Knot Theory and Related Applications

Dear Colleagues,

This special issue is devoted to fundamental work on the theory of knots, links, three dimensional manifolds and related topology in dimensions three and four.

Knot theory and physical science have been related for millennia in the sense that discoveries of effective methods of knot tying and weaving have been part of many human cultures from the very beginning of society. In the 19th century knots became of direct interest to physicists such as James Clerk Maxwell and Sir William Thompson (Lord Kelvin).

Kelvin had a theory of vortex atoms that suggested that atoms of matter could be identified with knotted vortices in the luminiferous aether. Aspects of the electromagnetic field were seen to be related to the linking of field lines, and Maxwell and Carl Friedrich Gauss found field related formulas for the topological linking numbers of curves in three dimensional space.

The Kelvin theory of vortex atoms eventually lost favor among physicists due to the replacement of the aether by more abstract spatial models and the problems of understanding the stability of knotted vortices. But this early involvement of knot theory with physics has continued to the present day. In the 19th century, partially at the behest of Lord Kelvin, knot tables were constructed by Peter Guthrie Tait and others. These tables were the beginning of knot tabulation continuing to the present day. In the early 20th century mathematicians used the work of Poincare on the fundamental group of topological spaces to study knots and obtained important results. For example Max Dehn gave the first proof that the trefoil knot and its mirror image are topologically distinct. This problem of chirality of knots is still under investigation. The question of knotted aether has transposed to the question of possibly knotted fields in physics such as knotted electromagnetic fields and knotted gluon fields.

The use of gauge fields in the hands of Edward Witten enabled mathematicians and physicists to make topological invariants of knots and links by examining transport along them in a space endowed with a gauge field. The work of Witten gave a unified point of view on knot polynomial invariants  that had emerged, first in the 1920’s through the work of Alexander and later in the 1980’s through the pioneering work of Vaughan Jones and many others. Witten’s work hailed all the way back to Gauss and Maxwell and can be seen as a generalization of their work on linking numbers. Generalizations of these invariants have led to new topological approaches such as Khovanov homology and other theories of link homology. Deep relationships with string theory have brought back the questions of knots and physics in new forms.

The work of Witten, Reshetikhin and Turaev led to new invariants of three dimensional manifolds and  to relationships with other methods for obtaining invariants of knots in three-manifolds. Virtual knot theory is a formulation of knot theory in thickened surfaces and has many combinatorially defined generalizations of the Kauffman bracket, Jones polynomial, other quantum invariants and Khovanov homology, as well as new invariants directly related to this class of three-manifolds. There are deep questions (the Kashaev conjectures and their generalizations) relating geometric structures on three manifolds and properties of quantum invariants. The theory of skein modules gives methods to define invariants of three manifolds in terms of knots and links embedded within them.

We now have a very active interpenetration of knot theory, topology and mathematical physics that includes all the problems begun in the 19th century and much more. In this special issue we invite papers on low dimensional topology that address basic issues in either the purely topological domain or in the relationships of knot theory with physics, biology, molecular biology and natural sciences.

We invite papers of wide interest in this field and we encourage contributors to write papers that include new research that is enfolded in an exposition that can reach a wide audience of mathematicians and scientists.

Prof. Vassily Olegovich Manturov
Prof. Dr. Louis H. Kauffman
Dr. Roger Fenn
Guest Editors

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