Explicit Formulas for the Alexander Polynomial of Pretzel Knots

TL;DR

This paper derives explicit formulas for the Alexander polynomial of pretzel knots, characterizes when these polynomials are trivial, and constructs knots that are topologically but not smoothly slice.

Contribution

It introduces explicit formulas for pretzel knot Alexander polynomials and identifies conditions for triviality, also constructing new examples of topologically slice but not smoothly slice knots.

Findings

Explicit formulas for Alexander polynomials of pretzel knots

Characterization of pretzel knots with trivial Alexander polynomial

Construction of knots that are topologically slice but not smoothly slice

Abstract

We provide explicit formulas for the Alexander polynomial of pretzel knots and establish several immediate corollaries, including the characterization of pretzel knots with a trivial Alexander polynomial. As an application, we construct a new family of knots that are topologically slice but not smoothly slice.