MAS344 Knots and surfaces What is a knot? Not a knot What is a - PowerPoint PPT Presentation

MAS344 Knots and surfaces

What is a knot? Not a knot

What is a knot? Not a knot A knot

What is a knot? Not a knot A knot Not a knot

Variants of knots a link (the Borromean rings) a tangle a braid

Another picture of the Borromean rings

Knotted DNA

Closing a braid to make a knot

A table of knots and links

The Perko pair

Deforming a knot These two knots are the same:

Deforming a knot

Deforming a knot

Deforming a knot

Deforming a knot

Deforming a knot

Deforming a knot

Deforming a knot

Deforming a knot

Deforming a knot

Jones polynomial example The Jones polynomial of this knot is A 16 − 4 A 12 + 6 A 8 − 7 A 4 + 9 − 7 A − 4 + 6 A − 8 − 4 A − 12 + A − 16 .

Some knot theorists Vaughan Jones Louis Kauffman John Conway

Which of these knots are equivalent?

Most basic invariant: number of components c = 1 c = 2 c = 3

Not an invariant: number of crossings ≃ ≃ n = 6 n = 0 n = 4 n = 3

Minimal crossing number Minimal crossing number is an invariant, but not a very useful one.

Universes and diagrams link diagram link diagram link universe (trefoil knot) (unknot) oriented link diagram oriented link diagram (positive Hopf link) (negative Hopf link)

All diagrams for the trefoil universe

Reidemeister moves Type 1 Type 2 Type 3

Reidemeister moves

Reidemeister example

Reidemeister example

Reidemeister example

Reidemeister example

Reidemeister example

Reidemeister example

Reidemeister example

Reidemeister example

Reidemeister example

Oriented link diagrams

Oriented Reidemeister moves

Sign of crossings ⊕ ⊖ ⊖ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊕ If you approach a positive crossing along the top strand, then you see the other strand pass underneath you from right to left.

The skein relation D + D 0 D − A 4 f ( D + ) − A − 4 f ( D − ) = ( A − 2 − A 2 ) f ( D 0 ) .