1. Characteristics of knots
Potential invariants???
Chirality,Twist and Writhe

2. Chirality Right-handed and left handed knots
Achiral knot– same as mirror image
Not necessarily even no of crossings that makes a knot achiral

3. Prime Knots and Connected Sums (#)
Any knot can be represented as the sum of prime knots
Add like surfaces--- punch a hole and connect

4. Unknotting Changing at most half the crossings of a knot unknots it
Unknotting Number– smallest number of crossing changes necessary to unknot it
Still unknown for some knots with 9 or more crossings

5. Twist Knots and Twisting Number
aka. “stevedore’s knot”
Unknotting number of 1
what’s a stevedore? Go through unknotting

6. Writhing number Sum of the signs of the crossings in a knot diagram
Suggests potential energy in physical knots

7. Other Knots Alternating knots Wild knots– infinite sums of knots
Knot doubles

8. Knot Invariants and Isotopy
Homeomorphism vs. Isotopy
Isotopy– deformations of the string
e.g. twisting isotopic to writhing (the Whitney trick)
Homeomorphism is more powerful than isotopy… being in 1-D limits stretching

9. Examples of Invariants
Coloring
Spanning Surfaces
Polynomials

10. 3-Coloring Easiest (but weakest) knot invariant
A Knot is 3-colorable if you can:
color each part
total of 3 colors
at each vertex, all strands are the same color or different colors
The first way to prove the trefoil is knotted (see next slide)

11. 3-Coloring the Trefoil

12. 3-coloring Preserved Under Redermeister Moves
Moves R1 and R2