Quandle homotopy invariants of knotted surfaces

TL;DR

This paper introduces a new quandle homotopy invariant for knotted surfaces in 4-sphere, extending classical link invariants, and computes related homotopy groups for specific quandles, revealing connections to known cocycle invariants.

Contribution

It defines a universal quandle homotopy invariant for knotted surfaces and computes key homotopy groups for regular Alexander quandles, linking to existing cocycle invariants.

Findings

The invariant is valued in the third homotopy group of the quandle space.

Computed the second and third homotopy groups for regular Alexander quandles.

Showed that dihedral quandle cocycle invariants are scalar multiples of Mochizuki's invariant.

Abstract

Given a finite quandle, we introduce a quandle homotopy invariant of knotted surfaces in the 4-sphere, modifying that of classical links. This invariant is valued in the third homotopy group of the quandle space, and is universal among the (generalized) quandle cocycle invariants. We compute the second and third homotopy groups, with respect to "regular Alexander quandles". As a corollary, any quandle cocycle invariant using the dihedral quandle of prime order is a scalar multiple of the Mochizuki 3-cocycle invariant. As another result, we determine the third quandle homology group of the dihedral quandle of odd order.