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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

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,±1,±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.

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
Received: 19 December 2017
Revised: 12 August 2018
Accepted: 16 October 2018
Published: 20 October 2019
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