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Knot Tricolorability Danielle Brushaber & Mckenzie Hennen

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Knot Tricolorability Danielle Brushaber & Mckenzie Hennen Knot Tricolorability Danielle Brushaber & McKenzie Hennen Faculty Mentor: Carolyn Otto University of Wisconsin-Eau Claire Problem & Importance Definitions Knot Theory, a field of Topology, can be used to Basic Knot Definitions Linear Algebra Definitions model and understand how enzymes (called topoi- Knot- a curve in R3 that is non-intersecting and closed, similar to a circle but somerases) work in DNA processes to untangle or Matrix of a knot- each intersection’s equation determines a row in a matrix. For example, using WH3 , the equation could be tangled up. Consider a tangled string attached at the ends. 1 repair strands of DNA. In a human cell nucleus, the 2a − b − c = 0 would make one row of a matrix. a Knot invariant b DNA is linear, so the knots can slip off the end, - a characteristic of a knot. Useful for distinguishing when two c Determinant- A property of a square matrix. We used the program Maple to find the determinant, as the matrices are n and it is difficult to recognize what the enzymes do. knots are different. Tricolorable-For a knot to be tricolorable, the three lines involved in an intersec- as large as 26 × 26. In the case of the Trefoil knot, finding the determinant by hand is conducted in the following way: a b c First, look at the number in the top left corner and disregard the adjacent column and row. Repeat this step However, the DNA in mito- tion must either be 3 different colors or all 1 color . Tricolorability is a 2 − − =0 2 −1 −1 b a c with the second number in the first row and once more with the last number in the first row. The matrices chondria is circular, along with topological invariant. The unknot (a circle) is not tricolorable, so therefore anything 2 − − =0 becomes −1 2 −1 2c − a − b =0 below show these steps: prokaryotic cells (bacteria), so that is tricolorable cannot be the unknot. Colorability is the minor determinant of −1 −1 2 the enzyme processes are more the knot’s matrix. Additionally, the factors of this number are the colorability. For Step 1: Step 2: Step 3: For the basic 2 × 2 matrix, noticable in knots in this type example, if a knot is 21 colorable, it is also 3 colorable (tricolorable), and 7 colorable. 2 -1 -1 of DNA. Image from paper by De Witt Full positive twist- a change within the link of a Whitehead Double. We use n 2 2 −1 −(−1) −1 −1 +(−1) −1 2 a b = ad − bc Determinant= Sumners. to denote the number of full positive twists. Images of n = 1 through n = 6 are 2 −1 −1 −1 −1 2 −1 2 −1 2 −1 −1 Invariants prove to be a useful tool in studying when c d pictured in the Conjectures section. two knots are different. Tricolorability is an easily un- −1 2 −1 2 −1 −1 Doubling operator- an operation on a knot doubling Minor determinant- the determinant of the matrix found when the first column and first row are disregarded. For any n × n matrix, the minor derstood invariant that we will use to distinguish dou- the knot segments and adding a link. The Whitehead bles (replications) of certain prime knots. determinant is of an n − 1 × n − 1 matrix. For knots, the minor determinant of the knot’s matrix is the knot’s colorability. (WH) doubling of 31 is pictured to the right. Conjectures Proof for W H51 Conjecture As the number of full positive twists (n) changes, the minor determinant of the knot changes. We have discovered a pattern for knots WH31,WH41, and WH51 For WH51 with link n, n − WH51 is tricolorable when n = 1+3k where k∈ WH31 WH51 N n =1 n =1 is11 WH41 n =1 is21 ∪{0}. n =2 is17 n =2 is31 n =2 Consider WH5 n =3 is23 Regardless of the n, n =3 is41 1 WOLOG, let WH5 n n with link n being 1 n =3 n =4 is29 the colorability of 4 is always n =4 is51 be colored in this 1 the number of full n n n =4 n n =5 is35 1 . n =5 is61 way, excluding color- positive twists n n =6 is41 n =6 is71 ing the link: =5 (n =1...n = k). n =6 colorability is 5 + 6n colorability is 11 + 10n If the entrances to the twisting are colored in this way , then for every n = 1+3k As the number of full positive twists increases in the Whitehead double of a knot, the matrix behaves in N The matrices for WH3 where n =1, 2 and 3 are shown below respectively. where k∈ ∪{0}, the twisting, and therefore the knot, is tricolorable. the following way. (This pattern occurs when the intersections of the link are represented by the last 1 2 0 −1 0 −100000000000 2 0 −1 0 −10000000000 2 0 −1 0 −1000000000 2 0 0 −1 0 −10000000000 Base Case: k = 0, n = 1+3(0) = 1. Link n =1 is tricolorable. rows in its matrix.) 2 0 0 −1 0 −1000000000 2 0 0 −1 0 −100000000 0 2 0 0 −1000000 −10 0 0 0 0 2 0 0 −1000000 −10 0 0 0 2 0 0 −1000000 −10 0 02000 −10 0 0 0 −100000 For n =1+3(k + 1): n =1+3(k +1)=1+3k +3. Adding 1 to k adds 3 full 02000 −10 0 0 0 −10 0 0 0 General Pattern: n n 02000 −10 0 0 0 −10 0 0 00200000 −10 0 0 0 −10 0 When creating the matrix for = 2 from = 1, a row and column are added. The 00200000 −10 0 0 0 −1 0 00200000 −10 0 0 0 −1 002000000 −10 0 −10 0 0 002000000 −10 0 −10 0 002000000 −10 0 −1 0 −10020000 −10000000 positive twists to n, which looks like . last two rows, an added row, and an added column are the only changing components of the matrix. The −10020000 −1000000 −10020000 −100000 0 −10200000 −1000000 0 −10200000 −100000 0 −10200000 −10 0 0 0 0 0 0 −10 0 −1000200000 Placing multiples of this within a link does not disrupt the entrance coloring, as new column will contain all zero’s in the rows above the blue line. The new row is inserted below the 0 0 0 −10 0 −100020000 0 0 0 −10 0 −10002000 0 0 −10 0 0 0 −100200000 0 0 −10 0 0 0 −10020000 0 0 −10 0 0 0 −1002000 0 −10 0 0 0 −1000020000 the pattern is simply extended. So, adding 3 full positive twists does not change highlighted line of the matrix in n = 1. Besides a zero being added to the end as the new column entry, 0 −10 0 0 0 −100002000 0 −10 0 0 0 −10000200 −1000000 −100020000 −1000000 −10002000 −1000000 −1000200 0000000000 −10 2 0 −1 0 the link’s colorability, and does not change the colorability of the knot. All knots this row will not change to make n = 3. In the last row, a zero is added within the sequence [...2, −1, −1] 0000000000 −10 2 0 −1 0000000000 −1 −12 0 000000000000 −10 2 −1 00000000000 −1 −10 2 000000000002 −1 −1 00000000000 −10 0 −1 2 colored in this way with the link n being n = 1+3k where k∈ N∪{0} are making it [...2, 0, −1, −1]. After n = 3, as n increases, a zero entry is placed between the (−1)’s. 0000000000020 −1 −1 0000000000020 −1 0 −1 tricolorable. Future Research Directions References and Acknowledgments • We will move forward to prove our conjectures, the equations listed above. • Office of Research and Sponsored Programs, UW-Eau Claire • We will attempt to generalize this process to other knots and doubling operators. • Department of Mathematics, UW-Eau Claire • Additionally, we will try to apply these findings more directly to DNA. • Graphics computed and rendered using Inkscape and LATEX • Adams, Colin Conrad. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W.H. Freeman, 1994. Print. • Sumners, De Witt. ”Lifting the Curtain: Using Topology to Probe the Hidden Action of Enzymes.” Notices of the AMS 42.5(1995): n. pag. American Mathematical Society, May 1995. Web. Apr. 2015..
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