Unifying 4-Dimensional Knot Theory (19w5118)


(Western Washington University)

(University of South Alabama)

(University of Nebraska-Lincoln)


The Banff International Research Station will host the "Unifying 4-Dimensional Knot Theory" workshop in Banff from November 3, 2019 to November 8, 2019.

Knot theory is the study of knotted loops in 3--dimensional space, considered up to smooth deformations which do not break the loop. Advances in knot theory have informed our collective understanding of a diverse range of topics, from the chemical reactions that can take place when replicating DNA becomes knotted, to physical properties of different conformations of knotted molecules. Knot theory also has deep connections to high energy physics. Although we live in 3--dimensional space, the evolution of a 3--dimensional object over time is inherently a 4--dimensional structure, since including time involves an extra degree of freedom. Thus, studying 3--dimensional and 4--dimensional spaces are integrally related to the world in which we live.

This workshop proposes to develop a better understanding of knotting in dimension four. In this realm, loops can never be knotted -- there is simply too much "room" in which to perform a smooth deformation. Instead, the interesting knotted objects are surfaces. One way to view a knotted surface is to imagine a knotted loop that evolves over time, although there are many ways to visualize these objects. We propose to bring together a variety of different perspectives in the theory of knotted surfaces; notably, we will connect classical decompositions with several brand new breakthroughs in order to formulate a more unified theory of knotted surfaces and spur a renaissance in this field of study.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).