A Characterization of T2g+1,2 among Alternating Knots

Abstract

Let K be a genus g alternating knot with Alexander polynomial \({\Delta _K}(T) = \sum\nolimits_{i = - g}^g {{a_i}{T^i}}\). We show that if |ag| = |ag−1|, then K is the torus knot T2g+1,±2. This is a special case of the Fox Trapezoidal Conjecture. The proof uses Ozsváth and Szabó’s work on alternating knots.