The Alexander Polynomial of a Rational Link

Mark E. Kidwell; Kerry M. Luse

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arXiv · 1705.05901

The Alexander Polynomial of a Rational Link

Mark E. Kidwell·Kerry M. Luse

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Abstract

We relate some terms on the boundary of the Newton polygon of the Alexander polynomial $Δ(x,y)$ of a rational link to the number and length of monochromatic twist sites in a particular diagram that we call the standard form. Normalize $Δ(x,y)$ so that no $x^{-1}$ or $y^{-1}$ terms appear, but $x^{-1}Δ(x,y)$ and $y^{-1}Δ(x,y)$ have negative exponents, and so that terms of even total degree are positive and terms with odd total degree are negative. If the rational link has a reduced alternating diagram with no self crossings, then $Δ(-1, 0) = 1$. If the standard form of the rational link has $m$ monochromatic twist sites, and the $j^{\textrm{th}}$ monochromatic twist site has $\hat{q}_j$ crossings, then $Δ(-1, 0) = \prod_{j=1}^{m}(\hat{q}_j+1)$. Our proof employs Kauffman's clock moves and a lattice for the terms of $Δ(x,y)$ in which the $y$-power cannot decrease.

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BibTeX RIS

Mark E. Kidwell, Kerry M. Luse. 2017-05-16. The Alexander Polynomial of a Rational Link. https://arxiv.org/abs/1705.05901

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