Topology Proceedings 30 No. 1 (2006), pp. 251-263

Non-Classicality and Quandle Difference Invariants

Natasha Harrell and Sam Nelson

Non-classical virtual knots may have upper and lower quandles which are not isomorphic. We exploit this property to define quandle difference invariants, which can detect non-classicality by comparing the numbers of homomorphisms into a finite quandle from a virtual knot's upper and lower quandles. The invariants for small-order finite quandles detect non-classicality in several interesting virtual knots. We compute the difference invariant with the six smallest connected quandles for all non-evenly intersticed Gauss codes with four crossings. For non-evenly intersticed Gauss codes with four crossings, the difference invariants detect non-classicality in 86% of codes which have non-trivial upper or lower counting invariant values.

Keywords: finite quandles, non-classicality, virtual knots

Mathematics Subject Classification: 57M27

Document formats: PDF file 317.0 Kb

Topology Proceedings, Volume 30, No. 1 (2006)
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