On the Second Twist Number

It has been shown that the twist number of a reduced alternating knot can be determined by summing certain coefficients in the Jones Polynomial. In the discovery of this twist number, it became evident that there exist higher order twist numbers which are the sums of other coefficients. Some relations between the second twist number and the first are explored while noting special characteristics of the second twist number. 1 Background Information We begin this paper with a short review of knot theory and some important definitions. Mathematically, a knot is a closed curve in space. Further, a link is a collection of one or more interlinked knots. Here we see that a knot is just a link with one component. The simplest link is just a circle or the unknot, since it has no crossings. If we consider a link with just one crossing, we realize that this is really the unknot again with a kink in it. Now in considering a link with two crossings we have a choice: either we can get an unknot with two kinks in it or we can get a link composed of two unknots as in figure 1, called the Hopf link. The simplest nontrivial knot is called the trefoil, and it has three crossings (see figure 1). The Hopf link and trefoil are examples of alternating links, or links in which crossings alternate. Figure 1: A Hopf link and a trefoil. 1 Figure 2: The Reidmeister moves. Figure 3: A four-crossing twist. Much of Knot theory is devoted to finding ways that can tell if two different link projections, or pictures of links, represent different links or the same link. If two links are topologically identical, they are called ambient isotopic. If we have two different projections of a specific link, it has been proven that there is a sequence of Reidemeister moves (see figure 2) from one to the other. In more intuitive terms this means that two links are equivalent if one can be deformed into the other without tearing or allowing strands to pass through other strands. A twist or an integral tangle in a link, is a section where two strands tangle around themselves one or more times, as seen in figure 3. The minimum number of twists taken over all projections of a link L is the twist number of that link and is denoted T(L). Figure 5 provides an example of a knot with twist number 2. For further discussion on twists see Lin [3, p. 7].