the evaluation of the Jones polynomial of an alternating link at $ t= -1 $.

Question

I've been looking at some graph polynomials and I found a very nice relation between the famous Tutte polynomial of graphs and the no less famous Jones polynomial of links.

Using this relation I was able to show, that for an alternating link $ L $ with an alternating diagram $D$:

$$ \lvert V_{L} (-1) \rvert = \# \{ \text{spanning trees of the Tait graph of } D \}. $$

(I used the Tutte polynomial of the Tait graph of $D$.)

Then I found in this paper, the equality: $$ \lvert V_{L} (-1) \rvert = \det (L) $$ for $ L $ an alternating link.

So my question is:

For an alternating link $ L $ with an alternating diagram $ D $, how do I prove that $$ \det (L) = \# \{ \text{spanning trees of the Tait graph of } D \}? $$

Thank you in advance for your help.

Answer 1

I think the equality you have shown is well known, though I don't know where its written down (this is somewhat close but orthogonal to my interests so don't take this opinion too seriously). Actually, I think the first author of the paper you linked mentioned it in a recent talk I attended.

I am not sure what you want to show if you already believe the equalities you've written down, but in fact $V_L(-1)=\Delta_L(-1)$ is true for any link (where $\Delta_L(t)$ is the Alexander polynomial and the determinant is usually defined as $|\Delta_L(-1)|$).

According to Wolfram.Mathworld, the equality $V_L(-1)=\Delta_L(-1)$ is present in Jones' 1985 paper where he introduced the polynomial, so that might be a good place to start.

For entertainment purposes: there is an "interesting" interpretation to the relationship you proved by a quite famous mathematician outside of knot theory http://www.math.rutgers.edu/~zeilberg/Opinion1.html

Answer 2

In A spanning tree expansion of the Jones polynomial, Morwen Thistlethwaite (journal link here, pdf link here) proves that if $G$ is the Tait graph of an alternating diagram $D$ of a link $L$, then the Tutte polynomial of $G$ evaluated at $x=-t$ and $y=-t^{-1}$ is the Jones polynomial of $L$ (up to multiplication by $\pm t^k$ for some $k$). Your result relating $|V_L(-1)|$ to the number of spanning trees in the Tait graph $G$ is obtained by letting $t=1$ in Thistlethwaite's result.

It is worth noting that in this paper Thistlethwaite uses the relationship between the Tutte polynomial and the Jones polynomial to prove some of Tait's conjectures on the properties of alternating links.