Algebraic structures in knot theory

A virtual knot is a smooth, simple closed curve in a thickened compact oriented surfaces considered up to ambient isotopy, stabilisation/destabilisation and orientation preserving homeomorphism of surfaces. Kuperberg proves that every virtual link has a unique representative as a link up to ambient isotopy in a thickened surface of the minimal genus. A classical knot theory is the study of smooth embedding of circles in the 3-sphere up to ambient isotopy. Considering classical theory as the study of links in the thickened 2-sphere, the preceding result implies that classical knot theory is embedded inside virtual knot theory. One of the fundamental problems in knot theory is the classification of knots. In the classical case, the fundamental group of the knot complement space is a well known invariant. But there are examples where it fails to distinguish distinct knots. Around the 1980s, Matveev and Joyce introduce a complete classical knot invariant (up to the orientation of the knot and the ambient space) using distributive groupoids (quandles), known as the knot quandle.
In the talk, I will describe the construction of knot quandle given by Matveev. I will introduce the notion of residually finite quandles and prove that all link quandles are residually finite. Using this, I will prove that the word problem is solvable for link quandle. I will discuss the orderability of quandles, in particular for link quandles. Since all link groups are left-orderable, it is reasonable to expect that link quandles are left (right)-orderable. In contrast, I will show that orderability of link quandle behave quite differently than that of the corresponding link groups. I will also introduce a recent combinatorial generalisation of virtual links to which we name as marked virtual links. I will associate groups and peripheral structures to these diagrams and study their properties.