The third term in lens surgery polynomials

TL;DR

This paper characterizes lens space knots in $S^3$ with a specific Alexander polynomial coefficient condition, showing they correspond to $(2,2g+1)$-torus knots and their surgery polynomials.

Contribution

It establishes a unique link between the third coefficient of the Alexander polynomial and the specific torus knot surgeries in $S^3$.

Findings

The second Alexander polynomial coefficient of lens space knots is always -1.

Non-zero third coefficient condition restricts the knot to a $(2,2g+1)$-torus knot.

The lens surgery polynomial matches that of the $(2,2g+1)$-torus knot.

Abstract

It is well-known that the second coefficient of the Alexander polynomial of any lens space knot in $S^3$ is $-1$. We show that the non-zero third coefficient condition of the Alexander polynomial of a lens space knot $K$ in $S^3$ confines the surgery to the one realized by the $(2,2g+1)$-torus knot, where $g$ is the genus of $K$. In particular, such a lens surgery polynomial coincides with $\Delta_{T(2,2g+1)}(t)$.