The Knot Theory MA3F2 page

• Prerequisites Little more than linear algebra plus an ability to visualise objects in 3-dimensions. Some knowledge of groups given by generators and relations, and some basic topology would be helpful. The lectures and mind map that follow (from 2006) will be updated as we go through 2007.
• Mind map Course structure.
• Lectures 1 Writhe and linking numbers, 2 Reidemeister moves and colouring, 3 Colouring, 4 Splittable links and chess boarding, 5 Quadrilateral decomposition, 6 Application of Cramer's rule, 7 The determinant of a link, 8 The colouring group, 9 The number of colourings, 10 Mirrors and codes, 11 The Alexander polynomial, 12 Knot sums, 13 Bridge number and plats, 14 Daisy chains and braids, 15 Braids and Seifert circles, 16 Alexander's theorem: links to braids, 17 Seifert circles and trees, 18 The bracket polynomial, 19 The Jones polynomial, 20 The skein relations, 21 Alternating links, 22 Span(V) = number of crossings, 23 Tangles, 24 Rational tangles and continued fractions, 25 Tangled DNA, 26 Genus and knot sum, 27 Genus of a numerator, 28 Conway polynomial, 29 Conway, Alexander, Jones and HOMFLY, 30 Not Knot video,
• Examples 1, 2, 3, 4, 5, 6, 7, 8,
• Solutions 1, 2, 3, 4, 5, 6, 7 , 8, 9, 10, 11, 12
• Plan your degree course entry
• Recommended for the course:
  • Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, 1994. 306p
  • Livingston, Charles. Knot Theory Washington, DC: Math. Assoc. Amer., 1993. 240 p.
  • Cromwell, Peter Knots and Links Cambridge University Press, 2004. 346 p.

• The Rolfsen knot table.
• Can you see the isotopy between the Perko pair?
• An unknotting of a `Gordian knot'.
• Pictures of knots with 10 crossings taken from the tables of Tate and Little. The Perko pair is 5II and 6VI. Here 6VI is omitted. Where crossings are not indicated assume alternating (over, under, over, under ...).
• 8_17 is achiral
• 8_17 with Seifert circles
• Jones polynomials of knots up to 10 crossings together with their Dowker-Thistlethwaite codes.
• Pictures of rational knots 3₁ through 8₁ with their Conway notation and another view of 3₁ to 7₇.
• From plats to bridges.
• Simplifying a 4-plat.
• From knot to braid.
• A table of braid representations of knots.
• A list of Alexander and Jones polynomials.
• Finding the maximal degree term in the bracket polynomial.
• A Maple worksheet for calculating Alexander polynomials and determinants of knots and links.
• A DOS program to calculate knot polynomials.
• A java program to calculate the Jones polynomial.
• Knotted DNA and a DNA page from the Knots Exhibition at Bangor
• The trefoil knot bounds a torus with a hole.
• An aid to understanding rational/2-bridge knot construction and notation.
• Visual aids for the Not Knot video.
• Knots from the Alambra and elsewhere.
• Celtic knots, and a quote from Tait.

• The 1997 exam.
• The 1998 exam.
• The 1998 exam solutions.
• The 1999 exam.
• The 1999 exam solutions.
• The 2000 exam.
• The 2000 exam solutions.
• The 2001 exam.
• The 2001 exam solutions.
• The 2002 exam.
• The 2002 exam solutions.
• The 2003 exam.
• The 2003 exam solutions.
• The 2004 exam.
• The 2004 exam solutions.
• The 2005 exam.
• The 2005 exam solutions.
• The 2006 exam.
• The 2006 exam solutions.
• The 1997/98 Syllabus
• The 1998/99/00 Syllabus