# Tag Info

8

Let $\overline{L}$ denote the mirror image of a link $L$. The (reduced) Alexander-Conway polynomial $D_L(t)$ of an $n$-component link $L$ satisfies $D_{\overline{L}}(t)=(-1)^{n+1}D_L(t)$. More generally, the multivariable potential function $\nabla_L(t_1,\ldots,t_n)$ also satisfies $\nabla_{\overline{L}}(t_1,\ldots,t_n)=(-1)^{n+1}\nabla_L(t_1,\ldots,t_n)$. ...

7

I will give an example that is likely not the simplest one. The example comes from the paper Behavior of knot invariants under genus 2 mutation by Dunfield, Garoufalidis, Shumakovitch, and Thistlethwaite (link here). The construction of their example uses cabled mutation. A mutation in a knot diagram removes a round disk intersecting the diagram ...

3

The relation is $\Delta_L(t)\stackrel{.}{=}\Delta_L(t,...,t)(t-1)$ (no need to take the normalised version). One reference is Proposition 7.3.10 of Kawauchi's "survey of knot theory". Arguably, a more conceptual understanding also follows from the use of Reidemeister torsion, see for instance Paragraphs 1.1 and 1.2 of Turaev's "Reidemeister torsion in knot ...

3

Both polynomials arise by considering the first homology of covering spaces of the link exterior $X_L$. If the link $L$ has $n$ components, then $\Delta_L(t_1,...,t_n)$ is extracted from the free abelian cover of $X_L$. The one variable polynomial comes from the infinite cyclic cover corresponding to the kernel of the map $\pi_1(X_L) \stackrel{ab}{\to} H_1(... 2 I think this is the one you get from HOMFLY polynomial. Recall that for the usual Conway triple$(L_+,L_-,L_0)$HOMFLY polynomial$P$has this relation:$lP(L+)+l^{-1}P(L_-)+mP(L_0)=0$Setting$l=i$and$m=-i(t-t^{-1})$would produce Alexander polynomial. Also recall that Reshetikhin-Turaev construction of HOMFLY-equivalent polynomial family$\text{v}_n, ...

1

Yes, there is such a one-variable polynomial. In fact, in my understanding, this came earlier than the multi-variable version. Gwénaël Masseyeau has some notes on the subject, with references to work of Conway (for the Conway polynomial and skein relations), Kauffman (for fixing Conway's approach in the single-variable case), and Hartley (for the multi-...

Only top voted, non community-wiki answers of a minimum length are eligible