Abstracts of talks

KNOTS in WASHINGTON IX

The Conference on Theory and its Ramications

September at the George Washington University

Organizers

JozefPrzytycki przytyckgwuedu

Yongwu Rong ronggwuedu

Friday September Time PM

Sp eaker Witold Rosicki Gda nsk University

Title Coloring and other Quandle invariants of surfaces in manifolds

Place Funger Hall Ro om

Coee and co okies will b e served in Funger b efore the talk

ab out we will go with the sp eaker for lunch

Abstract

The ncoloring is a very useful simple invariant in classical We

can dene analogous invariant for the surfaces embedded in space We

dene bridges for surfaces after SCarter and MSaito

4

Denition If K F R is an embedding and p is a pro jection such that

the comp osition p K is in general p osition then the lower decker set A

fx F exist y F p K x p K y and K x is the l ow er pointg

separates the surface into regions bridges connected comp onents of K F

A The connected comp onents of the set of exactly double p oints are called

edgecrossings They are edges of the graph A

4 4 3

Theorem Let K F R and p R R b e as ab ove Let n b e an

integer n To every bridge of the pro jection we assign a number smaller

then or equal to n We say that such an assignment is an ncoloring i for

every edgecrossing in p K F the sum of numbers assigned to lower regions

is congruent mo d n to twice the number assigned to the upp er region Then

the number of p ossible ncoloring is an invariant of the knotting

If we verify that the Roseman moves preserve this number we prove the the

orem We can dene the FennRourke core group for surfaces embedded in

4

R and check in the same way that it is invariant of the p osition

I will discuss the joint work with JHPrzytycki giving the top ological inter

pretation of this group

I will describ e also JSCarter DJeslovsky SKamadaLLangford and MSaito

work on coloring of surfaces by quandle and its rich consequences

Saturday September Funger seminar ro om

am Coee and refreshments

Ted Stanford US Naval Academy

Knots and links mo dulo braid subgroups

Abstract

Since every may b e realized as the closure of a braid it is natural

to ask what sort of corresp ondence exists b etween equivalence relations

on braids and equivalence relations on links More sp ecically consider

a sequence of groups Q Q Q etc where Q is a quotient of the braid

1 2 3 i

group B and where the sequence satises some kind of compatibility

i

condition with the standard inclusions from B to B For any such

i i+1

sequence Q an induced equivalence relation on links is obtained n

i

equivalence of knots dual to Vassiliev invariants of order n arises

this way but so do other interesting equivalence relations which do not

factor through nequivalence for any n I will present some elementary

results and examples

Mike Veve GWU

Op en problems in billiard knots

Abstract

While mathematical billiards have b een studied quite extensively for

many years the same cannot b e said for billiard knots Billiard knots

are a sp ecial type mathematical billiard namely p erio dic tra jectories

without selfintersections inside some billiard ro om a billiard ro om is

3

manifold inside R with a piecewise smo oth b oundary

Some of the op en problems involving billiard knots that will b e dis

cussed are

Classify the billiard knots of a sp ecic ro om or a family of ro oms

according to knot type

Is there a billiard ro om such that every knot type is equivalent to

some billiard knot in this ro om If it exists what are its prop er

ties

By a simple construction one can show that for any tame knot

there is a billiard ro om which contains it as a billiard knot Usually

this ro om will not b e convex For any tame knot is there a convex

ro om that contains a billiard knot that is equivalent to the tame

knot

Mark Kidwell US Naval Academy

Solution to a conjecture of Stoimenow ab out plane curves

Abstract

A prime curve in the plane with distinct endp oints no nugatory

crossings and all crossings transverse has crossing number c Stoimenow

introduced a new quantity the endp oint distance d it is the minimum

intersection number of a curve with the same endp oints as with

Stoimenow proved that d max c and conjectured that d

F l oor c We will prove this conjecture and give some applications

to bridges in knot diagrams

Lunch

Maxim Sokolov GWU Dichromatic skein mo dules and

invariants of manifolds Abstract

Deloups invariants M G q essentially introduced by his teacher

V Turaev via a simple example of mo dular categories are an inter

esting generalization of the MurakamiOhtsukiOkada invariants for

manifolds Here G is a nite ab elian group and q G QZ is a

quadratic form Deloup showed that these invariants dep end only on

b M and the linking form of the manifold Recently Deloup and Gille

1

proved that these invariants keep all the information ab out the link

ing form and b M Moreover they proved that all the information

1

n

n

p can in fact b e recovered from two sp ecial cases M Z

p

n

n n n

p coincides K Here M Z Z for prime p and M Z

p 2 2

n

with the MurakamiOhtsukiOkada invariant of order p and K is the

n

quadratic form dened by K x y xy We showed earlier that

MurakamiOhtsukiOkada invariants can b e obtained using Lickorishs

construction on the skein mo dule S M q q analog of the rst homol

2

n n

K can b e similarly dened Z ogy It turned out that M Z

2 2

using another simple skein mo dule which we will call the dichromatic

skein mo dule and which turned out to b e the tensor pro duct of two

copies of S M q

2

Tatsuya Tsukamoto GWU and Waseda Univ

The fourth skein mo dule the MontesinosNakanishi conjecture and n

algebraic links

Abstract

We analyze the concept of the fourth skein mo dule of manifolds that

is a skein mo dule based on the skein relation b L b L b L

0 0 1 1 2 2

(1)

b L b b invertible and a framing relation L aL We nd

3 3 0 3

the necessary conditions for trivial link to b e linearly indep endent in

3

the mo dule In the case of classical links ie links in S our skein

mo dule suggests three p olynomial invariants of unoriented framed or

unframed links One of them generalize the Kauman p olynomial

of links and another one can b e used to analyze amphicheirality of

links and may work b etter than the Kauman p olynomial Using

the idea of mutants and rotors we show that there are dierent links

representing the same element in the skein mo dule We show also

that that algebraic links in the sense of Conway and closed braids

are linear combinations of trivial links We introduce the concept of

an nalgebraic and link and analyze the skein mo dule for

algebraic links As a byproduct we prove the MontesinosNakanishi

moves conjecture for algebraic links including bridge links

It is a joint work with Jozef Przytycki

Coee break

Adam Sikora UMD

Skein mo dules at the th ro ots of unity

Abstract

The skein mo dule of a manifold M is a mo dule over a ring R with

a sp ecied element A in R The mo dule is generated by links in M

considered up to Kauman bracket skein relations The connections

b etween skein mo dules and quantum invariants of manifolds suggest

that the most interesting choice of A is a ro ot of unity or A treated as a

formal element The problem of calculating the skein mo dule of a given

manifold is very dicult and it has b een solved only for A

giving a wonderful connection with the theory of character varieties

In this talk we calculate the skein mo dules of rational homology spheres

for the th ro ots of We also describ e the skein algebra of a surface

4

F for A as the SL character variety of F deformed along a

2

co cycle which is the symplectic form on F

JozefHPrzytycki GWU and UMD

Positive knots and their prop erties

Abstract

We analyze a relation on links dened by L L i L can b e

1 2 2

obtained from L by changing some p ositive crossings of L

1 1

We prove in particular

If K is a p ositive knot then K p ositive unless

K is a p ositive pretzel knot k k k it generalizes the

1 2 3

result of Co chran and Gompf

a If a p ositive knot has one then it is a

p ositive

If K is a almost p ositive knot ie it has a diagram with no

more then negative crossings then either

i K right handed trefoil or

ii K mirror image of knot in Rolfsens b o ok notation or

2

iii K is a twist knot with the negative clasp

a If K is a almost p ositive knot dierent then a twist knot with

negative clasp then K n ie n surgery on K n is

a homology sphere that do es not b ound a compact smo oth

homology ball it extends a result of Co chran and Gompf

and uses the recent in result of Akbulut

b If K is a nontrivial almost p ositive knot dierent than the

stevedores knot then K is not a

c If K is a nontrivial almost p ositive knot dierent than the

gure eight knot then K is not amphicheiral

It is a old joint work with Kouki Taniyama

Yongwu Rong GWU

Mutations and the Alexander p olynomial

Abstract

The MelvinMorton conjecture proved by BarNatan and Garoufa

lidis states that the colored Jones p olynomial of a knot determines

its Alexander p olynomial A consequence of it is that the Alexander

p olynomial of a knot is preserved under a along a genus two

surface This is surprisingly a new theorem as noted by Co op er and

Lickorish who also gave a classical pro of for it In this talk we will ex

plain how their result can b e slightly more general based on our earlier

work on mutation