On calculations of the twisted Alexander ideals for spatial graphs, handlebody-knots and surface-links

TL;DR

This paper advances the calculation of twisted Alexander invariants for complex spatial objects like graphs, handlebody-knots, and surface-links, providing new insights and correcting existing data.

Contribution

It introduces methods for calculating twisted Alexander ideals for these objects and presents new invariant calculations, including corrections to existing tables.

Findings

Nontrivial invariants for Suzuki's theta-curves with trivial Alexander ideals

Calculated invariants for handlebody-knots up to six crossings

Corrected Yoshikawa's surface-link table and computed invariants

Abstract

There are many studies about twisted Alexander invariants for knots and links, but calculations of twisted Alexander invariants for spatial graphs, handlebody-knots, and surface-links have not been demonstrated well. In this paper, we give some remarks to calculate the twisted Alexander ideals for spatial graphs, handlebody-knots and surface-links, and observe their behaviors. For spatial graphs, we calculate the invariants of Suzuki's theta-curves and show that the invariants are nontrivial for Suzuki's theta-curves whose Alexander ideals are trivial. For handlebody-knots, we give a remark on abelianizations and calculate the invariant of the handlebody-knots up to six crossings. For surface-links, we correct Yoshikawa's table and calculate the invariants of the surface-links in the table.