CUHK MATH Alternating Knots

Tie me up, Tie me down
 
Equivalence
Trefoil
Figure Eight
Whitehead
Trivial Knot
Perko Pairs
 
Classifications
Torus Knots
Alternating Knots
 
Invariants
Crossing number
Bridge number
Unknotting number
Tricolorability
Linking number
Polynomials


Example of Alternating Knot

Do you see why the following is a alternating knots?

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for an answer.


Whether they are Alternating Knots or not?

Knot with 3 crossings:

Knot with 5 crossings:

Knot with 6 crossings:

Knot with 7 crossings:

Are there non-alternating knots?


Non-alternating Knots

The first non-alternating knot discovered!

Actually, all knots with less than 8 crossings (except composite knots) are alternating.

Here are the brothers and sisters of the above knot:




Decide whether the
left three are
alternating or not?


Composite Knots

Composite knots are made up from simpler knots:


Square Knot

Square knot is non-alternating as it have 6 crossing, not a prime!


More Examples I would like to try other examples on classifications of knots.
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