COURSE SUMMARY
Course Title: 
Theory of Knots and Braids
Course Code: 
18MAT737
Year Taught: 
2019
Semester: 
2
Type: 
Elective
Degree: 
Doctoral Programs
School: 
School of Arts and Sciences
Campus: 
Kochi

'Theory of Knots and Braids' is an elective course offered in the second semester of M.Phil. in Mathematics program (Full Time/Part Time) offered by School of Arts & Sciences, Kochi.

Unit 1: Knots, Knot arithmetic and Invariants:

Knots- Definition, Simple examples, Elementary properties, Polygonal links and Reidemeister moves Knot arithmetics and Seifert surfaces; Knots in Surfaces- Torus Knots, The linking coefficient, The Arf invariant and The colouring invariant.

Unit 2: Fundamental group, Quandle and Conway’s algebra

Examples of unknotting, Fundamental group, Calculating knot groups, Quandle- Geometric description of the quandle, Algebraic description of the quandle, Completeness of the quandle, Special realisations, Conway algebra and polynomials, Realisations of the Conway algebra, Matrix representation.

Unit 3: Jones polynomial and Khovanov’s polynomial

Kauffman’s bracket, Jones’ polynomial and skein relations, Kauffman’s two–variable polynomial, Jones’ polynomial. Khovanov’s complex, Simplest properties, Tait’s first conjecture and Kauffman– Murasugi’s theorem, Classification of alternating links, The third Tait conjecture, A knot table, Khovanov’s polynomial, The two phenomenological conjectures.

Unit 4: Braids, links and representations

Definitions of the braid group- Geometrical definition, Topological definition, Algebro– eometrical definition and Algebraic definition; Equivalence of the four definitions, The stable braid group , Pure raids, Links as braid closures, Braids and the Jones polynomial; Representations of the braid groups- The Burau representation, The Krammer–Bigelow representation, Krammer’s explicit formulae, Bigelow’s onstruction, Alexander’s theorem, Spherical and cylindrical braids; Vogel’s algorithm.

Unit 5: Vassiliev’s invariants and the Chord diagram algebra

Vassiliev’s invariants-Definition and Basic notions, Singular knots, Invariants of orders zero and one, Examples of higher–order invariants, Conway polynomial coefficients, Other polynomials and Vassiliev’s invariants, An example of an infinite-order invariant, The Chord diagram algebra - Chord diagram algebra- Basic structures, Bi algebra structure, The four colour theorem.

  1. Knot Theory, V. O. Manturov, CRC Press; 1st edition (2004).
  2. Knot theory and its applications, K. Murasugi, Springer.
  3. Braid and Knot theory in dimension four, S. Kamada, American Mathematical Society.