QUANTUM GROUPS AND KNOT INVARIANTS

We discuss the Jones-Conway polynomial, also known as Homfly polynomial. It is a knot invari-ant, and we prove its existence and uniqueness given some simple axioms (value on the unknot and the so-called skein relations). The proof is following [2]. Definition 1.1. Informally, a knot is obtained by gluing together the endpoints of a shoelace in R3, considered up to isotopy (but remembering orientation). A link is the same as a knot, but multiple shoelaces are allowed. A tangle consists of (oriented) shoelaces and two parallel sticks, an upper and a lower, which we may glue some endpoints to. In addition, we assign + and − signs to the points where the end of a shoelace is attached to the stick, dependent on the orientation of the shoelaces, in the following way: on the upper stick, assign + to all the shoelaces leaving it, and − to all the shoelaces arriving to it, and on the lower stick, assign − to all the shoelaces leaving it, and + to all the shoelaces arriving to it. The meaning of the sticks here is that the order of + and − is fixed. The shoelaces have ends attached to them, but they do not wind around the sticks or interact with them in any other way. In fact, we can assume that the entire tangle is in R3 between planes z = 1 and z = 0, and that the sticks are parallel lines at heights 0 and 1. We are originally interested in distinguishing knots, and partially achieve it by constructing a strong invariant of oriented links. The main theorem we prove, using quantum groups and R-matrices, is the following. Theorem 1.2 (The Jones–Conway Polynomial). There is a unique map from the set of oriented links in R3 to the polynomial ring Z[x, x−1, y, y−1], denoted by L 7 → PL(x, y), such that (1) If L and L ′ are isotopic, then PL = PL′. (2) P (x, y) = 1 where is the unknot. (3) (Skein relations) If (X+, X−, X0) is a Conway triple of oriented links, meaning a triple where all their links have one distinguished crossing, with all orientations pointing down, such that on that crossing X+ contains X with / on top of \, X − contains X with \ on top of /, and X0 contains ||, and the rest of the links are the same; then polynomials associated to them satisfy the skein relation xPX+ − x−1PX − = yPX0. Example 1.3. Let n be n disjoint copies of the unknot. Then P n(x, y) = ( x − x−