Crosscap number and the partial order on two-bridge knots Article Swipe

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Jim Hoste , Patrick D. Shanahan , Cornelia A. Van Cott ·

YOU? · · 2020 · Open Access ·

We consider the relationship between the crosscap number $\gamma$ of knots and the partial order on the set of all prime knots, which is defined as follows. For two knots $K$ and $J$, we say $K \geq J$ if there exists an epimorphism $f:\pi_1(S^3-K) \longrightarrow \pi_1(S^3-J)$. We prove that if $K$ and $J$ are 2-bridge knots and $K> J$, then $\gamma(K) \geq 3\gamma(J) -4$. We show that the inequality is sharp and that it does not hold in general for all prime knots. A similar result relating the genera of two knots has been proven by Suzuki and Tran. Namely, if $K$ and $J$ are 2-bridge knots and $K >J$, then $g(K) \geq 3 g(J)-1$, where $g(K)$ denotes the genus of the knot $K$.

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Knot (papermaking) Combinatorics Epimorphism Mathematics Order (exchange) Prime (order theory) Economics Engineering Chemical engineering Finance

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Type: preprint
Language: en
Landing Page: https://arxiv.org/pdf/2010.05009
OA Status: green
Related Works: 20
OpenAlex ID: https://openalex.org/W3092510900

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Title: Crosscap number and the partial order on two-bridge knots

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Type: preprint

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Language: en

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Publication year: 2020

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Publication date: 2020-10-10

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Authors: Jim Hoste, Patrick D. Shanahan, Cornelia A. Van Cott

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Landing page: https://arxiv.org/pdf/2010.05009

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Open access: Yes

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OA status: green

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OA URL: https://arxiv.org/pdf/2010.05009

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Concepts: Knot (papermaking), Combinatorics, Epimorphism, Mathematics, Order (exchange), Prime (order theory), Economics, Engineering, Chemical engineering, Finance

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