Quandles and algebra of Knots.

This intensive MAM research frontier lectures PhD course aims to present an introduction to Quandle Theory and Knot Theory. The lectures are organised by the research environment in Mathematics and Applied Mathematics, MAM at Mälardalen University, Västerås, Sweden, in November, 2018.

Introduction to Quandle Theory and Knot Theory

Dr. Mohamed Elhamdadi

Department of Mathematics and Statistics,

University of South Florida, Tampa, USA

Intensice MAM research frontier lectures, PhD course

Quandles and algebra of Knots (pdf 96 kB)

 

Time and place:

13.15-18.00, November 27, 2018, Place:  U2-158, Västerås 

13.15-14.00, Thursday, November 29, Place R2-605, Västerås

About the course

Quandles and racks are non-associative algebraic structures whose axioms correspond to the axiomatization of the three Reidemeister moves in knot theory. The earliest known work on racks is contained in the 1959 correspondence between John Conway and Gavin Wraith who studied racks in the context of the conjugation operation in a group.  Around 1982, Joyce [5] (used the term quandle) and Matveev [7] (who call them distributive groupoids) introduced independently the notion of a quandle. Joyce and Matveev associated to each oriented knot K a quandle Q(K ) called the knot quandle. The knot quandle is a complete invariant up to orientation. Since then quandles and racks have been investigated by topologists in order to construct knot and link invariants and their higher analogues (see for example [2] and references therein). In 1992, Fenn and Rourke [3] showed that any codimension-two link has a fundamental rack which contains more information than the fundamental group. They gave some examples of computable link invariants derived from the fundamental rack and explained the connection of the theory of racks with that of braids. In

2003, Fenn, Rourke and Sanderson [4] introduced rack homology. This (co)homology was modified in 1999 by Carter et al. [1] to give a cohomology theory for quandles. This cohomology was used to define state-sum invariant for knots in three space and knotted surfaces in four space. In this course, we will give a gentle introduction to the theory of quandles and explain its relation to Knot theory and mention recent works including connections to Hom-algebra structures [6].

References

[1] Carter, J.S.; Jelsovsky, D.; Kamada, S.; Langford, L.; Saito, M., Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math.  Soc. 355 (2003), 3947-3989.

[2] Elhamdadi, M. and Nelson S., Quandles-an introduction to the algebra of knots, Student Mathematical Library, 74. AMS, 2015. x+245 pp.

[3] Fenn R., and Rourke C., 1992. Racks and links in codimension two, J. Knot Theory Ramifications, 1, 343-406.

[4] Fenn R., Rourke C. and Sanderson B., 1992. The rack space, Trans. Amer. Math. Soc. 359 (2007), no. 2, 701-740.

[5] Joyce, D., A classifying invariant of knots, the knot quandle, J. Pure Appl. Alg., 23, (1982), 37–65.

[6] Makhlouf, A. and Silvestrov S., Hom-algebra structures, J. Gen. Lie Theory Appl. 2 (2008), no. 2, 51-64.

[7] Matveev, S., Distributive groupoids in knot theory, (Russian) Mat. Sb. (N.S.) 119(161) (1982), no. 1, 78–88, 160.