Lectures in Knot Theory : An Exploration of Contemporary Topics

This text is based on lectures delivered by the first author on various, often nonstandard, parts of knot theory and related subjects. By exploring contemporary topics in knot theory including those that have become mainstream, such as skein modules, Khovanov homology and Gram determinants motivated by knots, this book offers an innovative extension to the existing literature. Each lecture begins with a historical overview of a topic and gives motivation for the development of that subject. Understanding of most of the material in the book requires only a basic knowledge of topology and abstract algebra. The intended audience is beginning and advanced graduate students, advanced undergraduate students, and researchers interested in knot theory and its relations with other disciplines within mathematics, physics, biology, and chemistry.

Inclusion of many exercises, open problems, and conjectures enables the reader to enhance their understanding of the subject. The use of this text for the classroom is versatile and depends on the course level and choices made by the instructor. Suggestions for variations in course coverage are included in the Preface. The lecture style and array of topical coverage are hoped to inspire independent research and applications of the methods described in the book to other disciplines of science. An introduction to the topology of 3-dimensional manifolds is included in Appendices A and B. Lastly, Appendix C includes a Table of Knots.

Jozef H. Przytycki (1953-) is Professor of Mathematics at the George Washington University and summer visiting Professor at Gdansk University.

A distinguished mathematician, he received the Kazimierz Kuratowski Prize (1982), the Trachenberg prize (2010) and OVPR Distinguished Researcher Award (GWU). Przytycki received his Ph.D. from Columbia University in 1981, and wrote his dissertation on incompressible surfaces in 3-manifoldsunder Joan S. Birman. He received his Master degree in Mathematics in 1977 from Warsaw University, in his native Poland. He was a postdoctoral fellow with Dale Rolfsen at the University of British Columbia, Kunio Murasugi at the University of Toronto and Vaughan Jones at University of California, Berkeley. He also spent a semester as a member at Institute for Advanced Study, 1990. In 1995 he started working at George Washington University. He obtained his Habilitation at Warsaw University, in December 1994 and Presidential Professorship in Poland in 2013. Over the years he has traveled throughout the word and spent time as a visiting professor in many fine universities (e.g. Berkeley, Toronto, Vancouver).

His research includes, classical knot theory, topology and geometry of 3- manifolds, algebraic topology based on knots, skein modules and algebras, homology theories motivated by knot theory, to name several. Przytycki has been invited to international conferences, and has served in a variety of capacities: as a speaker, co-organizer, and member of scientific committees. He is the co-organizer of a series of conferences: Knots in Washington (started in 1995) as well as Knots in Poland (I, II, III) and Knots in Hellas conferences (I and II). He has served as editor of several journals such as Fundamenta Mathematicae, journal of Knot Theory and Its Ramification, and Involve. Every December he organizes an intensive workshop for his students, "Mathathon", where some open problem is presented and usually at least partially solved.

Till date he has supervised sixteen PhD students.

Rhea Palak Bakshi was born and brought up in India. She did most of her schooling at Welham Girls' School in Dehradun. She received her B.Sc. Honors in Mathematics in 2014 from Lady Sri Ram College for Women at the University of Delhi and her M.Sc. in Mathematics in 2016 from the University of Mumbai. Thereafter, she moved to the United States and got a Ph.D. in Mathematics in 2021 from the George Washington University. Her research interests lie at the confluence of low-dimensional topology, quantum topology, and knot theory. She is particularly interested in the theory of skein modules and algebras, various related conjectures such as the volume conjecture and the AJ conjecture, TQFTs, Khovanov homology, and categorification. She is also the coauthor of two chapters on skein modules and algebras in the Encyclopedia of Knot Theory. She currently works as a Research Fellow at the Institute for Theoretical Studies at ETH Zürich in Switzerland.

Dionne Ibarra is a Research Fellow at Monash University Clayton campus, Australia. She was born in Fresno, California USA and obtained her BA in 2010 and MA in 2012 from California State University, Fresno under the supervision of Professor Carmen Caprau.

In the winter semester of 2020 she was a Junior fellow at Institut Mittag-Leffler in Djursholm, Sweden. In 2022, she received a PhD in Mathematics from the George Washington University and wrote her dissertation on framed links in 3-manifolds and algebraic approaches to knot theory under the supervision of Professor Jozef H. Przytycki. Her main research area is low-dimensional topology, knot theory, diagrammatic algebras, and quantum invariants of links and 3-manifolds. She has also conducted research in interpolation methods using unit quaternions through the National Science Foundation (NSF) Mathematical Sciences Graduate Internship (MSGI) Program.

Gabriel Montoya-Vega was born in Barranquilla, Colombia where in 2015 he obtained a BS in Mathematics from the Universidad del Atlántico. In 2017 he completed a MS in Mathematics at the University of Puerto Rico-Mayaguez. He received a PhD in Mathematics in 2022 from the George Washington University. His main research area is knot theory, including invariants of links such as Fox colorings and Khovanov homology, and the history of knot theory.

After completing his PhD, he was awarded a National Science Foundation Mathematical and Physical Sciences Ascend Postdoctoral Research Fellowship affiliated to the City University of New York Graduate Center and to the University of Puerto Rico at Río Piedras.