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Department of Mathematical Sciences

# MATH2627 Geometric Topology II

This course gives an introduction to topology in an intuitive, visual way by studying knots, links and surfaces. Apart from everyday life, knot theory has applications in many sciences (e.g. in quantum physics or molecular biology) and in various branches of mathematics. Knots and links give rise to exciting geometric objects and the course provides tools to study them. Starting with combinatorial moves we introduce sophisticated invariants, like the Conway- and Jones-polynomials, which are effectively computable and carry important topological information.

Surfaces are also very intuitive objects appearing in many branches of mathematics, and we will show applications to knot theory, and study properties of vector fields on surfaces.

## Outline of Course

• Knots, equivalence, knot diagrams, Reidemeister moves, 3-colouring, writhe and linking numbers, alternating knots.
• Knot polynomials, Jones polynomial, trefoil not isotopic to mirror image, Alexander-Conway polynomial, absolute polynomial.
• Surfaces, triangulation of surfaces, Euler characteristic, classification of surfaces, Seifert surfaces, genus of knots.
• Plane vector fields, index round a closed curve, non-zero index means field vanishes somewhere inside curve. Brouwer Fixed Point Theorem. Degree of map of circle into punctured plane, vector fields on surfaces, Poincare-Hopf theorem.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.

### Examination Information

For information about use of calculators and dictionaries in exams please see the Examination Information page in the Degree Programme Handbook.