## MAS344 Knots and Surfaces

Semester 2, 2021/22 | 10 Credits | ||||

Lecturer: | Dr Fionntan Roukema | Timetable | Reading List | ||

Aims | Outcomes | Teaching Methods | Assessment | Full Syllabus |

The course studies knots, links and surfaces in an elementary way. The key mathematical idea is that of an algebraic invariant: the Jones polynomial for knots, and the Euler characteristic for surfaces. These invariants will be used to classify surfaces, and to give a practical way to place a surface in the classification.

**Prerequisites:** MAS114 (Numbers and Groups)

No other modules have this module as a prerequisite.

## Outline syllabus

- Knots and links
- The Jones polynomial
- Surfaces
- The Euler characteristic

## Aims

- To present a classification, that of compact surfaces, beginning from definitions and basic examples
- To instill an intuitive understanding of knots and compact surfaces
- To introduce and illustrate discrete invariants of geometric problems
- To show that adding extraneous structure may give information independent of that structure
- To develop the theory of the Euler characteristic
- To illustrate how a general mathematical theory can apply to quite different physical objects, and solve very specific problems about them

## Learning outcomes

- Recognize when a subset of
**R**^{2}(**R**^{3}) is a 1-dimensional (2-dimensional) manifold. - Show knowledge of basic surfaces.
- Perform word operations on words representing compact surfaces and decide which surface, up to homeomorphism, a given word represents.
- Compute the Euler characteristic of a plane model or a pattern of polygons on a compact surface and determine the underlying surface up to homeomorphism.
- Use Reidemeister moves to show two simple links are equivalent.
- Compute the Jones polynomial of a link and use it to obtain information about chirality, and to distinguish inequivalent links.

## Teaching methods

Lectures, problem solving

20 lectures, no tutorials

## Assessment

One formal 2.5 hour written examination. All questions compulsory.

## Full syllabus

**1. Knots and links.**

(3 hours)

**3. The Jones polynomial.**

(7 hours) Invariants. The Jones polynomial. Calculations from the skein relation; uniqueness. Construction (states of a universe, expectation, disconnectedness, Kauffman bracket, writhe). Applications to chirality.

**4. Surfaces.**

(2 hours) Basic surfaces (cylinder, Möbius band, sphere, torus, Klein bottle, projective plane). Orientability. Open, closed, bounded and compact sets in

**R**

^{n}. Homeomorphisms. Manifolds in

**R**

^{n}. Connected sum of two surfaces.

**5. Standard forms for surfaces (plane models).**

(4 hours) Definition of plane models and surface words. Relationship with compact surfaces. Orientability of plane models and surface words. Word operations which preserve a surface up to homeomorphism. Equivalent words. Standard form for words. Words and connected sums. Every orientable compact surface is homeomorphic to a sphere or a connected sum of tori; every non-orientable compact surface is homeomorphic to a connected sum of projective planes.

**6. Identifying and distinguishing surfaces (Euler characteristic).**

(4 hours) Patterns of polygons covering compact surfaces. Vertices, edges, faces and the Euler number. Plane models as patterns of polygons. All patterns of polygons covering a compact surface have the same Euler number. The Euler Characteristic χ(M) of a compact surface M. χ(S)=2, χ(T)=χ(K)=0, χ(P)=1. Euler Characteristic of a connected sum of two surfaces. Euler Characteristic of a connected sum of n tori or projective planes. Classification of compact surfaces using Euler Characteristic and orientability. Genus. Classification of compact surfaces using orientability and genus.

## Reading list

Type | Author(s) | Title | Library | Blackwells | Amazon |
---|---|---|---|---|---|

C |
Callier and Xu | A Guide to the Classification Theorem for Compact Surfaces | Blackwells | Amazon | |

C |
Comwell | Knots and Links | Blackwells | Amazon | |

C |
Firby and Gardiner | Surface topology | Blackwells | Amazon | |

C |
Gilbert and Porter | Knots and surfaces | Blackwells | Amazon | |

C |
Kauffman | On knots | Blackwells | Amazon |

(

**A**= essential,

**B**= recommended,

**C**= background.)

Most books on reading lists should also be available from the Blackwells shop at Jessop West.