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Ty Callahan Lord Kelvin thought that atoms could be knots Mathematicians create table of knots Organization sparks knot theory Knot A loop in R3 Unknot Arc Portion of a knot Diagram Depiction of a knot’s projection to a plane OK NOT OK Two knots are equivalent if there is an isotopy that deforms one link into the other Isotopy Continuous deformation of ambient space Able to distort one into the other without breaking Nothing more than trial and error can demonstrate equivalence Can mathematically distinguish between nonequivalence Choice of the sense in which a knot can be traversed Orientation results in two possible crossings Right and Left Two Principles 1) Assign a value of 1 to any diagram representing an unknot 2) Skein Relation: Whenever three oriented diagrams differ at only one crossing, the Jones Polynomial is governed by the following equation 1 t R[t] tL[t] (t 1 2 t 1 2 )Q[t] 1 t R1[t] t (t R1[t] (t 3 2 1 2 1 t 1 2 )Q1[t] t )Q1[t] t 2 2 1 t R2[t] tL2[t] (t R2[t] t L2[t] t 2 1 2 3 2 t 1 2 t 1 2 ) 1 t t (t t 1 2 t (t 1)(t 1 2 3 2 1 2 t 1 2 )Q3[t] (t 1)Q3[t] t 1 2 Q3[t] t ) (t 1)Q3[t] 1 2 t 1 2 L2[t] Q3[t] t R2[t] t (t 1 R2[t] t 5 2 2 2 t 1 3 2 1 2 )t t 2 t R2[t] t 5 2 t 3 t 2 1 2 3 1 2 2 t t 1 2 1 2 Q1[t] R2[t] t R1[t] (t 3 2 1 t )(t 2 5 5 2 2 t 1 2 1 2 t )t R1[t] t t t t t 4 2 3 R1[t] t t t 4 3 2 2 Right Left R1[t] t t t 4 4 3 3 R1[t] t t t 1 The Jones Polynomial of the Right Trefoil knot does not equal that of the Left Trefoil knot The knots aren’t isotopes “KNOT” EQUAL!!