Divide knot presentation of sporadic knots of Berge's lens space surgery

TL;DR

This paper extends the divide knot presentation method to include sporadic Berge knots in lens space surgeries, using generalized L-shaped curves to represent these knots and providing formulas relating surgery coefficients.

Contribution

It introduces a generalized L-shaped curve framework to represent sporadic Berge knots, expanding the divide knot presentation to previously unrepresented cases.

Findings

All sporadic Berge knots can be represented by generalized L-shaped divide curves.

A generalized formula relates surgery coefficients to the divide knot presentations.

The method unifies the representation of major and sporadic Berge knots.

Abstract

Divide knots and links, defined by A'Campo in the singularity theory of complex curves, is a method to present knots or links by real plane curves. The present paper is a continuation of the author's previous result that every knot in the major subfamilies of Berge's lens space surgery (i.e., knots yielding a lens space by Dehn surgery) is presented by an L-shaped curve as a divide knot. In the present paper, L-shaped curves are generalized and it is shown that every knot in the minor subfamilies, called sporadic examples, of Berge's lens space surgery is presented by a generalized L-shaped curve as a divide knot. A formula on the surgery coefficients and the presentation is also generalized.