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Introduction to Knot Theory 28/05/2018 Introduction to Knot Theory 28/05/2018 5.3. Tait conjectures. We will now discuss the so-called Tait conjec- tures. They were formulated in late 19th century by P. G. Tait, when we was creating tables of knots. It took about 100 years to prove these conjectures (after some necessary modifications). Definition 5.15. A link diagram D is alternating if, when traveling along any component, one goes through crossings alternately over and under. An alternating link is one with an alternating diagram. Example 5.16. The standard diagrams of the trefoil and figure-eight knot are alternating: The following diagrams are not alternating: The right one is the minimal diagram of the knot 819, which is the torus knot T3,4. It is an example of a non-alternating knot—it does not have an alternating diagram. Definition 5.17. An isthmus in a link diagram D is a crossing at which three regions meet. In other words, there is a path in the complement of D that connects opposite sides of the crossing: D0 A diagram D is reduced if it is has no isthmus. 1 2 Tait Conjectures (1) A connected reduced alternating diagram of a link has the min- imal number of crossings. (2) If a connected reduced alternating diagram D is equivalent to its mirror image, then w(D) = 0. (3) Two connected reduced alternating diagrams of the same link are related by the flip move: F ↔ F Remark 5.18. Originally the second conjecture was formulated for all diagrams. However, it is not true: Hoste, Thistlewaite and Weeks found a counter-example in 1998, which is a diagram with 15 crossings: Definition 5.19. The A-state (resp. B-state) of a link diagram D is the one with each crossing replaced with its horizontal (resp. vertical) resolution. We say that D is adequate if switching any resolution in its A-state as well as B-state merges two loops together. Place around each crossing labels A and B in a way, such that A-labeled (resp. B-labeled) regions get connected in the A-state (resp. B-state): B B B A B A A ←−− A A −−→ A A B B B In case a link diagram D is connected and alternating, no region contain labels of both type. Indeed neighboring crossings have the following configuration: 3 B A A B A B B A Hence, we can assign the labels A and B to regions of D. Then the A- state (resp. B-state) is a collection of loops that bound regions of D labeled B (resp. A). Lemma 5.20. A reduced alternating diagram is adequate. Proof. We may assume that the diagram is connected. Suppose it is not adequate. Then there must be a crossing c in the interior of some loop in the A-state, such that switching its resolution to the vertical one splits the loop into two pieces: Because the diagram is alternating, it follows that all other crossings at this loop are inside (see above), so that c is an isthmus and the diagram is not reduced. We analyze the B-state likewise. Proposition 5.21. Let D be an connected reduced alternating link diagram. Then spanhDi = 4c(D) + 4. Proof. Let s and s′ be two states of D that coincide everyone but a single crossing c, where s(c)= −1 and s′(c) = 1. Then ′ ′ ′ n(s )= n(s) − 1 p(s )= p(s) + 1 ℓ(s )= ℓ(s) ± 1 where the sign in the right equality depends on whether going from s to s′ merges two loops together or splits a loop into two. The following table lists the extremal degrees from hD|si and hD|s′i for both cases: 4 hD|si hD|s′i merge n(s) − p(s) − 2ℓ(s) n(s) − p(s) − 2ℓ(s) min deg split n(s) − p(s) − 2ℓ(s) n(s) − p(s) − 2ℓ(s) − 4 merge n(s) − p(s) + 2ℓ(s) n(s) − p(s) + 2ℓ(s) − 4 max deg split n(s) − p(s) + 2ℓ(s) n(s) − p(s) + 2ℓ(s) Therefore, ′ ′ max deghD|si > max deghD|s i and mindeghD|si > min deghD|s i with strict inequality in the first (resp. second) case if s′ has less (resp. more) circles than s. In particular, the A-state and B-state contribute towards hDi unique monomials of extremal degrees, because D is ade- quate. Thence, spanhDi = maxdeghD|sAi− min deghD|sBi where we write sA and sB for the A-state and B-state of D respectively. Finally, because loops in A-state (resp. B-state) are in one-to-one cor- respondence with regions in D labeled B (resp. A), we have that ℓ(sA)+ ℓ(sB)= r(D) = 2 − c(D) + 2c(D)=2+ c(D) where we use the well-known formula for planar graphs #vertices − #edges + #regions = 2 with vertices—crossings (hence, c(D)) and edges—semi-arcs (four edges at each crossing, so that twice as much in total). Thence, spanhDi =(n(sA) − p(sA) + 2ℓ(sA)) − (n(sB) − p(sB) − 2ℓ(sB)) =(c(D) − 0 + 2ℓ(sA)) − (0 − c(D) − 2ℓ(sB)) == 4c(D) + 4 as desired. Corollary 5.22. A reduced alternating diagram is minimal. Proof. Consider first the case of a connected diagram D. By Proposi- tion 5.21 1 c(D)= (spanhDi) − 1, 4 so that c(D) cannot get lower. This proves that D is minimal. In case D is not connected, apply the above to each connected component separately. 5 Remark 5.23. In fact a stronger result holds: spanhDi 6 4c(D) for any link diagram D. To prove this one needs the so called Dual State Lemma, which says that for any state s, ℓ(s)+ ℓ(ˆs) 6 r(D), wheres ˆ is the dual state (i.e.s ˆ(c)= −s(c)). In particular, the A-state is dual to the B-state. 5.4. Other polynomials. Soon after the discovery of the Jones poly- nomial a new polynomial was discovered that specialize to both Jones and Conway–Alexander polynomial. ±1 Proposition 5.24. There is a unique invariant PL(a,q) ∈ Z[a ,z] of framed links satisfying the skein relations P (a,z)= aP (a,z) P (a,z) − P (a,z)= zP (a,z) − P (a,z)= a 1P (a,z) P (a,z) = 1 As in the case of Kauffman and Jones polynomial, we can transform PL into an invariant of link by scaling it with a power of a. Indeed, −w(L) HL(a,q)= a PL(a,q) is invariant under all Reidemeister moves. It is called the HOMFLYPT polynomial, following the initials of people that discovered it. The first six letters stand for J. Hoste, A. Ocneanu, K. Millett, P. J. Freyd, WB.˙ R. Lickorish, and D. N. Yetter, who discovered the polynomial in May 1985, whereas the last two letters are for J. Przytycki and P. Traczyk form Poland, who published their paper a few months later. The HOMFLYPT polynomial satisfies the relation − aH − a 1H = zH . Notice the following substitutions: • a = 1 recovers the Alexander–Conway polynomial, • a = q2 and z = q − q−1 recovers the Jones polynomial. It is common to consider the case a = qN for N > 2. This results in the so-called slN -polynomial, because it is closely related to the repre- sentation theory of the Lie algebra slN . In fact, the Jones polynomial arose from examining certain properties of representations of sl2..
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