Geometric Topology – The Jones Polynomial at Specific Values of $t$

gt.geometric-topologyknot-theoryquantum-topology

I've been calculating some Jones polynomials lately and I was just curious if there was a "physical" (or, rather, geometric) meaning to evaluating the Jones polynomial at a particular value of $t$.

For example, if I take the Jones polynomial for the (right) Trefoil knot, I have

$J(t) = t + t^3 – t^4$.

Is there some way I can interpret $J(0)$? $J(1)$?

I understand that the Jones polynomial is a laurent polynomial, so I don't expect $J(0)$ to make sense for a lot of knots (for example the left trefoil has $J(t) = t^{-1} + t^{-3} – t^{-4}$), but I thought it was worth asking.

I also know that $J(t^{-1})$ gives the Jones polynomial of the mirror image knot. Is there a way to interpret $J(-t)$? $J(t^2)$? How about $J(t) = 0$?

Edit to clarify what I mean when I say "physical meaning":
Since the Jones polynomial is a link invariant, $J(0)$ is also a link invariant (if it exists). Does this invariant correspond to a property of the knot that you can visualise, such as, say, the linking number or the crossing number?

Best Answer

The evaluation of the Jones polynomial at $e^{i\pi/3}$ is related to the number of 3-colourings $tri(K)$ of $K$ (see also here) as well as to the topology of the branched double cover $\Sigma(K)$:

$$tri(K) = 3\left|V^2_K(e^{i\pi/3})\right| = 3^{\dim H_1(\Sigma(K);\mathbb{Z}/3\mathbb{Z})+1}$$

This was proved by Przytycki in this paper (Theorem 1.13) and Lickorish-Millet here. I don't know whether similar relations hold for more general Fox colourings.

This is not really an answer to the precise questions you're asking, but it's a pretty result.

UPDATE (Aug 19, 2014): I have found some more references and some more info in this problem list: the third remark on page 383 (page 11 of the PDF) covers what was known in 2004. In particular, it says that computing $V_K(\omega)$ is $\#P$-hard (see Neil Hoffman's comment below) unless $\omega$ is a power of $e^{i\pi/3}$ or $\omega = \pm i$, and it gives the interpretation for $V_K(\omega)$ in the four remaining cases (the first two have been mentioned by Jim Conant in the comments above). If $L$ is a link, I will call $\ell$ the number of components, and $\Sigma(L)$ the double cover of $S^3$ branched over all components of $L$.

  • $V_L(1) = (-2)^{\ell - 1}$; for a knot, $V_K(1) = 1$;
  • $\left|V_L(-1)\right| = \left|H_1(\Sigma(L))\right|$ if $H_1(\Sigma(L))$ is torsion, and is 0 otherwise; for a knot, $\left|V_K(-1)\right| = \left|\det(K)\right|$;
  • $V_L(i) = (-\sqrt2)^{\ell-1}(-1)^{\mathrm{Arf}(L)}$ if $L$ is a proper link (i.e. ${\rm lk}(K,L\setminus K)$ is even for every component $K$ of $L$), and vanishes otherwise (Murakami); notice that the Arf invariant is defined only for proper links.
  • $V_L(e^{2i\pi/3}) = 1$.
Related Question