Finite type invariants of nanowords and nanophrases

TL;DR

This paper extends the concept of finite type invariants from virtual knots to nanowords and nanophrases, analyzing their properties and providing new invariants of low degrees.

Contribution

It generalizes finite type invariants to nanowords and nanophrases, including new invariants of degrees 1, 2, and 4 for specific cases.

Findings

Linking matrix is a finite type invariant of degree one.

T invariant is a finite type invariant of degree two.

A degree 4 finite type invariant for open homotopy of Gauss words.

Abstract

Homotopy classes of nanowords and nanophrases are combinatorial generalizations of virtual knots and links. Goussarov, Polyak and Viro defined finite type invariants for virtual knots and links via semi-virtual crossings. We extend their definition to nanowords and nanophrases. We study finite type invariants of low degrees. In particular, we show that the linking matrix and T invariant defined by Fukunaga are finite type of degree one and degree two respectively. We also give a finite type invariant of degree 4 for open homotopy of Gauss words.