids

The target space:

Definition




b γ δ ε

X φ ψ Ξ

mabc

−−−−−−→µB1−β

ta ,tb→tc



(λ1 X1

X1 A1,λ2 X2

X2 A2

)∗−−→

λ1λ2 X1 X2

X1 A1 0X2 0 A2

Proposition



Metamonoids

The target space:

Definition




b γ δ ε

X φ ψ Ξ

mabc

−−−−−−→µB1−β

ta ,tb→tc



(λ1 X1

X1 A1,λ2 X2

X2 A2

)∗−−→

λ1λ2 X1 X2

X1 A1 0X2 0 A2

Proposition



The invariant

Theorem (Bar-Natan)There exists a unique meta-monoid morphismZ : {Pure, regular, X-labelled tangles} −→ BX such that:

a b7→

1 a ba 1 1 − tab 0 ta b a

7→

1 a ba 1 1 − t−1

ab 0 t−1

a

a 7→1 aa 1


The invariant

Example:


The invariant

Example:

cd

ab


The invariant

Example:

cd

ab7→

1 a ba 1 1 − t−1

ab 0 t−1

a

∗

1 c dc 1 1 − t−1

cd 0 t−1

c


The invariant

Example:

cd

ab

7→

1 a b c da 1 1 − ta 0 0b 0 ta 0 0c 0 0 1 1 − tcd 0 0 0 tc


The invariant

Example:

c

ab

7→

1 a b ca tc −(ta − 1)tc 0b 0 ta 0c 1 − tc (ta − 1)(tc − 1) 1


The invariant

Example:

c

a

7→

ta + tc − ta tc a c

a −tc−ta−tc+ta tc

(ta−1)tc−ta−tc+ta tc

c ta(tc−1)−ta−tc+ta tc

−ta−ta−tc+ta tc


The invariant

Properties:

Satisfies “Overcrossings Commute”, so is a welded tangleinvariant.

For a long knot K , produces (λ, 1) where λ is the Alexanderpolynomial of K (modulo units).

For a long link L , the result (λ,A) gives the MVA (modulounits): λ det(A − I)/(1 − tl).

Efficient computationally (polynomial time in the number ofstrands).

Reduction of an invariant:

{ribbon-knotted S2 and S1 in R4} −→ {Free Lie and cyclic words}


Reducing the Archibald invariant

Goal: Relate the two invariants, “A → B”.Strategy: Reduce the Archibald invariant to matrix form.

The Hodge star operator

Consider again, for∣∣∣Xout

∣∣∣ =∣∣∣X in

∣∣∣ = n:

AHD(X in,Xout) = Λn(Xout

)⊗ Λn

(Xout ∪ X in

)For a fixed w ∈ Λn

(Xout

), apply the Hodge star operator ∗w :

n⊕k=0

Λn−k(Xout

)∧ Λk

(X in

) ∗w−−→

n⊕k=0

Λk(Xout

)∧ Λk

(X in

)defined by: ∗w (α ∧ β) = γ ∧ β⇔ α ∧ γ = w



Goal: Relate the two invariants, “A → B”.Strategy: Reduce the Archibald invariant to matrix form.

The Hodge star operator

Consider again, for∣∣∣Xout

∣∣∣ =∣∣∣X in

∣∣∣ = n:

AHD(X in,Xout) = Λn(Xout

)⊗ Λn

(Xout ∪ X in

)For a fixed w ∈ Λn

(Xout

), apply the Hodge star operator ∗w :

n⊕k=0

Λn−k(Xout

)∧ Λk

(X in

) ∗w−−→

n⊕k=0

Λk(Xout

)∧ Λk

(X in

)defined by: ∗w (α ∧ β) = γ ∧ β⇔ α ∧ γ = w



TheoremFor an n-tangle T, let:

λ = the zero-graded component of ∗w (tMVA(T))

A = the matrix of one-graded components of ∗w (tMVA(T))

If λ , 0, the pair (λ,A) determines tMVA(T).

This is achieved using the formula:

det M(T){1,...,n}\{i1,...,ik };{j1,...,jk } = (−1)kn+k(k−1)/2−∑

p ipdet A j1,...,jk

i1,...,ik

λk−1

M(T){1,...,n}\{i1,...,ik };{j1,...,jk } = M(T) excluding columns ci1 , . . . , cikfrom Xout and including cj1 , . . . , cjk from X in.A j1,...,jk

i1,...,ik= submatrix of A with columns j1, . . . , jk and rows i1, . . . , ik .



TheoremFor an n-tangle T, let:

λ = the zero-graded component of ∗w (tMVA(T))

A = the matrix of one-graded components of ∗w (tMVA(T))

If λ , 0, the pair (λ,A) determines tMVA(T).

This is achieved using the formula:

det M(T){1,...,n}\{i1,...,ik };{j1,...,jk } = (−1)kn+k(k−1)/2−∑

p ipdet A j1,...,jk

i1,...,ik

λk−1

M(T){1,...,n}\{i1,...,ik };{j1,...,jk } = M(T) excluding columns ci1 , . . . , cikfrom Xout and including cj1 , . . . , cjk from X in.A j1,...,jk

i1,...,ik= submatrix of A with columns j1, . . . , jk and rows i1, . . . , ik .



Corollary

The pair (λ,A) is an invariant in its own right, with the inducedmaps:

λ a b X in

a α β θ

b γ δ ε

Xout φ ψ Ξ

mabc

−−−−−−→ta ,tb→tc

λ + β c X in

c γ + βγ−αδλ ε + βε−δθ

λ

Xout φ + βφ−αψλ Ξ + βΞ−ψθ

λ(λ1 X in

1Xout

1 A1,

λ2 X in2

Xout2 A2

)∗−−→

λ1λ2 X in1 X in

2Xout

1 λ2A1 0Xout

2 0 λ1A2

Let AX =the space of such pairs for n-tangles labelled by the set X(X in and Xout).



Corollary

The pair (λ,A) is an invariant in its own right, with the inducedmaps:

λ a b X in

a α β θ

b γ δ ε

Xout φ ψ Ξ

mabc

−−−−−−→ta ,tb→tc

λ + β c X in

c γ + βγ−αδλ ε + βε−δθ

λ

Xout φ + βφ−αψλ Ξ + βΞ−ψθ

λ(λ1 X in

1Xout

1 A1,

λ2 X in2

Xout2 A2

)∗−−→

λ1λ2 X in1 X in

2Xout

1 λ2A1 0Xout

2 0 λ1A2

Let AX =the space of such pairs for n-tangles labelled by the set X(X in and Xout).


Making the connection

Theorem

The map BX → AX taking a pair (λ,A) 7→ (λ,−λA) respects theoperations on both sides.

Corollary

The above map induces a partial trace operation on the Bar-Nataninvariant, coming from the Archibald invariant:

λ c Xc α θ

X ψ Ξ

trc−−−−−→

λ(1 − α) XX Ξ + ψθ

1−α


Making the connection

Theorem

The map BX → AX taking a pair (λ,A) 7→ (λ,−λA) respects theoperations on both sides.

Corollary

The above map induces a partial trace operation on the Bar-Nataninvariant, coming from the Archibald invariant:

λ c Xc α θ

X ψ Ξ

trc−−−−−→

λ(1 − α) XX Ξ + ψθ

1−α


Still to investigate:

How are the algebraic structures related (circuit algebra vsmeta-monoid)?

This invariant reduces to the Gassner (resp.) Buraurepresentation on braids.

What is the domain space of the trace map?

Possibly leading to a non-commutative generalization of theMVA.

Possibly categorifiable.


The End

Thank you!

Documents

Two generalizations of the Multivariable Alexander Polynomial