I have been reading the book 'Grid Homology for Knots and Links' (see https://web.math.princeton.edu/~petero/GridHomologyBook.pdf) - in Section 3.3 it provided a way to compute the Alexander polynomial using grid diagrams. Specifically, it gave an explicit formula for the Alexander polynomial in Definition 3.3.4 on Page 53. However, when I tried it on a left-hand trefoil knot, it does not seem to work. Please could I have some help:
The grid diagram I have been working on is as follows.
From here I have obtained the matrix
$$ M(\mathbb{G})= \begin{pmatrix} 1 & 1 & t & t & t\\ 1 & t & t^2 & t^2 & t\\ 1 & t & t^2 & t & 1\\ 1 & t & t & 1 & 1\\ 1 & 1 & 1 & 1 & 1 \end{pmatrix},$$
of which the determinant is $t^6-5t^5+11t^4-14t^3+11t^2-5t+1$ (see https://www.wolframalpha.com/input/?i=det(%7B%7B1,1,t,t,t%7D,+%7B1,t,t%5E2,t%5E2,t%7D,+%7B1,t,t%5E2,t,1%7D,+%7B1,t,t,1,1%7D,%7B1,1,1,1,1%7D%7D) for verification).
Since it is a $5 \times 5$ grid, we have $n=5$, so multiplying the above expression with $(t^{-1/2}-t^{1/2})^{-4}$, which gives $(t^4-t^3+t^2)$. So by looking at the formula in the book, we would need $a(\mathbb{G})$ to be $-3$ to give the correct Alexander polynomial $\Delta(t)=t^{-1}-1+t$ (see http://mathworld.wolfram.com/TrefoilKnot.html). But I am not sure how to obtain that based on the instruction above Definition 3.3.4:
By summing these contributions for all O's and X's and dividing the result by 8, we get a number $a(\mathbb{G})$ associated to the $n \times n$ grid.
This is part of my project but unfortunately I cannot find many resources online. I have also found a master's thesis (by Nancy Scherich) on this topic (see https://nancyscherich.files.wordpress.com/2018/01/the-alexander-polynomial.pdf, where the 'Minesweeper matrix' on Page 32 would give the same determinant as $M(G)$, however I don't think the Theorem 5.8 is correct as I have tried some examples...)
Many thanks in advance!
Let's compute $a(\mathbb{G})$ for your example.
The winding numbers at the intersections of horizontal and vertical lines (including the top and rightmost lines) are given by the array $$\begin{array}{r r r r r r} 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & -1 & -1 & 0\\ 0 & -1 & -2 & -2 & -1 & 0\\ 0 & -1 & -2 & -1 & 0 & 0\\ 0 & -1 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}$$
I'll color code the entries in that array as follows: a black entry is not in the corner of any $X$ or $O$, a red entry is in the corner of exactly one $X$ or $O$, and a blue entry is in the corner of exactly two $X$'s or $O$'s.
$$\begin{array}{r r r r r r} 0 & \color{red}{0} & \color{red}0 & 0 & \color{red}0 & \color{red}0\\ \color{red}0 & \color{blue}0 & \color{red}{-1} &\color{red}{-1} & \color{blue}{-1} & \color{red}{0}\\ \color{red}{0} & \color{red}{-1} & \color{red}{-2} & \color{blue}{-2} & \color{blue}{-1} &\color{red}{0}\\ 0 & \color{red}{-1} & \color{blue}{-2} & \color{blue}{-1} & \color{blue}{0} & \color{red}{0}\\ \color{red}{0} & \color{blue}{-1} & \color{blue}{-1} & \color{blue}{0} & \color{red}{0} & 0\\ \color{red}{0} & \color{red}0 & \color{red}0 & \color{red}0 & 0 & 0 \end{array}$$
Then $$a(\mathbb{G}) = \frac{1}{8}\left(\sum\text{red entries} + 2\sum\text{blue entries}\right)=-3,$$ as desired.