The volume-determinant inequality for Turk's head links

Abstract

I prove the volume–determinant inequality for Turk's head links with p ≥ 3 strands and q ≥ 2 repetitions, including links with several components. The proof compares a geometric upper bound for volume with an analytic lower bound for the determinant. The diagram's periodic rotation constrains the ideal vertices in a decomposition of the link complement. Its compact fixed circle excludes parabolic endpoints of an elliptic lift, allowing a uniform bound on the oriented volume of each three-column piece. For p,q ≥ 4, these estimates give an explicit positive gap between 2π log det(L) and vol(L). The cases q = 2 and q = 3 are treated separately. The geometric pieces need not be convex or embedded.