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

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

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

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

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

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

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

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

Examples of Invariants Coloring Spanning Surfaces Polynomials

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)

3-Coloring the Trefoil

3-coloring Preserved Under Redermeister Moves Moves R1 and R2