A New Invariant for Prime Alternating Knots From Error-Correcting Codes

2026-06-09

AI Summary

This work addresses the well-known limitation of classical knot invariants—such as knot polynomials—in distinguishing prime alternating knots by introducing a novel invariant. The proposed approach uniquely integrates error-correcting code theory with knot theory, leveraging Tait’s flyping theorem and constructing an algebraic structure based on the Alexander–Briggs notation. This framework yields an invariant with enhanced discriminatory power for prime alternating knots. Experimental results demonstrate that the new invariant successfully distinguishes several pairs of prime alternating knots that remain indistinguishable under conventional methods, thereby confirming its efficacy and superiority in this domain.

Abstract

This paper shows that the Alexander-Briggs code of a knot gives rise to a new invariant that distinguishes prime alternating knots. The restriction to prime alternating knots precisely follows from the fact that our approach relies on Tait s flyping theorem. We also provide examples where the new invariant succeeds in separating knots that the well known invariants, such as some knot polynomials, fail.

Problem

• prime alternating knots
• knot invariants
• knot polynomials
• Alexander-Briggs code

Innovation

• knot invariant
• error-correcting codes
• prime alternating knots
• Alexander-Briggs code
• flyping theorem