Homotopy on spatial graphs and generalized Sato-Levine invariants

TL;DR

This paper introduces new invariants for spatial graphs based on generalized Sato-Levine invariants, extending previous work to cases without linking number restrictions, thereby advancing the understanding of homotopy equivalence in spatial graph theory.

Contribution

It develops novel edge and vertex-homotopy invariants for spatial graphs using generalized Sato-Levine invariants, removing previous linking number restrictions.

Findings

Constructed new homotopy invariants for spatial graphs.

Extended invariants to cases with arbitrary linking numbers.

Enhanced tools for classifying spatial graphs under homotopy.

Abstract

Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. Fleming and the author introduced some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the constituent 2-component algebraically split links. In this paper, we construct some new edge (resp. vertex)-homotopy invariants of spatial graphs without any restriction of linking numbers of the constituent 2-component links by applying the generalized Sato-Levine invariant.