Vol. 8, No. 3, 2015

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Colorability and determinants of $T(m,n,r,s)$ twisted torus knots for $n\equiv\pm 1\pmod m$

Matt DeLong, Matthew Russell and Jonathan Schrock

Vol. 8 (2015), No. 3, 361–384
Abstract

We develop theorems to compute the $p$-colorability of the families of $T\left(m,n,r,s\right)$ twisted torus knots for $n\equiv ±1\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}m\right)$ by finding their determinants. Instead of the usual method of reducing crossing matrices to find the determinant, we describe a new method that is applicable for braid representations with full cycles and twists.

Keywords
knot theory, determinants, colorability, twisted torus knots
Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 11C20, 05C15