Project Mentors: Allison Henrich, Daniel Heath, and AJ Stewart
During the first two weeks of the SUMmER 2016 program, students in the knot theory group learned about the foundations of knot theory, studying knot equivalence, knot invariants, knot families, unknotting operations, pseudoknots, multi-crossing knot diagrams, and knot games. Following this introduction, students explored the following projects.
J. Cantarella, A. Henrich, E. Magness*, K. Perez*, E. Rawdon, B. Zimmer*, "Knot Fertility and Lineage," Journal of Knot Theory and its Ramifications, 26 (2017), no. 13, 1750093, 20 pages.
Abstract: Is your favorite knot fertile? We define a knot K to be a parent knot of a knot H if some number of crossings in a minimal crossing projection of K can be resolved to produce a diagram of H. We say that K fertile if it is a parent knot of every knot with a smaller crossing number than itself. We explore families of knots and their relative fertility. We also explore ways to find the trefoil in every knot.
J. Bryant*, D. Heath, and A. Robkin*, "Free will = determinism (on the Lorentz template)," submitted, 2016.
J. Bryant*, D. Heath, and A. Robkin*, "Knots on Lorentz-like templates," in preparation, 2016.
Abstract: Knots on the Lorenz template have been well studied when the flow is oriented downwards. In this presentation we investigate the effects of reversing the flow on the knots produced in the template. We use Lyndon to words describe knots on the Lorenz template. Extending these results to twisted Lorenz-like templates allows us to then use Lyndon words to classify various families of torus knots.
F. Guan*, M. Mian*, A. Morvant*, and A. Stewart*, "Extensions of the link smoothing game," in preparation, 2016.
Abstract: We study the Parity Link Smoothing Game introduced by Henrich and Johnson in 2013. In this game, players take turns smoothing the pre-crossings of a link shadow diagram, resulting in a diagram with some number of links. One player’s goal is to resolve the shadow into an even number of components, and the other player’s goal is to resolve it into an odd number. We investigate whether or not the Parity Link Smoothing Game is balanced (meaning that both players have equal numbers of winning outcomes) and determine which player has a winning strategy on a given diagram. In addition, we explore the possibility of creating a balanced game for more than two players. Finally, we attempt to answer open questions about the Link Smoothing Game by applying techniques of combinatorial game theory, specifically the Grundy function.
Project Mentors: Tom Edgar and Steven Klee
If x and y are two whole numbers, we say that y dominates x if every digit in y is at least as large as its corresponding digit in x, meaning the ones digit of y is at least as large as the ones digit in x; the tens digit of y is at least as large as the tens digit of x; and so on for the hundreds digits, thousands digits, etc. If y dominates x, we write x<<y.
But there is nothing special about the way that we choose to represent numbers in base 10 with a ones digit, tens digit, hundreds digit, thousands digit, and so on. We can also talk about dominance for integers that are represented in binary: for example, 1001001 << 1101011 because each bit in 1101011 is at least as large as the corresponding bit in 1001001.
Beyond representations in base two, base three, or base ten, we can also explore representations of integers in rational bases. In each of these contexts, there is a way to define "digits" and so we can also define domination for these various representations of numbers. Students in the SUMmER 2016 REU explored these notions of digital dominance, specifically in base-p when p is prime and in base-3/2. This resulted in the following projects.
P. de Castro*, D. Domini*, T. Edgar, D. Johnson*, S. Klee, R. Sundaresan*, "Digital representations of rows of Pascal's triangle with no entries divisible by a fixed prime power," Pi Mu Epsilon Journal, 14 (2017), no. 7, 431-442.
Abstract: A famous result of Kummer shows that none of the binomial coefficients (n choose k) are divisible by a fixed prime p if and only if n is one less than a power of p. For a fixed prime power p^m, we characterize the digital representations of all numbers n such that none of the binomial coefficients (n choose k) are divisible by p^m.
P. de Castro*, D. Domini*, T. Edgar, D. Johnson*, S. Klee, R. Sundaresan*, "Counting binomial coefficients divisible by a prime power," American Mathematical Monthly, to appear, 2018.
Abstract: We define a new class of integer partitions generalizing hyperbinary partitions. We then utilize these partitions to provide a new, relatively simple formula for the number of binomial coefficients in a particular row of Pascal's triangle that are divisible by a fixed power of a prime. Our formula specializes to a famous result due to N.J. Fine from 1947. Furthermore, we demonstrate a connection between the binomial coefficients congruent to zero modulo a fixed prime power and the so-called digital dominance orders on the natural numbers.
A. Bland*, Z. Cramer*, P. de Castro*, D. Domini*, T. Edgar, D. Johnson*, S. Klee, J. Koblitz*, R. Sundaresan*, "Happiness is integral, but not rational," Math Horizons, 25 (2017), no. 1, 8-11.
Abstract: The digital sum of squares function is defined by taking the sum of the squares of the digits of a given number. A number N is called happy if, by repeatedly applying the digital sum of squares function, we eventually reach the number 1. In this paper, we explore happy numbers in base-3/2.