Ty Callahan

 Lord Kelvin thought that atoms could be knots  Mathematicians create table of knots  Organization sparks knot theory

 Knot  A loop in R 3  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

Right Left

 The Jones Polynomial of the Right Trefoil knot does not equal that of the Left Trefoil knot  The knots aren’t isotopes “KNOT” EQUAL!!