An Elementary Proof of the Signature of Satellite Knots

TL;DR

This paper provides a straightforward linear algebra-based proof of Litherland's formula for the Tristram-Levine signature of satellite knots, simplifying the understanding of this knot invariant.

Contribution

It offers a new elementary proof of a known formula, making the result more accessible by avoiding advanced algebraic methods.

Findings

Proof of Litherland's formula using linear algebra

Simplifies understanding of satellite knot signatures

Accessible approach for knot theory researchers

Abstract

We present a proof of Litherland's formula for the Tristram-Levine signature of a satellite knot in terms of its constituents. Litherland's original proof used more advanced algebraic techniques, while ours uses only linear algebra and some basic results in knot theory.