Multivariable Alexander polynomials of twisted torus links via Dehn surgery

Abstract

Let $T(p,q;r,s)$ be the twisted torus link obtained from the torus link $T(p,q)$ by adding $s$ full twists to $r$ adjacent strands, and let $C$ be an unknotted circle encircling those strands. For all $p$, $q$, and $r$ we compute the multivariable Alexander polynomial of $T(p,q)\cup C$ in closed form. A relation of Silver and Williams between the Alexander polynomials of a link and of its $1/n$ Dehn fillings then gives a closed formula for the multivariable Alexander polynomial of every twisted torus link $T(p,q;r,s)$, answering a question of Adnan and Park. When $\gcd(p,q)=1$, so that $T(p,q;r,s)$ is a knot, the formula is a sum of $pr$ monomials with coefficient $+1$, divided by cyclotomic factors, and its exponents are governed by an elementary counting function on the orbit of $q$ modulo $p$. Because the numerator has no cancellation, we are able to give the degree for all $s$ and a simple non-fiberedness criterion.