Genwicclosed curves

closed curve 8 S1 1122

self intersection Jlt Ht

transverse For smallenoughsub curves Ht e d

YZCI c.ttare homeomorphic to two grithoggonal

A curve isgeneric ifEvery self int is transverse u

Every self int is pairwiseno 814 24 7 84

Ypgself intersections

Any curve approximated by polygonperturb

generic polygon11 vets no vertex

genericclosed distinct on edge do

curve 3wayinto

Two curves are isotopic if they are

homotopic through Eitives ith same

orientationpreserEEgselfintersectionsEquiv intr homeomorphism H 112 ITE

s t Hoy

Imagegraph_vertices _self ints toedges_snbueezry.gs between

Almostalways connected4 regularplane graph

Simple 0 no vertices

But not every coven 4 reg planargraphis image of a curve

F Every curve is an0 Euler tour of its image

Gaussian EulertoursoundU After entering any nodeI leave thru oppositeedge

image ofmulticurven8 stws.tw Ust ITE

Faeces of a curve faces of imagegraphcomponents of 11221

open disks except outer face comp ofcloseddisk

monogon µbi5q

iangle

Homotopy moves Change a small neighborhoodof one face with vertices

I o se

n a

1 30 2 30 3 33

Every homotopy betweenpolygons vertex moves

1Every h py betweengeneric curves

finite sequence of homotopymoves

genericTheoremi Every planar curve with n vertices

has n 12 faces Euler'sformula

Proof Fix Y Je Jf oVttF 2E Z

An un n Itinverts n Ivets n n Z n nf faces f I faces f F Z f f

is preserved FIE D

eatsa the imagegraph Timpossible

Note added after lecture: This is the correct convention.The vertex signs and Gauss code above right are also correct.Note added after lecture: These instructions are for tracing a face counterclockwise (on the left of a moving point), but the two examples in the Gauss diagrams are tracing the face clockwise (on the right).

05 0Knottdiagram bffIIII.tn OR

Gausscodeesequence of vertex labelsalong curve

signed code labelvertices9 ABcdeFGChaIgD

Stpreimages

Gauss diagram ofcrossing

Tracing faces I

featIt Enter 1 Leave

forward backwardsback forwardforward forward A Babback 1 back

in 0Gt time extract all ntt facesSigned Gausscode is consistent with aplanarcurveIFFFIn12