The topological biquandle of a link

Abstract

To every oriented link L, we associate a topologically defined biquandle B L , which we call the topological biquandle of L. The construction of B L is similar to the topological description of the fundamental quandle given by Matveev. We find a presentation of the topological biquandle and explain how it is related to the fundamental biquandle of the link.Date: November 12, 2019.

Knot Quandle Decompositions

“…5. Several generalizations of the quandle structure as virtual quandles (see [13]) or biquandles (see [9,10]), have been introduced to study classical knots and links, virtual ones or knots and links in manifolds different from S 3 (see [5]). Is it possible to extend the fundamental quandle functor in these cases?…”

Knot quandle decompositions

“…or biquandles (see [9,10]), have been introduced to study classical knots and links, virtual ones or knots and links in manifolds different from S 3 (see [5]). Is it possible to extend the fundamental quandle functor in these cases?…”

On the axioms of singquandles

“…The definition of the (oriented) singquandle associated to a singular link is purely combinatorial. We wonder if there is also a topological construction for such an object as for the fundamental quandle for classical links [Joy82,Mat82] or alternatively for one of its proper quotients as for the fundamental biquandle of a link [Hor19].…”