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 R2 (R3) 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)

Examples (to include the unlinks, trefoil, figure eight, simple chains and Borromean rings). Equivalence of links. The Reidemeister moves. Link universes, crossings, components. Orientation and the right hand rule. Amphicheiral and reversible knots.
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 Rn. Homeomorphisms. Manifolds in Rn. 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.