#### Volume 19, issue 5 (2019)

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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
##### Abstract

We prove that if the lens space $L\left(n,1\right)$ is obtained by a surgery along a knot in the lens space $L\left(3,1\right)$ that is distance one from the meridional slope, then $n$ is in $\left\{-6,±1,±2,3,4,7\right\}$. This result yields a classification of the coherent and noncoherent band surgeries from the trefoil to $T\left(2,n\right)$ 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 $L\left(3,1\right)$. 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