Constructing Seifert surfaces from n-bridge link projections

TL;DR

This paper introduces a new algorithm for constructing minimal Seifert surfaces from n-bridge link projections, effectively handling complex links and a specific knot family where canonical genus exceeds the knot genus.

Contribution

The paper presents a novel algorithm that constructs minimal Seifert surfaces for large classes of links and generalizes Seifert's algorithm for broader homology class representations.

Findings

Algorithm produces minimal surfaces for large classes of braids and alternating links.

Successfully constructs surfaces realizing the knot genus g(K).

Generalizes Seifert's algorithm for arbitrary relative second homology classes.

Abstract

This paper presents a new algorithm "A" for constructing Seifert surfaces from n-bridge projections of links. The algorithm produces minimal complexity surfaces for large classes of braids and alternating links. In addition, we consider a family of knots for which the canonical genus is strictly greater than the genus, (g_c(K) > g(K)), and show that A builds surfaces realizing the knot genus g(K). We also present a generalization of Seifert's algorithm which may be used to construct surfaces representing arbitrary relative second homology classes in a link complement.