Invariants of Grace (Lihexuan) Yuan University of Melbourne

Objectives Tricolourability X polynomials and Jones polynomials saves the day! The aim of the project is to investigate various invariants A 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 and its . What is a knot? guish the 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 . 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() = 4n. Figure 3:Tricolourability is invariants under Reidemeister moves. •The span is the difference between the highest power and Figure 8:The writhe of knot projections under RI. such that f(K1, 0) = K1 and f(K1, 1) = K2. the lowest power that occurs in a polynomial. It is an invariant or knots. • We can use the writhe of the knot to define a new polyno- How tell if two knots are ambient isotopic? •The highest power in the bracket polynomial is n+2(W-1) mial, Two knots are the equivalent if they differ by a composition −w K when we do all A-split at each crossing; The lowest power is X(K) = (−A3) ( ) < K > of Reidemeister movement or planar isotopy. -n-2(D-1). W(or D) denotes the number of white (or dark) so that it is invariants under RI. area indicated below. Thus, the span = 4n.

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() = 4n. Bracket polynomial for the alternating Since span of the bracket polynomial is an invariants, then P2 knots also have n crossings, i.e. span() = 4n = span(). Hence both alternating reduced projections of the same knot Figure 1:Reidemeister moves. •A-split and B-split: Area A at a crossing is the have the same number of crossings.  area swiped over when rotating the overstrand counterclock- Figure 5:The three rules for bracket polynomial. wise.An A-split opens a channel between the two area A. B Acknowledgements split is analogous to A split. The bracket polynomial is invariant under RII and RIII, but I would like express my gratitude to my supervisor Marcy Robertson for not invariant under RI. her extraordinary support, patient guidance and kind invitation to the seminar. I would also like to thank the School of Mathematics and Statis- Figure 2:Ambient isotopic of trefoil. Figure 10:(a) Area A and Area B. (b) A split. (c) B split tics for offering the project.

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 . [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 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 : 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.