Smoothed Invariants

TL;DR

This paper introduces two new knot invariants: one based on linking numbers and another for flat knots using smoothing pairs, both utilizing crossing parity, expanding the toolkit for knot classification.

Contribution

The paper presents novel knot invariants derived from crossing parity, including a sum with linking numbers and a flat knot invariant based on smoothing, enhancing knot analysis methods.

Findings

First invariant uses linking numbers for knot classification

Second invariant applies smoothing to flat knots

Both invariants leverage crossing parity for construction

Abstract

We construct two knot invariants. The first knot invariant is a sum constructed using linking numbers. The second is an invariant of flat knots and is a formal sum of flat knots obtained by smoothing pairs of crossings. This invariant can be used in conjunction with other flat invariants, forming a family of invariants. Both invariants are constructed using the parity of a crossing.