KNOT INVARIANTS: THE JONES POLYNOMIAL AND KHOVANOV HOMOLOGIES

· What is a knot invariant and why are they useful?

· What is the Jones polynomial and how is it calculated?

· What is the Khovanov homology and how is it an improvement over the Jones polynomial?

Q: What is a knot invariant and why are they useful?

A: A knot invariant is an object or a quantity that can be calculated from a knot or knot diagram, and is the same for any two equivalent knot diagrams. Many times, it is difficult to tell whether two knot diagrams are equivalent—the Reidemeister moves will eventually transform one into the other if they are; however, the Reidemeister moves do not provide a proof that two knots are different (there is no upper bound on how many Reidemeister moves could be required). Hence, it is necessary to find certain quantities that are the same for equivalent knot diagrams, but distinguish between almost every distinct knot (It would be asking too much for it to distinguish between EVERY knot). Also, if the knot invariants can be found through some algorithm, it may be simpler for a computer to find them than computer a stack of Reidemeister moves. This is the importance of knot invariants. Two knot invariants that distinguish between almost every known distinct knot are the Jones polynomial and the Khovanov homologies.

Q: What is the Jones polynomial and how is it calculated?

A: The Jones polynomial is maybe the most common knot invariant, due to its efficiency at distinguishing knots and the ease of its calculation. We will not calculate the Jones polynomial in the way most books do, but the result is the same, and the method we use closely relates to that which Khovanov uses to find his invariant. As an example, we will examine the Hopf link, the simplest non-trivial link:

First we introduce the concept of 0-smoothings and 1-smoothings. The idea is to “break” both strands at a crossing and reconnect the ends in such a way that eliminates the crossing. Such a connection is called a “smoothing.” If you do this for each crossing in the link, it is called a “complete smoothing.” There are two ways to smooth a crossing. You can either connect the “over” strands clockwise or counterclockwise to the “under” strands. A clockwise connection is called a 0-smoothing, while a counterclockwise connection is called a 1-smoothing.

We now look at all complete smoothings of the link. To index them, we number the crossings in the link, and look at the complete smoothings as the vertices of the n-dimensional unit cube, the i-th coordinate being 0 if the i-th crossing is a 0-smoothing, 1 if it is a 1 smoothing.

Next, for each closed loop in each diagram, we assign a factor of (q + q^-1), where q is an indeterminate variable. Next, we multiply each term by a factor of (-q)^s where s is the number of 1-smoothings in the corresponding diagrams. Finally, we sum the resulting terms.

Technically, this is the unnormalized Jones polynomial. To get the normalized Jones polynomial, there is an addition factor to multiply through by, but for our purposes, this polynomial is good enough.

Q: What is the Khovanov homology and how is it an improvement over the Jones polynomial?

A: Under construction.