The Alexander polynomial of twisted torus knots in the complementary range

Abstract

A twisted torus knot T(p,q;r,s) is obtained from the flat-torus diagram of T(p,q) by inserting s full twists along a disk meeting r adjacent strands. Our previous work determined its Alexander polynomial for 1<r<=max{|p|,|q|}. In this paper, we compute it in the complementary range max{|p|,|q|}<r<=|p|+|q| by placing the knot on a standard genus-two Heegaard surface and applying Fox calculus. We then combine the two calculations into a unified Alexander-polynomial formula valid for 1<r<=|p|+|q|. In the complementary range, we prove monicity and give an explicit formula for the breadth. This determines the Seifert genus for nonnegative twists and gives a genus lower bound for negative twists. We also exhibit infinite two-parameter families of non-L-space knots in the complementary range.