Inequivalent surface-knots with the same knot quandle

2005/12/05 by Kokoro Tanaka, Tanaka, Kokoro · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Connective tissue disorders research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Primary 57Q45 #Secondary 57M25 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.math/0512099

openalex publication_date 2005/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We have a knot quandle and a fundamental class as invariants for a surface-knot. These invariants can be defined for a classical knot in a similar way, and it is known that the pair of them is a complete invariant for classical knots. In this paper, we compare a situation in surface-knot theory with that in classical knot theory, and prove the following: There exist arbitrarily many inequivalent surface-knots of genus g with the same knot quandle, and there exist two inequivalent surface-knots of genus g with the same knot quandle and with the same fundamental class.

Citations

• A classifying invariant of knots, the knot quandle

Cited by

• On biquandle-based invariant of immersed surface-links, Yoshikawa oriented fifth move, and ribbon 2-knots

Related

• The quandary of quandles: The Borel completeness of a knot invariant
• Characterizations of knot groups and knot symmetric quandles of surface-links
• Knot invariants derived from quandles and racks
• Quandle homotopy invariants of knotted surfaces
• The triple point number of surface-knots of genus one is at least four