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LENS SPACE SURGERIES ALONG CERTAIN 2-COMPONENT LINKS RELATED WITH PARK’S RATIONAL BLOW DOWN, AND REIDEMEISTER-TURAEV TORSION

Part of: Algebraic number theory: global fields Low-dimensional topology

Published online by Cambridge University Press:  15 October 2013

TERUHISA KADOKAMI  and
YUICHI YAMADA
TERUHISA KADOKAMI*
Affiliation:
Department of Mathematics, East China Normal University, Dongchuan-lu 500, Shanghai, 200241, PR China
YUICHI YAMADA
Affiliation:
Department of Mathematics, The University of Electro-Communications, 1-5-1, Chofugaoka, Chofu, Tokyo email yyyamada@sugaku.e-one.uec.ac.jp
*
✉mshj@math.ecnu.edu.cn
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Abstract

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We study lens space surgeries along two different families of 2-component links, denoted by ${A}_{m, n} $ and ${B}_{p, q} $, related with the rational homology $4$-ball used in J. Park’s (generalized) rational blow down. We determine which coefficient $r$ of the knotted component of the link yields a lens space by Dehn surgery. The link ${A}_{m, n} $ yields a lens space only by the known surgery with $r= mn$ and unexpectedly with $r= 7$ for $(m, n)= (2, 3)$. On the other hand, ${B}_{p, q} $ yields a lens space by infinitely many $r$. Our main tool for the proof are the Reidemeister-Turaev torsions, that is, Reidemeister torsions with combinatorial Euler structures. Our results can be extended to the links whose Alexander polynomials are same as those of ${A}_{m, n} $ and ${B}_{p, q} $.

Keywords

Dehn surgeryAlexander polynomialReidemeister torsioncombinatorial Euler structuregeneralized rational blow down

MSC classification

Secondary: 57M25: Knots and links in $S^3$ 57M27: Invariants of knots and 3-manifolds 57M50: Geometric structures on low-dimensional manifolds 11R04: Algebraic numbers; rings of algebraic integers 11R27: Units and factorization
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 96 , Issue 1 , February 2014 , pp. 78 - 126
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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