# Fundamentals of Numerical Analysis and Symbolic Computation

Coordinates:

Lecturer: Dr. Christoph Koutschan

Lecture number: 326003

Weekly lecture hours: 2

ECTS: 3

**Knot Theory and Computer Algebra **

This course gives a gentle introduction to knot theory, the subfield of mathematics that studies knots (as in real life, a mathematical knot can be imagined as a knotted string, with the difference that the two ends of the string are joined together). Central questions in this field include: Can a given knot be unknotted, without cutting the string? Are two given knots equivalent, i.e., can the first be transformed into the second, again without cutting the string or passing it through itself?

To address these questions, many different knot invariants have been found (colorability, knot polynomials, knot genus, knot groups, various homology theories, etc.), some of which will be discussed in the lecture. At the same time, we introduce concepts from computer algebra which support the computation of these knot invariants. A special focus is placed on symbolic summation techniques related to the holonomic systems approach, which allows, e.g., to compute the colored Jones polynomial and the non-commutative A-polynomial of a knot.

The first lecture takes place on Tuesday, October 8th, from 10.15 - 11.45 a.m. in room K223B.

Dates:

Every Tuesday, starting 22 October 2013, 10:00 - 11:30, S3 058

Exceptions: 12 November 2013, 10:00 - 11.30, **S3 057**

14 January 2014, 10:00 - 11:30, **S2 Z74**