# Alexander Polynomial

## Seifert surfaces

### Examples  ## Definition

A Seifert surface for an oriented link in $S^3$ is a compact connected oriented surface smoothly embedded in $S^3$ with oriented boundary equal to the link.

## Existence

Existence can be shown by an algorithm to construct a Seifert surface from a given link projection.

### Remove crossings ### Close curves ### Collection of disks ### Attaching bands   ## Construction

1. Remove crossings.
2. Connect strands following orientation without creating new crossings.
3. Fill interior of resulting disks.
4. Connect disks with “twists” matching crossing orientation.

## Bands $$\to$$        ## Genus of a Seifert surface

We have then that as an abstract surface, a Seifert surface for a link is a disc with a number of “handles”

($D^1\times D^1$) added. That number is its genus.

We take the smallest genus of possible Seifert surfaces for a link as the genus of the link.

### Computing the genus of a surface

$2g=2-s-n+c$

1. $g$: Genus
2. $s$: Number of Seifert circles
3. $n$: Number of components
4. $c$: Number of Crossings # Seifert Matrix  $\text{Lk}\LP \mathscr{L}\RP=\text{#} \img{/presentations/Alex_Poly/crossing/Crossing_+.svg}-\text{#} \img{/presentations/Alex_Poly/crossing/Crossing_-.svg}$ We can put oriented simple closed curves through each of the bands. A Seifert surface is oriented, it has a top side and bottom side. We can take a push off of each of the curves in the “up” (blue) direction We can take an $\LP i,j\RP$ pair and compute $a_{i,j}=\text{Lk}\LP f_i,\ f_j^+\RP$    This populates a matrix:

$$\begin{bmatrix} \text{Lk}\LP f_1,\ f_1^+\RP & \text{Lk}\LP f_1,\ f_2^+\RP & \cdots & \text{Lk}\LP f_1,\ f_{2g}^+\RP\\ \text{Lk}\LP f_2,\ f_1^+\RP & \text{Lk}\LP f_2,\ f_2^+\RP & \cdots & \text{Lk}\LP f_2,\ f_{2g}^+\RP\\ \text{Lk}\LP f_3,\ f_1^+\RP & \text{Lk}\LP f_3,\ f_2^+\RP & \cdots & \text{Lk}\LP f_3,\ f_{2g}^+\RP\\ \vdots & \vdots & \ddots & \vdots\\ \text{Lk}\LP f_{2g},\ f_1^+\RP & \text{Lk}\LP f_{2g},\ f_2^+\RP & \cdots & \text{Lk}\LP f_{2g},\ f_{2g}^+\RP\\ \end{bmatrix}$$

## Example $$\to$$

$$\begin{bmatrix} -1 & 1 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 1\\ 0 & -1 & 0 & 1\\ \end{bmatrix}$$

## Alexander Polynomial

For an oriented link $\mathscr{L}$ and it’s associated Seifert matrix $S$ we define the Alexander polynomial of $\mathscr{L}$ as $\Delta_\mathscr{L}\LP t\RP=\text{det}\LP t^{\frac{1}{2}}S-t^{-\frac{1}{2}}S^T\RP$

$\operatorname{det}\LP t^{\frac{1}{2}}\begin{bmatrix} -1 & 1 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 1\\ 0 & -1 & 0 & 1\\ \end{bmatrix}-t^{\frac{-1}{2}}\begin{bmatrix} -1 & 1 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 1\\ 0 & -1 & 0 & 1\\ \end{bmatrix}^T\RP=-t^4+3 t^3-3 t^2+3 t-1$  $\begin{bmatrix} -1 & 1 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 1\\ 0 & -1 & 0 & 1\\ \end{bmatrix}$
$$-t^4+3 t^3-3 t^2+3 t-1$$

## Invariant

$\Delta_\mathscr{L}\LP t\RP$ is unique up to stabilization, a method for adding bands to the surface. Results in $\Delta_\mathscr{L}\LP t\RP$ being unique up to a $\pm t^k$.

## Limitations: Example    $$1-t+t^2$$
$$1-t+t^2$$

## Bound on genus

$\operatorname{deg}\LP\Delta_\mathscr{L}\LP t\RP\RP\leq 2\large g$

## Sources

1. Livingston, C. (1993). Knot Theory. Mathematical Association of America. https://doi.org/10.5948/UPO9781614440239
2. Lickorish, W. B. R. (1997). An Introduction to Knot Theory. In Graduate Texts in Mathematics. Springer New York. https://doi.org/10.1007/978-1-4612-0691-0
3. Saveliev, N. (2011). Lectures on the Topology of 3-Manifolds. DE GRUYTER. https://doi.org/10.1515/9783110250367
4. Dale Rolfsen, Knots and links, Mathematics Lecture Series, vol. 7, Publish or Perish, Inc., Houston, TX, 1990, Corrected reprint of the 1976 original.
5. van Wijk, J. J., & Cohen, A. M. (2006). Visualization of Seifert surfaces. In IEEE Transactions on Visualization and Computer Graphics (Vol. 12, Issue 4, pp. 485-496). Institute of Electrical and Electronics Engineers (IEEE). https://doi.org/10.1109/tvcg.2006.83
6. van Wijk, J. J., & Cohen, A. M. (n.d.). Visualization of the Genus of Knots. In VIS 05. IEEE Visualization, 2005. VIS 05. IEEE Visualization, 2005. IEEE. https://doi.org/10.1109/visual.2005.1532843