On Alexander Polynomials of Certain (2,5) Torus Curves

TL;DR

This paper computes Alexander polynomials for a specific class of (2,5) torus curves, showing they match generic cases when the curve is irreducible with a unique singularity at the origin.

Contribution

It provides explicit Alexander polynomial calculations for (2,5) torus curves with particular singularity conditions, extending understanding of their topological invariants.

Findings

Alexander polynomial matches that of generic torus curves for irreducible cases.

Unique inner singularity at the origin influences the polynomial calculation.

The polynomial remains invariant under certain irreducibility conditions.

Abstract

In this paper, we compute Alexander polynomials of a torus curve C of type (2, 5), C : f(x, y) = f_2(x, y)^5 + f_5(x, y)^2 = 0, under the assumption that the origin O is the unique inner singularity and f2 = 0 is an irreducible conic. We show that the Alexander polynomial remains the same with that of a generic torus curve as long as C is irreducible.