Finite type invariants for cyclic equivalence classes of nanophrases

TL;DR

This paper introduces finite type invariants for cyclic equivalence classes of nanophrases, constructs their universal forms, and clarifies their relation to linking matrices and Arnold's invariants.

Contribution

It defines finite type invariants for nanophrases, constructs universal invariants, and analyzes their relation to existing invariants like linking matrices and Arnold's invariants.

Findings

Universal finite type invariant of degree 1 is essentially the linking matrix.

Extended Arnold's invariants are finite type of degree 2 but not universal.

Provided a new proof that Arnold's invariants do not form the universal degree 2 invariant.

Abstract

In this paper, we define finite type invariants for cyclic equivalence classes of nanophrases and construct the universal ones. Also, we identify the universal finite type invariant of degree 1 essentially with the linking matrix. It is known that extended Arnold's basic invariants to signed words are finite type invariants of degree 2, by Fujiwara. We give another proof of this result and show that those invariants do not provide the universal one of degree 2.