Equations for HOMFLY polynomial of $(2,2k+1)$ Torus knots?

Question

Equations for HOMFLY polynomial of $(2,2k+1)$ Torus knots?

I'd appreciate a reference... Been looking around but can't find anything. $(2,2k+1)$ torus knots are a very well understood class of notes so I'm sure this formula is out there. Thank you!

Answer 1

You can calculate this by studying the HOMFLY polynomial for tangles. First, to set a convention, I'm going to use the HOMFLY polynomial with this skein relation:

A fact about the skein relation is that every tangle can be reduced into a linear combination $L_0$ and $L_+$. To that end, let's study the adding-a-right-hand-twist operator:

With respect to the $\{L_0,L_+\}$ basis, then, the matrix of $T$ is $$[T] = a^{-1}\begin{bmatrix} 0 & a^{-1} \\ a & z \end{bmatrix}.$$ This allows us to see what twisting something $k$ times would do by calculating $[T]^k$.

Another ingredient is closing a tangle:

Since a right-hand twist $L_+$ is represented by the vector $(0,1)$, we have that the HOMFLY polynomial of a $T(2,2k+1)$ torus knot is $$P(T(2,2k+1)) = \begin{bmatrix} \frac{a-a^{-1}}{z} & 1 \end{bmatrix} [T]^{2k} \begin{bmatrix} 0 \\ 1 \end{bmatrix}.$$

This matrix $[T]$ is diagonalizable, so you can compute this matrix power as $$P(T(2,2k+1)) = a^{-2k} \begin{bmatrix} \frac{a-a^{-1}}{z} & 1 \end{bmatrix} A \begin{bmatrix} \frac{z-\sqrt{4+z^2}}{2} & 0 \\ 0 & \frac{z+\sqrt{4+z^2}}{2} \end{bmatrix}^{2k} A^{-1}\begin{bmatrix} 0 \\ 1 \end{bmatrix}$$ with $$A=\begin{bmatrix} -\frac{z+\sqrt{4+z^2}}{2a} & -\frac{z-\sqrt{4+z^2}}{2a} \\ 1 & 1 \end{bmatrix}.$$ Note that it doesn't matter which branch cut you take to compute the square roots --- all that matters is that the square roots are negatives of one another. It's probably best to just interpret them purely symbolically (i.e., use only the fact that the square of $\sqrt{4+z^2}$ is $4+z^2$).

After some simplification, we can get $$P(T(2,2k+1)) = \frac{(2-a^2(2+z x))x^{2k} + (-2+a^2(2+z y))y^{2k}}{4^ka^{2k+2}z(y-x)}$$ where \begin{align*} x&=z-\sqrt{4+z^2} & y&=z+\sqrt{4+z^2}. \end{align*} I've checked that this expression matches a table of HOMFLY polynomials for $1\leq k\leq 8$.

I wouldn't be surprised if someone already calculated it in this way, but unfortunately I'm not aware of a reference, other than things like The Knot Atlas and KnotInfo, which contain HOMFLY polynomials for individual knots.

Edit: After much more simplification, I got it into this form: $$P(T(2,2k+1)) = a^{-2k-1} \sum_{n=0}^k \left( a\binom{k+n+1}{2n+1} - a^{-1}\binom{k+n}{2n+1} \right) z^{2n}$$

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Edit: I found some potential references.

Duzhin and Shkolnikov explain how to compute HOMFLY polynomials for 2-bridge knots (closures of rational tangles). Corollary 1 in section 4 purports to give a formula for $T(2,n)$ torus links, but it doesn't seem to match the tables: http://www.pdmi.ras.ru/~duzhin/papers/rat_knots_e3.pdf

Labastida and Marino use Chern-Simons gauge theory to obtain a formula for all torus links. See Theorem 3.1: https://cds.cern.ch/record/259127/files/9402093.pdf