Lefschetz Fibrations on Knot Traces of Alternating and Extended Alternating Knots

TL;DR

This paper constructs explicit positive allowable Lefschetz fibrations on knot traces of alternating and extended alternating knots, revealing minimal genus fibers and broadening the class of knots with such structures.

Contribution

It introduces a method to produce PALFs on knot traces of a new class of knots, including extended alternating knots, with explicitly determined minimal genus fibers.

Findings

Knot traces of positive pretzel knots with s rows admit PALFs with genus s-1.

Knot traces of positive torus knots admit PALFs with genus 1.

Knot traces of positive torus-pretzel knots admit PALFs with genus s-1.

Abstract

In our previous work, we introduced a simple and explicit method for constructing a positive allowable Lefschetz fibration (PALF) from a $2$-handlebody decomposition of any given compact Stein surface. In this paper, we apply this construction to knot traces whose attaching circles are either alternating knots or \emph{extended alternating knots} (a generalized class introduced herein). We demonstrate that each such knot trace admits a PALF whose regular fiber has a genus exactly equal to the number of white regions in the associated planar graph, yielding PALFs whose regular fibers have a significantly small genus. As immediate corollaries, we prove that knot traces of positive pretzel knots with $s$ rows admit PALFs with regular fibers of genus $s-1$, and those of positive torus knots admit PALFs with regular fibers of genus $1$. Furthermore, we define \emph{positive torus-pretzel knots} by replacing each twist block of a positive pretzel knot with the crossings of a positive torus knot, and we establish that their knot traces also admit PALFs with regular fibers of genus $s-1$.