Invariants of Knots Grace (Lihexuan) Yuan University of Melbourne
Objectives Tricolourability X polynomials and Jones polynomials Writhe saves the day! The aim of the project is to investigate various invariants A knot is tricolourable if each of the strands in the projection of the knots. can be coloured by one of the three colours so that at each • Pick an orientation of the knot projection, at each crossing, crossing, there will be one of the two scenarios: we have a either +1 or -1 defined below. (+1 if the strands 1. The three strands are of three different colours, can be lined up by rotating the under strand clockwise.) The How to tell two knots are the same? 2. The three strands have the same colour. writhe of the oriented knot projection is the sum of all the +1 Tricolourability is an invariants for knots, so we can distin- or -1 at each crossing. Figure 11:States of the trefoil knot and its bracket polynomial. What is a knot? guish the unknot from the trefoil. But we cannot use tri- 3 A knot is a smooth embedding of the circle S1 into R up colourability to distinguish trefoil and 74 knot. to isotopy. i.e. a map f : S → R3 without singularity such Conjecture 1 Figure 7:+1 and -1 crossing. that any projection to R2 is considered to be equivalent up Now we can use bracket polynomial to prove a conjecture. to ambient isotopy. • It turned out that the writhe is an invariants under RII and RIII. RI changes the writhe by ±1. "Equivalent" knots? Conjecture: Two reduced alternating projec- Informally, two knots are equivalent if one knot can be con- tions of the same knot have the same number of tinuously deformed into another without breaking it. crossings. Two knots K1 and K2 are ambient isotopic if there is an isotopy Lemma: If K has a reduced alternating pro- f : R3 × [0, 1] → R3 (1) jection of n crossings, then span(
Figure 4:Tricolourabibility of the unknot, trefoil and 74
Bracket (Kauffman) Polynomial Figure 12:White and dark area. The bracket polynomial is defined by the following three rules: Figure 9:X polynomial is invariants under RI. Suppose P1, P2 are two reduced alternating projection of If we replace each A by t−1/4, we obtain the Jones polynomials. the knot K, and P1 has n crossings, then by the lemma, span(
How to distinguish knots? •At each crossing, we make an A-split or B-split. The choice References of how to to split all of n crossings in the knot projection is Invariants Let K be the set of knots, S be any unspecific called a state. We will end up with various unknots. [1] Colin C. Adams. |S| The knot book. set. Figure 6:Bracket polynomial is not invariants under RI. •We denote as the number of unknots in the projection V K → S after splitting.Each time we have a A split, we multiply the American Mathematical Society, Providence, RI, 2004. : An elementary introduction to the mathematical theory of knots, So the bracket polynomial is not an invariants, but we can resultant polynomial by A. Revised reprint of the 1994 original. is an invariants, when V (K1) = V (K2) if K1 is ambient iso- •So the bracket polynomial for the alternating knot K is build a polynomial invariants based on bracket polynomial. [2] Dominic Goulding. topic to K . X 2 < K >= Aa(S)A−b(S)(−A2 − A−2)|S|−1 Knot Theory: The yang-baxter equation, quantum groups and S computation of the homfly polynomial. https://www.maths.dur.ac.uk/Ug/projects/highlights/ PR4/Goulding_Knot_Theory_report.pdf, 2010.