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

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

Nov 03, 2022 188 likes •621 views

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

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MAS344 Knots and surfaces

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What is a knot?

Not a knot

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What is a knot?

Not a knot A knot

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What is a knot?

Not a knot A knot Not a knot

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Variants of knots

a link (the Borromean rings) a tangle a braid

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Another picture of the Borromean rings

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Knotted DNA

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Closing a braid to make a knot

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A table of knots and links

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The Perko pair

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Deforming a knot

These two knots are the same:

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Deforming a knot

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Deforming a knot

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Deforming a knot

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Deforming a knot

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Deforming a knot

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Deforming a knot

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Deforming a knot

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Deforming a knot

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Deforming a knot

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Jones polynomial example

The Jones polynomial of this knot is A16 − 4A12 + 6A8 − 7A4 + 9 − 7A−4 + 6A−8 − 4A−12 + A−16.

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Some knot theorists

Vaughan Jones John Conway Louis Kauffman

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Which of these knots are equivalent?

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Most basic invariant: number of components

c = 1 c = 2 c = 3

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Not an invariant: number of crossings

n = 6 n = 0 n = 4 n = 3 ≃ ≃

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Minimal crossing number

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

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Universes and diagrams

link universe link diagram (trefoil knot) link diagram (unknot)

  • riented link diagram

(positive Hopf link)

  • riented link diagram

(negative Hopf link)

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All diagrams for the trefoil universe

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Reidemeister moves

Type 1 Type 2 Type 3

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Reidemeister moves

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Reidemeister example

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Reidemeister example

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Reidemeister example

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Reidemeister example

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Reidemeister example

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Reidemeister example

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Reidemeister example

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Reidemeister example

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Reidemeister example

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Oriented link diagrams

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Oriented Reidemeister moves

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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.

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The skein relation

D+ D0 D− A4f (D+) − A−4f (D−) = (A−2 − A2)f (D0).