Quandle and Biquandle Homology Calculation in R

Abstract

In knot theory several knot invariants have been found over the last decades. This paper concerns itself with invariants of several of those invariants, namely the Homology of racks, quandles, biracks and biquandles. The software described in this paper calculates the rack, quandle and degenerate homology groups of racks and biracks. It works for any rack/quandle with finite elements where there are homology coefficients in 'Z'k. The up and down actions can be given either as a function of the elements of 'Z'k or provided as a matrix. When calculating a rack, the down action should coincide with the identity map. We have provided actions for both the general dihedral quandle and the group quandle over 'S'3. We also provide a second function to test if a set with a given action (or with both actions) gives rise to a quandle or biquandle. The program is provided as an R package and can be found at https://github.com/ansgarwenzel/quhomology. AMS subject classification: 57M27; 57M25

Keywords

• Homology
• Quandle
• Biquandle
• Rack
• Birack
• R
• Knot Theory