## Algebraic & Geometric Topology

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

#### 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 ${−6,±1,±2,3,4,7}$. 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

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

Dates
Revised: 12 August 2018
Accepted: 16 October 2018
First available in Project Euclid: 26 October 2019

https://projecteuclid.org/euclid.agt/1572055260

Digital Object Identifier
doi:10.2140/agt.2019.19.2439

Mathematical Reviews number (MathSciNet)
MR4023320

Zentralblatt MATH identifier
07142610

#### Citation

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. https://projecteuclid.org/euclid.agt/1572055260

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