Distance one lens space fillings and band surgery on the trefoil knot

Lidman, Tye; Moore, Allison; Vazquez, Mariel

doi:10.2140/agt.2019.19.2439

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Distance one lens space fillings and band surgery on the trefoil knot

Tye Lidman, Allison H Moore and Mariel Vazquez

Algebraic & Geometric Topology 19 (2019) 2439–2484

DOI: 10.2140/agt.2019.19.2439

Abstract

We prove that if the lens space $L (n, 1)$ is obtained by a surgery along a knot in the lens space $L (3, 1)$ that is distance one from the meridional slope, then $n$ is in ${- 6, \pm 1, \pm 2, 3, 4, 7}$ . This result yields a classification of the coherent and noncoherent band surgeries from the trefoil to $T (2, n)$ torus knots and links. The main result is proved by studying the behavior of the Heegaard Floer $d$ –invariants under integral surgery along knots in $L (3, 1)$ . The classification of band surgeries between the trefoil and torus knots and links is motivated by local reconnection processes in nature, which are modeled as band surgeries. Of particular interest is the study of recombination on circular DNA molecules.

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Keywords

lens spaces, Dehn surgery, Heegaard Floer homology, band surgery, torus knots, $d$–invariants, reconnection, DNA topology

Mathematical Subject Classification 2010

Primary: 57M25, 57M27, 57R58

Secondary: 92E10

References

Publication

Received: 19 December 2017

Revised: 12 August 2018

Accepted: 16 October 2018

Published: 20 October 2019

Authors

Tye Lidman
Department of Mathematics North Carolina State University Raleigh, NC United States

Allison H Moore
Department of Mathematics University of California Davis Davis, CA United States	Department of Mathematics and Applied Mathematics Virginia Commonwealth University Richmond, VA United States

Mariel Vazquez
Department of Mathematics, and Department of Microbiology and Molecular Genetics University of California Davis Davis, CA United States

Volume 19, issue 5 (2019)