FUNDAMENTAL BIQUANDLES OF RIBBON 2-KNOTS AND RIBBON TORUS-KNOTS WITH ISOMORPHIC FUNDAMENTAL QUANDLES

Abstract

The fundamental quandles and biquandles are invariants of classical knots and surface knots. It is unknown whether there exist classical or surface knots which have isomorphic fundamental quandles and distinct fundamental biquandles. We show that ribbon 2-knots or ribbon torus-knots with isomorphic fundamental quandles have isomorphic fundamental biquandles. For this purpose, we give a method for obtaining a presentation of the fundamental biquandle of a ribbon 2-knot/torus-knot from its fundamental quandle.