Milnor-Orr invariants from the Kontsevich invariant

TL;DR

This paper explores the relationships between Milnor, Orr, and Kontsevich invariants in knot theory, revealing equivalences and computational methods for these complex invariants.

Contribution

It establishes the equivalence between the Orr invariant and the tree reduction of the Kontsevich invariant, and discusses their relation to Milnor invariants.

Findings

Orr invariant of degree k equals the tree reduction of Kontsevich invariant of degree < 2k

Close relation between Orr and Milnor invariants

Provides a method for computing these invariants

Abstract

As nilpotent studies in knot theory, we focus on invariants of Milnor, Orr, and Kontsevich. We show that the Orr invariant of degree $ k $ is equivalent to the tree reduction of the Kontsevich invariant of degree $< 2k $. Furthermore, we will see a close relation between the Orr invariant and the Milnor invariant, and discuss a method of computing these invariants