Knots_Knotes-Justin_Roberts.pdf

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notes.dvi

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De nition 1.1.1 (Provisional). A knot is a closed loop of string in R

; two knots are equivalent

coincides with the other. Cutting and rejoining is not allowed.

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Question 1.2.2. How mightwe prove inequivalence of knots? To showtwoknots are equivalent,

remaining two separate: the entanglement of the rings is dependenton all threecomponents at the

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any diagram of that knot. (This is a natural measure of complexity.) A minimal diagram of K is

K of a knot K is obtained by re ecting it in a plane in R

De nition 1.3.1. The mirror-image

De nition 1.3.3. An oriented knot is one with a chosen direction or \arrow" of circulation along

orientation. One may also de ne the inverse r

By analogy with amphichirality, we have a notion of a knot being reversible or invertible if it is

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7! alternating diagram

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