Algebraic & Geometric Topology

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

Tye Lidman, Allison H Moore, and Mariel Vazquez

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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.

Article information

Algebr. Geom. Topol., Volume 19, Number 5 (2019), 2439-2484.

Received: 19 December 2017
Revised: 12 August 2018
Accepted: 16 October 2018
First available in Project Euclid: 26 October 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds 57R58: Floer homology
Secondary: 92E10: Molecular structure (graph-theoretic methods, methods of differential topology, etc.)

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


Lidman, Tye; Moore, Allison H; Vazquez, Mariel. Distance one lens space fillings and band surgery on the trefoil knot. Algebr. Geom. Topol. 19 (2019), no. 5, 2439--2484. doi:10.2140/agt.2019.19.2439.

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