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The Three-Variable Bracket Polynomial for Reduced, Alternating Links Rose-Hulman Undergraduate Mathematics Journal Volume 14 Issue 2 Article 7 The Three-Variable Bracket Polynomial for Reduced, Alternating Links Kelsey Lafferty Wheaton College, Wheaton, IL, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Lafferty, Kelsey (2013) "The Three-Variable Bracket Polynomial for Reduced, Alternating Links," Rose- Hulman Undergraduate Mathematics Journal: Vol. 14 : Iss. 2 , Article 7. Available at: https://scholar.rose-hulman.edu/rhumj/vol14/iss2/7 Rose- Hulman Undergraduate Mathematics Journal The Three-Variable Bracket Polynomial for Reduced, Alternating Links Kelsey Lafferty a Volume 14, no. 2, Fall 2013 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 Email: [email protected] a http://www.rose-hulman.edu/mathjournal Wheaton College, Wheaton, IL Rose-Hulman Undergraduate Mathematics Journal Volume 14, no. 2, Fall 2013 The Three-Variable Bracket Polynomial for Reduced, Alternating Links Kelsey Lafferty Abstract.We first show that the three-variable bracket polynomial is an invariant for reduced, alternating links. We then try to find what the polynomial reveals about knots. We find that the polynomial gives the crossing number, a test for chirality, and in some cases, the twist number of a knot. The extreme degrees of d are also studied. Acknowledgements: I would like to thank Dr. Rollie Trapp for advising me on this project, providing suggestions for what to investigate, and helping to complete several proofs. I would also like to thank Dr. Corey Dunn for his advice. This research was jointly funded by NSF grant DMS-1156608, and by California State University, San Bernardino. Page 98 RHIT Undergrad. Math. J., Vol. 14, no. 2 1 Introduction The mathematical study of knots began in the 1800s and was closely tied to the ether model of the universe. In the 1890s, physicist P.G. Tait created knot tables with the goal of making a table of elements [12]. At the time, it was thought that atoms were knots in the ether. Consequently, it became important to know whether or not two knots were isotopic. A knot invariant is a property of a knot where if two knots have different invariants, then they are not isotopic. In the 1980s, Vaughan Jones defined the Jones polynomial for links [5]. Others generalized this polynomial, and it was shown that the Jones polynomial was a knot invariant [4]. Kauffman defined a three-variable bracket polynomial for knot diagrams. He also showed that under certain conditions, this polynomial is equivalent to the Jones polynomial [8]. In this paper, we show that the three-variable bracket polynomial is an invariant for reduced, alternating links. We then attempt to find what the three-variable bracket polynomial reveals about knots. For more about knots, links, and invariants, see [2]. Section 2 provides the necessary background to understand the results of this paper and proofs thereof. It also gives the proof that the three-variable bracket polynomial is an invariant for reduced, alternating links. In Section 3, we show how several properties of knots can be determined by the three-variable bracket polynomial; we examine the minimum degree of d as well as the maximum degree of d. Section 4 contains some concluding statements and suggestions for further research. 2 Background For our purposes, a mathematical knot is any closed loop in R3. Knot diagrams are embeddings of three-dimensional knots into the plane, where over-crossings are represented by a solid line, and under-crossings are represented by a break in the line. A knot with no crossings is the unknot.A link is multiple knots that may or may not be joined in such a way that they can not be pulled apart without breaking one of them (see Figure 1). In this paper, the only knots considered are reduced, alternating knots. A knot is reduced if it cannot be redrawn with any fewer crossings. A knot is alternating if, were one to choose an arbitrary point on the knot and follow a line around the knot, every time one passed an over-crossing, the next crossing must be an under-crossing; similarly, under-crossings must be followed by over-crossings. An example of a reduced, alternating knot is shown in the knot diagram in Figure 1(a). A twist is a region of a knot in which only two strands cross each other. The twist number of an alternating knot diagram is the fewest number of twists in any diagram of the knot. In Figure 1(a), the uppermost left and uppermost right crossings could each be considered a twist. However, upon further examination, one can easily see that the two can be viewed as a single twist with two crossings. One can define A and B regions of an alternating knot. A regions are defined as the region to the left of an under-crossing strand when viewing the crossing from that strand; the B regions are to the right. An example of a knot with labeled regions is depicted in Figure 1(b). Using these regions, it is possible to color any reduced, alternating knot diagram with RHIT Undergrad. Math. J., Vol. 14, no. 2 Page 99 (a) A Reduced, Alternating, Figure-8 Knot (b) Labeled A and B Regions for the Figure-8 Knot Figure 1: Figure-8 Knot a checkerboard pattern by shading either the A or B regions. Smoothings are defined for each crossing as cutting the knot and gluing it so that either the A regions or B regions are connected at the crossing, as depicted in Figure 2. If the A regions are connected after the smoothing, then it is an A smoothing. Otherwise, it is a B smoothing. Another way to describe smoothings is by zero and infinity tangles. A tangle is any part of a knot where a loop can be drawn around it so that two strands are entering the loop and two are exiting. A zero tangle is a tangle where the strands enter from the right or left and exit the opposite side without any crossings. The \slope" of the strands is zero. Figure 2(c) shows a zero tangle. An infinity tangle is a tangle where two strands enter from the top or bottom and exit the opposite side without any crossings. The \slope" of the strands is infinite. An example of an infinity tangle is depicted in Figure 2(b). These concepts can be used to identify twists with multiple crossings. If a twist is aligned vertically (so that its crossings make a vertical line), then an A twist is a twist in which all A smoothings would give an infinity tangle; a B twist is a twist in which all B smoothings would give an infinity tangle. In Figure 11, the twist is a B twist. (a) Original (b) A Smooth- (c) B Smooth- Crossing ing of Crossing ing of Crossing Figure 2: Smoothing Options for a Crossing Using the notion of smoothings, polynomials can be derived from knots. The three- Page 100 RHIT Undergrad. Math. J., Vol. 14, no. 2 variable bracket polynomial turns out to be an invariant for certain links, meaning that if two links have different polynomials, then they are not isotopic. For the rest of this paper, we use the term bracket polynomial to refer to the three-variable bracket polynomial, although typically it refers to a one-variable polynomial. Often a recursive definition for the bracket polynomial is used. The recursive definition of the bracket polynomial for a link L is depicted below (for more on the bracket polynomial see [9]). D E D E D E = A + B D E L = d hLi D E = 1 The Recursive Definition of the Bracket Polynomial In the recursive definition, it can be seen that A and B mean that either an A or a B smoothing has been applied at a crossing, respectively. The variable d is used to count the number of disjoint loops created as smoothings are applied to the knot. This paper uses the state model approach of defining the bracket polynomial. A state is a specific assigning of smoothings to crossings. The states of the Hopf Link are given in Table 1. In the state model approach, one considers every possible state of the knot. Each state is given a term AmBndp, where m is the number of A smoothings, n is the number of B smoothings, and p is one less than the number of loops created after the smoothings. By summing the terms from each state, one creates the bracket polynomial. This paper orders terms by decreasing degrees of A, with the largest degree of A coming first. An example of deriving the bracket polynomial is given in Table 1 for the Hopf Link. Note that if d were to equal one, the bracket polynomial would be equal to (A+B)r where r is the total number of crossings. The polynomial should follow the binomial expansion because one is assigning an A or B smoothing to each of the crossings. However, because d does not equal one, the bracket polynomial does not follow the binomial expansion exactly. For example, the bracket polynomial for the Figure-8 knot is given in expression (1). A4d2 + 4A3Bd + (5A2B2 + A2B2d2) + 4AB3d + B4d2 (1) We define a term that splits as one that has the same degrees of A and B, but different degrees of d. In (1), a term that splits is (5A2B2 + A2B2d2). Notice that if d equaled one, then the term would be 6A2B2, which is what would be expected from the binomial expansion.
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