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Computing Alexander polynomials for arborescent links

Haimiao Chen

Abstract

For arborescent links, we present an efficient method of computing their Alexander polynomials. Applying this method, we express the Alexander polynomials of Montesinos links in terms of certain functions associated to rational tangles which can be computed recursively. Specifically, we deduce explicit closed formulas for all pretzel links.

Computing Alexander polynomials for arborescent links

Abstract

For arborescent links, we present an efficient method of computing their Alexander polynomials. Applying this method, we express the Alexander polynomials of Montesinos links in terms of certain functions associated to rational tangles which can be computed recursively. Specifically, we deduce explicit closed formulas for all pretzel links.

Paper Structure

This paper contains 7 sections, 6 theorems, 61 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The method
  3. Some computations
  4. An illustrating example
  5. A family of non-Montesinos links
  6. Montesinos links
  7. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 2.2

Suppose $L=D(T)$ is an oriented arborescent link, and a decomposition of $T$ is given. For subtangles $S$ of $T$, recursively compute $z_v(S),z_h(S)$ according to the following rules: Then $\Delta_{L}\doteq z_h(T)$ if $L$ is a knot, and $\Delta_L\doteq z_h(T)/(1-t_{\rm ne})$ otherwise. $\blacktriangleleft$$\blacktriangleleft$

Figures (11)

  • Figure 1: From left to right: a tangle $T\in\mathcal{T}_2^2$; $N(T)$; $D(T)$; $T_1\ast T_2$; $T_1+T_2$.
  • Figure 2: Left: $[1]$. Right: $[-1]$.
  • Figure 3: The rational tangle $[p/q]=[[k_1],\ldots,[k_s]]$, with $s=3$ on the left, and $s=4$ on the right.
  • Figure 4: Left: $[[2h_1],[2h_2]]$. Right: $[[2h_1-1],[2h_2]]$.
  • Figure 5: The knot $Q$, with an orientation chosen.
  • ...and 6 more figures

Theorems & Definitions (17)

  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Example 2.4
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 7 more