Knots with many minimal genus Seifert surfaces

TL;DR

This paper identifies a specific subfamily of alternating, arborescent, prime knots that have exactly 2^{2n-1} minimal genus Seifert surfaces, matching the previously established lower bound.

Contribution

It provides an explicit subfamily of knots with precisely the maximum number of minimal genus Seifert surfaces, confirming the sharpness of Roberts' lower bound.

Findings

Subfamily of knots with exactly 2^{2n-1} minimal genus Seifert surfaces

Confirmation that Roberts' lower bound is sharp for this subfamily

Enhanced understanding of Seifert surface diversity in knot theory

Abstract

Roberts proved that a family of alternating, arborescent, prime knots each have at least $2^{2n-1}$ distinct minimal genus Seifert surfaces, where $n$ is the genus of the knot in question. We give a subfamily of these knots that have exactly this many minimal genus Seifert surfaces.