Lefschetz Fibrations on Knot Traces of Alternating and Extended Alternating Knots

Atsushi Tanaka

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arXiv · 2605.13017

Lefschetz Fibrations on Knot Traces of Alternating and Extended Alternating Knots

Atsushi Tanaka

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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 $s-1$.

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BibTeX RIS

Atsushi Tanaka. 2026-05-13. Lefschetz Fibrations on Knot Traces of Alternating and Extended Alternating Knots. https://arxiv.org/abs/2605.13017

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