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

TL;DR

This paper classifies certain lens space surgeries and band surgeries on the trefoil knot, using Heegaard Floer invariants, with applications to DNA recombination processes.

Contribution

It provides a classification of lens space surgeries and band surgeries on the trefoil knot based on Heegaard Floer $d$-invariants, linking topology with biological models.

Findings

Identifies possible lens space fillings from specific surgeries.

Classifies band surgeries transforming the trefoil into torus knots and links.

Connects topological results to DNA recombination models.

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 non-coherent 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.