quandles & q -colouring of knots

Quandles&

Q-colouring of knots

Krzysztof Putyra

Jagiellonian University

21st September 2006

Knots & diagrams

f

An embedding f : 13 is a knot if it is smooth or PL.

X = {f : 13 | f is a knot} – a knot space

Two knots are equivalent iff lies in the same path component of X.

Knots & diagrams

pbridgetunnels

This gives us a diagram of a knot.

Knots & diagrams

unknot trefoil

cinquefoilfigure-eight knot

Knots & diagrams

Knots & diagrams

R1 R2 R3

Theorem (K. Reidemeister, 1927).

Let K1, K2 be knots with diagrams D1,

D2. Then K1, K2 are equivalent iff D1

can be obtained from D2 by a finite

sequence of moves, called Reidemeister moves:

Kurt Reidemeist

er

Fox’s n-colourabilityf

crossing relations

Fox’s n-colourability

A knot is n-colourable if its diagram posses any non-trivial n-colouring.

What are the crossing relations?

A n-colouring is trivial if it uses only one colour.


10

Understand colouringsHow to compute n-colourings?

0

1

2 3

4a

b

c = ?

crossing relation:

2 moda b c n

1 1

2 2

3 3

4 4

5 5

2 0 1 1 00 2 0 1 11 0 2 0 11 1 0 2 00 1 1 0 2

c nbc nbc nbc nbc nb

Understand colourings

0

1

2 3

4

2 4 5 22c c c nb

4 1 2 42c c c nb 3 5 1 32c c c nb

5 2 3 52c c c nb

1 3 4 12c c c nb

Ac nb

c4

c2

c5c3

c1

Rows are lin.dep. delete first rowSet c1 = 0 delete first column

A

detK := |detA+| is invariant under Redemeister moves.It is called the determinant of a knot.

Understand colouringsTheorem. Knot K is n-colourable iff GCD(detK, n) ≠ 1.

Proof. We need to solve in n the equation:

A+c = 0There exist matrices B, C, D such that

D = BA+C D = diag(d1,…,dl)

where B, C are isomophisms and detD = detA+.

Now kerA+ kerD and

kerD ≠ 0 i: GCD(di, n) ≠ 1 GCD(detK, n) ≠ 1

31 & 51

Understand colourings

What can be changed to improve colourings?

Crossing relations!Crossing relations!

det = det = 5

Colourings cannot distinguish these knots!

Improving colouringsHaving the set of colours C, define the crossing relation

in the following way:

cl cov = cr

for some operation : C×C C.

cr

cl

cov

Which properties such an operation must have, to produce some natural

invariants?

cr

cl

cov

Improving colourings

cl cov = cr

Conditions for : C×C C:


cr

cl

cov

cl cov = cr


Q1: x x = x

x

x

x

x x


cr

cl

cov

cl cov = cr


Q1: x x = x

Q2: unique z: z x = y

x y x y

x y x a

where a x = y x


cr

cl

cov

cl cov = cr


Q1: x x = x


Q3: (z y) x = (z x) (y x)

x

y

x

yx x

wherez’ = (z x) (y x)z” = (z y) x

y x y x

z zz’ z”


cr

cl

cov

cl cov = cr


Q1: x x = x


Q3: (z y) x = (z x) (y x)To make computings easy, we like:

Q4: : C×C C is linear

Quandle

Quandles – definitionsA quandle is a set Q equipted with a binary operation

: Q×Q Q such that for all a,b,c Q:Q1: a a = a (idempotent)

Q2: exists unique x: x a = b (left-invertible)

Q3: (a b) c = (a c) (b c) (self-distributive)

A quandle (Q,) is called linear if Q is a ring and

Q4: : Q×Q Q is linear

Define :Q×QQ as follow:

(a b) b = a

The pair (Q, ) is called a dual quandle to (Q, ).

Quandles – properties

( ) ( ) ( )ax ay ay ax a x y y a x y ay

Theorem. A quandle dual to a linear quandle is linear.

Proof. Let (Q, ) be a linear quandle with dual (Q, ).Then for a, x, y Q we have:

( )ax ay a x y

what gives under Q2:

In a similar way one can check, that is additive.

Quandles – propertiesTheorem. An operation of duality is an involution.

( )x y y y x y

Proof. Let (Q, ) be a linear quandle with dual (Q, ).Then for x, y Q we have:

( )x y y x

what gives under Q2:

This shows that operation is dual to .

Quandles – examples

discrete quandleX – any set x x

background structure x y x y

conjugative quandleG – a group y-1xy yxy-1

dihedral quandlen – a ring 2y – x 2y – x

R – a ring, s – unit (1–s)y + sx (1–s-1)y + s-1x

Alexander quandle

= [t±1] (1–t)y + tx (1–t-1)y + t-1x

Q-colourabilityQ-colouring – a function from arcs of a diagram into a quandle Q.

a

b = ?

c = ?

crossing relations:

c a = b

b a = c

A diagram is Q-colourable if it posses any non-trivial Q-colouring.

A Q-colouring is trivial if it uses only one colour.

1

2

3

4

5

6 0 1 7 00 6 0 1 7

07 0 6 0 11 7 0 6 00 1 7 0 6

ccccc

Q-colourability

2 4 56 7 0c c c

4 1 26 7 0c c c 3 5 16 7 0c c c

5 2 36 7 0c c c

1 3 46 7 0c c c

0Ac

c4

c2

c5c3

c1colouring with

(22, )

x y = 7x – 6y

Rows are lin.dep. delete first rowSet c1 = 0 delete first column

A

detQK := detA+ is invariant under Redemeister moves (with

accurancy to units). It is the Q-determinant of a knot.

Q-colourability

D – a diagram with n arcs and m crossingsQ – a linear quandleA – a matrix generated by crossing relations

Module of Q-colurings:

colQD = kerA

Q-module of a diagram:

modQD = = (x1,…, xn : r1 = … = rm = 0)Qn

imA

Q-colourability

Theorem. For a diagram D and a linear quandle Q:

colQD Hom(modQD; Q)

Proof: Let modQD = (m1,…,mn : r1 = … = rm = 0).

For f: modQD Q define a Q-colouring f̃ as

f̃ (xi) := f (mi).

Also every Q-colouring f̃ induces a homomorphism

s.th.

f (mi) := f ̃(xi)

That is because for any relation ri = x y – z we have

f (ri) = f (x) f (y) – f (z) = f̃ (x) f̃ (y) – f ̃(z) = 0.

Q-colourability

Theorem. The Q-module of a diagram is invariant under Reidemeister moves.

Colloary. The module of Q-colourings of a diagram is a knot invariant.

Proof. Let D1 and D2 be diagrams of a knot K. Then

colQD1 Hom(modQD1; Q) Hom(modQD2; Q) colQD2

Q-colourability

Consider quandle 22 with an operation:

5 4a b a b

Cinquefoil does not posses non-trivial 22-colouring, in opposition to the figure-eight knot.

2

0 10

14

Q-colourability

Q-colourability

Knots colourable with no linear quandles exist!

Q-colourability

Further improvements:• modules with a Q-structure• non-commutative rings• relations of another type

References

• R. H. Crowell, R. H. Fox, An introduction to knot theoryGinn. and Co., 1963

• L. Kauffman, On knotsAnnals of Math. Studies, 115, Princeton University Press, 1987

• L. Kauffman, Virtual knots theoryEurop. J. Combinatorics (1990) 20, 663-691

• B. Sanderson, Knots theory lectures http://www.maths.warwick.ac.uk/~bjs/

• S. Nelson, Quandle theoryhttp://math.ucr.edu/~snelson/

http://www.maths.warwick.ac.uk/~bjs/

http://math.ucr.edu/~snelson/





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quandles & q -colouring of knots

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