Symmetric Extensions of Dihedral Quandles and Triple Points of Non-orientable Surfaces

TL;DR

This paper constructs new symmetric quandle extensions of dihedral quandles using signed permutation matrices, computes their homology groups, and applies these to analyze the minimal triple point numbers of non-orientable surface-knots.

Contribution

It introduces a novel method to create symmetric quandle extensions of dihedral quandles and applies homology computations to surface-knot invariants.

Findings

Constructed non-trivial good involutions on dihedral quandle extensions.

Computed symmetric quandle homology groups for the smallest example.

Applied homology results to bound the minimal triple point number of non-orientable surface-knots.

Abstract

Quandles with involutions that satisfy certain conditions, called good involutions, can be used to color non-orientable surface-knots. We use subgroups of signed permutation matrices to construct non-trivial good involutions on extensions of odd order dihedral quandles. For the smallest example of order 6 that is an extension of the three-element dihedral quandle, various symmetric quandle homology groups are computed, and applications to the minimal triple point number of surface-knots are given.