Crosscap number and epimorphisms of two-bridge knot groups
| Autor: | Patrick D. Shanahan, Jim Hoste, Cornelia A. Van Cott |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Algebra and Number Theory
Computer Science::Information Retrieval Astrophysics::Instrumentation and Methods for Astrophysics Geometric Topology (math.GT) Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) Epimorphism 57K10 Bridge (interpersonal) Mathematics::Geometric Topology Prime (order theory) Combinatorics Mathematics - Geometric Topology Knot (unit) FOS: Mathematics Computer Science::General Literature Order (group theory) Crosscap number Mathematics |
| Popis: | We consider the relationship between the crosscap number $\gamma$ of knots and a 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 also classify all pairs $(K,J)$ for which the inequality is sharp. 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 $K$. Comment: Version 2: updated to incorporate referee's comments |
| Databáze: | OpenAIRE |
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