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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.

SeifertView of $6_{2}$

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.

Genus of a Link

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

Link Crossings

Linking number

$\text{Lk}\LP \mathscr{L}\RP=\text{#} \img{/presentations/Alex_Poly/crossing/Crossing_+.svg}-\text{#} \img{/presentations/Alex_Poly/crossing/Crossing_-.svg}$

Links on a surface

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 $

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