Alexander Polynomial

Seifert surfaces

Examples 

<img class="centerImg" style="   max-width:500px; " src=" /presentations/Alex_Poly/bands/Band.svg"/>

<img class="centerImg" style="   max-width:500px; " src=" /presentations/Alex_Poly/alg/Alg_7.svg"/>

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 

<img class="centerImg" style="   max-width:500px; " src=" /presentations/Alex_Poly/alg/Alg_4.svg"/>

<img class="centerImg" style="   max-width:500px; " src=" /presentations/Alex_Poly/alg/Alg_5.svg"/>

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 

<img class="centerImg" style="   max-width:500px; " src=" /presentations/Alex_Poly/alg/Alg_7.svg"/>

<p>$$\to$$</p>

<img class="centerImg" style="   max-width:500px; " src=" /presentations/Alex_Poly/bands/Band.svg"/>

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 

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font-size: 150% !important; align-items: center; text-align: center; display: block; margin-left: auto; margin-right: auto;" –>

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

  1. $g$: Genus
  2. $s$: Number of Seifert circles
  3. $n$: Number of components
  4. $c$: Number of Crossings
<img class="centerImg" style="   max-width:500px; " src=" /presentations/Alex_Poly/alg/Alg_7.svg"/>

Seifert Matrix

Link Crossings 

<img class="centerImg" style="   max-width:500px; " src=" /presentations/Alex_Poly/crossing/Crossing_&#43;.svg"/>

<img class="centerImg" style="   max-width:500px; " src=" /presentations/Alex_Poly/crossing/Crossing_-.svg"/>

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 

<img class="centerImg" style="   max-width:500px; " src=" /presentations/Alex_Poly/bands/Band.svg"/>

<p>$$\to$$</p>

<!-- A_{rc} -->

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

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