Skein relations. Conway polynomial

Sasha Patotski

Cornell University [email protected]

November 6, 2014

Sasha Patotski (Cornell University) Deformations November 6, 2014 1 / 13 Surgery “transfer”

Figure : Careful: it can change the type of a knot!

Sasha Patotski (Cornell University) Deformations November 6, 2014 2 / 13 Surgery “resolution”

Figure : Careful: it also can change the type of a knot!

Sasha Patotski (Cornell University) Deformations November 6, 2014 3 / 13 Defining relations

Definition A Conway polynomial of a link is a function ∇ giving for any diagram D a polynomial ∇(D) in one variable x defined by the following two relations, called skein relations:

If links N and N0 are equivalent, we require ∇(N) = ∇(N0).

Sasha Patotski (Cornell University) Deformations November 6, 2014 4 / 13 Examples

Figure : Conway relations

Sasha Patotski (Cornell University) Deformations November 6, 2014 5 / 13 Examples

Figure : Conway relations

Sasha Patotski (Cornell University) Deformations November 6, 2014 6 / 13 Examples

Figure : Conway relations

Sasha Patotski (Cornell University) Deformations November 6, 2014 7 / 13 Conway invariant is nice

Sasha Patotski (Cornell University) Deformations November 6, 2014 8 / 13 Can we really compute Conway polynomial?

Theorem Yes, we can really compute Conway polynomial!

Sasha Patotski (Cornell University) Deformations November 6, 2014 9 / 13 More surgery

Lemma Any knot can be surgically changed to be an unknot.

Proof:

Sasha Patotski (Cornell University) Deformations November 6, 2014 10 / 13 Lemma For any split link L we have ∇(L) = 0.

Corollary We can compute Conway polynomial!

Sasha Patotski (Cornell University) Deformations November 6, 2014 11 / 13 Alternative name: LYMPHOTU L = Lickorish, Y = Yetter, M = Millet, P = Przytycki, H = Hoste, O = Ocneanu, T = Traczik, U = unknown discoverers

Figure : Skein relation for HOMFLY polynomial

HOMFLY polynomial

Most popular name: HOMFLY H = Hoste, O = Ocneanu, M = Millet, F = Freyd, L = Lickorish, Y = Yetter

Sasha Patotski (Cornell University) Deformations November 6, 2014 12 / 13 Figure : Skein relation for HOMFLY polynomial

HOMFLY polynomial

Most popular name: HOMFLY H = Hoste, O = Ocneanu, M = Millet, F = Freyd, L = Lickorish, Y = Yetter Alternative name: LYMPHOTU L = Lickorish, Y = Yetter, M = Millet, P = Przytycki, H = Hoste, O = Ocneanu, T = Traczik, U = unknown discoverers

Sasha Patotski (Cornell University) Deformations November 6, 2014 12 / 13 HOMFLY polynomial

Most popular name: HOMFLY H = Hoste, O = Ocneanu, M = Millet, F = Freyd, L = Lickorish, Y = Yetter Alternative name: LYMPHOTU L = Lickorish, Y = Yetter, M = Millet, P = Przytycki, H = Hoste, O = Ocneanu, T = Traczik, U = unknown discoverers

Figure : Skein relation for HOMFLY polynomial

Sasha Patotski (Cornell University) Deformations November 6, 2014 12 / 13 Is HOMFLY the biggest guy in the neighborhood?

No, HOMFLY can’t distinguish between the following knots:

Sasha Patotski (Cornell University) Deformations November 6, 2014 13 / 13