On the Jones Polynomial and Its Applications

This paper is a self-contained introduction to the Jones polynomial that assumes no background in knot theory. We define the Jones polynomial, prove its invariance, and use it to tackle two problems in knot theory: detecting amphichirality and finding bounds on the crossing numbers. 1. Preliminaries 1.1. Definitions. For the most part, it is enough to think of a knot as something made physically by attaching the two ends of a string together. Since knots exist in three dimensions, when we need to draw them on paper, we often use knot diagrams. Figure 1.1 contains examples of knot diagrams. (a) unknot (b) trefoil (c) trefoil (again) (d) figure-8 knot Figure 1.1. Examples of knot diagrams As we can see from Figure 1.1b and Figure 1.1c, different diagrams can represent the same knot. To see that these two are really the same knot, we could make Figure 1.1b out of a piece of string and move the string around in space (without cutting it) so that it looks like Figure 1.1c. There are some restrictions on knot diagrams: (1) each crossing must involve exactly two segments of the string and (2) those segments must cross transversely. (See Figure 1.2.) (a) triple crossing (b) non-transverse crossing Figure 1.2. Examples of invalid knot diagrams Date: May 7, 2013. 1 ON THE JONES POLYNOMIAL AND ITS APPLICATIONS 2 There are two ways to travel around a knot; these correspond to the orientations of the knot. An oriented knot is a knot with a specified orientation. On a knot diagram, we can indicate an orientation via an arrow. (See Figure 1.3.) (a) one orientation (b) the other orientation Figure 1.3. Two orientations of the trefoil Sometimes we’ll use more than one piece of string, so we define a link to be a generalization of a knot: links can be made by multiple pieces of string. For each string, we attach the two ends together. (Note that we do not attach the ends of two different strings together.) The number of components of a link is the number of strings used. (Observe that every knot is a link with one component.) A link diagram is a straightforward generalization of a knot diagram, and an oriented link is a link where all the components have specified orientations. Figure 1.4 contains examples of two-component links. (a) unlink (b) Hopf link (c) Whitehead link Figure 1.4. Examples of links with two components 1.2. More mathematically... Readers who are not satisfied with the definitions given above may prefer the definition of a knot given here. Definition 1.1. Let X be a topological space. An isotopy of X is a continuous map h : X × [0, 1] → X such that h(x, 0) = x and h(·, t) is a homeomorphism for each t ∈ [0, 1]. ♦ Definition 1.2. A knot is a smooth embedding f : S → R. Two knots are considered equivalent if they are related by a smooth isotopy of R. ♦ This definition of a knot has the advantage of placing knot theory on firm mathematical grounding. For more detailed definitions, see [Cro04], [Mur96], or [Lic97]. Remark 1.3. We need the embeddings in Definition 1.2 to be smooth to avoid pathological “wild” knots. Instead of requiring the maps to be smooth, we could require them ON THE JONES POLYNOMIAL AND ITS APPLICATIONS 3 to be to be piecewise linear. Either of these is enough to guarantee the non-existence of wild knots. See [Cro04, Chapter 1] for what can happen if we do not assume either regularity condition. ♦ 1.3. Knot invariants. A knot invariant is something (such as number, matrix, or polynomial) associated to a knot. A link invariant is defined similarly for links. Example 1.4. The unknotting number of a knot K is the minimum number of times that K must be allowed to pass through itself to get to the unknot. This is a knot invariant. ♦ Example 1.5. We will define something called the crossing number. Suppose for a knot K, we take a diagram D of K and count the number of crossings in D. This number is not an invariant of K because K has many different diagrams that differ in number of crossings. For example, in Figure 1.5, we see two different diagrams of the unknot. (a) 0 crossings (b) 3 crossings Figure 1.5. Two diagrams of the unknot, with different number of crossings. Thus, we have not yet successfully defined a knot invariant. However, if we consider all diagrams of K and take the minimum number of crossings over all diagrams, then we do have an invariant of K. This is called the crossing number of a knot. We will see this invariant again in section 4. ♦ 1.4. Reidemeister moves. In 1926, Kurt Reidmeister proved that given two diagrams D1 and D2 of the same knot, it is always possible to get from one diagram to the other via a finite sequence of moves, now called Reidmeister moves. These moves can be divided into three types: