Quandle homotopy invariants of knotted surfaces

Details Quandle homotopy invariants of knotted surfaces Quandle homotopy invariants of knotted surfaces

2010-11-28

arxiv.org/abs/1011.6035v3

Mathematics

Algebraic Topology

I rewrote the introduction, and corrected some grammatical mistakes

Scientific paper

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.

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