Lefschetz Fibrations on Knot Traces of Alternating and Extended Alternating Knots
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866916008607875072 |
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| author | Tanaka, Atsushi |
| author_facet | Tanaka, Atsushi |
| contents | 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 $s-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13017 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lefschetz Fibrations on Knot Traces of Alternating and Extended Alternating Knots Tanaka, Atsushi Geometric Topology 57K40, 57K10, 57K41 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 $s-1$. |
| title | Lefschetz Fibrations on Knot Traces of Alternating and Extended Alternating Knots |
| topic | Geometric Topology 57K40, 57K10, 57K41 |
| url | https://arxiv.org/abs/2605.13017 |