UW Bothell and Seattle University join forces to host a 2016 research symposium that will showcase a plenary talk and the results from various groups from the National Science Foundation Research Experiences for Undergraduates (REU) sites at that two campuses. The full program for August 11 is below.

9:00-9:20 Continental breakfast – DISC 252

9:30-9:50 DISC 252

**Hexagonal Mosaic Knots**

Malachi Alexander, Selina Foster, Gianni Krakoff

Abstract: Lomonaco and Kauffman introduced knot mosaics in their 2008 paper Quantum Knots and Mosaics. We generalize the tiles, terms, and theorems from squares to hexagons. Additionally, we investigate properties unique to hexagonal mosaics, including minimal hextile number, composition of knots, and saturating diagrams.

10:00-10:20 DISC 252

**Divisibility by Powers of 2 in Pascal’s Triangle**

Philip de Castro, Desiree Domini, Devon Johnson, Ranjani Sundaresan

Abstract: A famous theorem of Kummer provides a connection between the entries of Pascal’s Triangle and base-2 arithmetic. We utilize this result to describe the rows of Pascal’s Triangle with no entries divisible by a fixed power of 2.

10:30-10:50 DISC 252

**Classifying All k-Isotoxal, Equilateral, Convex Tiles**

Toryn Avery, Rebecca Claxton, Kayla Neal

Abstract: In this paper, we intend to fully complete a problem initially presented by Ruby Chick and Dr. Casey Mann in their paper Equilateral k-Isotoxal Prototiles: Find all equilateral and convex tiles that admit only edge-to-edge edge-k-transitive monohedral tilings. We investigate quadrilaterals, pentagons, and hexagons that tile the plane by restricting their edges to congruent S-, C-, or J-curves, as defined in the previous paper.

11:00-11:20 DISC 252

**Extensions of the Link Smoothing Game**

Frank Guan, Mahnoor Mian, Angelique Morvant

Abstract: We study the Parity Link Smoothing Game introduced by Henrich and Johnson in the paper The Link Smoothing Game (2013). In this game, players take turns smoothing the precrossings 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.

11:30-12:20 Invited talk in DISC 061

**The translation surface for the Bothell pentagon**

Jayadev Athreya, Associate Professor, University of Washington Seattle

Abstract: We describe how to build a surface out of the famous Bothell pentagon which tiles the plane. We describe a process that can be summarized as “from Billiards to Pac-Man”, except if we start with the Bothell pentagon we get a genus 11 surface! No knowledge of billiards or topology is required, we’ll cover it all in the talk!

12:30-1:30 **Lunch**

Pick up Panera lunch boxes and enjoy the sun outside!

1:30-1:50 DISC 061

**Knot Fertility and Lineage**

Elsa Magness, Kayla Perez, Briana Zimmer

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. In this talk, we will explore families of knots and their relative fertility. We also explore ways to find the trefoil in every knot.

2:00-2:20 DISC 061

**Multi-material Decomposition for Dual-energy Computed Tomography**

Amanda Alexander, Derek Thurmer, Zachery Viray

Abstract: Dual-energy computed tomography (DECT) is a promising imaging technique in which data generated from two X-ray spectra are acquired at multiple angles around a patient. From these two data sets, DECT is able to provide a decomposition of the patient body into two materials, such as soft tissue and bone. Recent work has shown that using the assumption of volume conservation, one can obtain a third equation which permits decomposition of a third material. This three-material decomposition is typically performed after images have been reconstructed from the two data sets.

One drawback of a post-reconstruction technique is that the images must be corrected for effects such as beam hardening, which degrade image quality and thus affect the accuracy of the material decomposition. An alternative is to use a pre-reconstruction approach which attempts to obtain a material decomposition directly from the X-ray data. In this talk we present a pre-reconstruction technique based on conservation of volume and compare it with a post-reconstruction technique. Using numerical experiments, we show that the pre-reconstruction approach may be preferable in cases where mixtures of materials occur in some regions of the image. We also show that it is possible to recover decompositions of more than three materials, under certain restrictions on the object being imaged.

2:30-2:50 DISC 061

**Digital Patterns and Dominance Order in Base-3/2**

Andre Bland, Zoe Cramer, Joseph Koblitz

Abstract: In this talk we explore the base-3/2 representations of non-negative integers. We explain how to construct these representations and identify digit patterns within them. A special matrix is recursively defined in order to describe and understand these patterns. Connections are drawn between this matrix and a partial order on non-negative integers. Properties of this order are investigated by using two visualization techniques.

3:00-3:30 Tea Time & Snacks – DISC 061

3:30-3:50 DISC 061

**Spectral Subgradient Method for Unconstrained Optimization**

David Kotval, David Richmond, Yiting Xu

Abstract: Unconstrained minimization problems with non-differentiable convex objective functions, are usually solved by using the subgradient

method, whose convergence is guaranteed if the optimal value of the objective function is known. In this work, we combine the subgradient method with the spectral step length, which does not require either exact or approximated estimates of the optimal value. Since subgradient methods are not descent methods, we add a nonmonotone globalization strategy to ensure sufficient progress is made. This work also presents numerical results on a set of non-differentiable test functions. These numerical results indicate that using the spectral step length furnishes significant improvement over other subgradient methods.

4:00-4:20 DISC 061

**Knots on Lorenz-like Templates**

Justin Bryant, Alex Robkin

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.

4:30 Depart for Elevated Sportz in Bothell for trampolining!!