Polynomial Invariants of Singular Knots and links

TL;DR

This paper introduces a new polynomial invariant for singular knots and links based on singquandles, generalizing existing invariants and enabling distinction of links that previous invariants could not differentiate.

Contribution

The authors extend the quandle polynomial to singquandles and develop a new singular link invariant that generalizes the singquandle counting invariant.

Findings

The singquandle polynomial is an invariant of finite singquandles.

The new invariant can distinguish singular links with identical singquandle counting invariants.

The polynomial generalizes previous invariants, providing finer distinctions among singular links.

Abstract

We generalize the notion of the quandle polynomial to the case of singquandles. We show that the singquandle polynomial is an invariant of finite singquandles. We also construct a singular link invariant from the singquandle polynomial and show that this new singular link invariant generalizes the singquandle counting invariant. In particular, using the new polynomial invariant, we can distinguish singular links with the same singquandle counting invariant.