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Invited Paper Session Abstracts - Knot Theory

Thursday, August 4, 8:30 a.m. - 9:50 a.m and 2:00 p.m. - 4:20 p.m.., Fairfield

With the increase in undergraduate research there is also an increased need for open and accessible problems for students to tackle. Knot theory is particularly fertile ground for such problems. Each speaker in this session will introduce a topic, pose three open questions that are accessible to undergraduate research, and place the questions in context of the topic.

Colin Adams, Williams College
Lew Ludwig, Denison University

Turning Knots into Flowers: Petal Number and related Problems

8:30 a.m. - 8:50 a.m.
Colin Adams, Williams College

Working with students, we found that every knot has a petal diagram, a projection with just a single multi-crossing and petals around it like a daisy. We will discuss past work of students and the variety of open problems about petal projections, petal numbers and the generalizations to ubercrossing number and multi-crossing number.

Knot Mathematical Fiddlestix: An Introduction to Lattice Knots

9:00 a.m. - 9:20 a.m.
Jennifer McLoud-Mann, University of Washington, Bothell

This talk will be an introduction to lattice knots in the simple cubic, simple-hexagonal, face-centered, and body-centered cubic lattices. An appealing feature of working with lattice knots is their discrete nature - students get to play with stix to formulate conjectures! Examples of projects used to introduce undergraduates to mathematical research in this accessible area will be given as well as ideas for future work.

Problems Related to Spanning Surfaces of Knots

9:30 a.m. - 9:50 a.m.
Cynthia Curtis, College of New Jersey

Surfaces in knot complements with boundary the knot play a large role in understanding both knots and 3-dimensional spaces. These surfaces are used in the construction of many sophisticated, modern knot invariants. Using a collection of such surfaces, we discuss three concrete, fairly combinatorial problems accessible to undergraduates which are motivated by more advanced questions in knot theory.

Rope Magic and Topology

2:00 p.m. - 2:20 p.m. Louis Kauffman, University of Illinois, Chicago

In this talk the speaker will perform rope magic that appears to contradict topological theorems.
Rope will be supplied to the members of the audience so that they can participate in this event. There are a number of possible outcomes for this talk.

  1. Members of the audience may become convinced that basic knot theory is utterly wrong. For example, they will see a demonstration of the cancellation of the connected sum of two non-trivial knots. They will see a demonstration of the connected sum of two non-trivial knots coalescing to a single non-trivial prime knot. Knots will appear and disappear on a rope without sliding out the ends of the rope. Hands will penetrate small loops by what seems to be an access to higher dimensional spaces.

  2. The lecturer may disappear into 4-space.

  3. The lecturer may not disappear into 4-space, but the meeting grounds of the MAA may disappear into 4-space.

  4. None of the above.

  5. All of the above.

Khovanov Homology Mod 2 Detects Adequate Homogeneous States (NEW)

2:30 p.m. - 2:50 p.m.
Thomas Kindred, The University of Iowa

Given a adequate homogeneous state of a link diagram, we construct a sum of this state's enhancements (in the sense of Viro) which is nonzero in Khovanov homology over \(\mathbb{Z}/2\mathbb{Z}\). This talk is intended for a general audience. No background knowledge is assumed. Expect lots of pictures and concrete examples.

Accessible Problems for Undergraduates in Knot Coloring (CANCELED)

2:30 p.m. - 2:50 p.m.
Candice Price, Sam Houston State University

This talk will include a brief discussion of two invariants for knots: Fox-coloring and N-colorability. We include various examples and conclude by posing some problems concerning these invariants that can be explored. The invariants introduced here are arithmetic and algebraic and thus provide easily attainable projects for undergraduates.

Computer Algorithms for Counting Knot Mosaics

3:00 p.m. - 3:20 p.m.
Lew Ludwig, Denison University

In 2006, Lomonaco and Kauffman introduced another way to depict knots, knot mosaics, which uses 11 distinct 1x1 tiles. A relatively new topic, knot mosaics provide a treasure trove of open questions for undergraduate research. In this presentation, we will discuss my work with undergraduate students to use computer algorithms to count the number of distinct knot mosaics that can occur on square mosaic board. This presentation is intended for a general mathematical audience.

Gamifying Knot Theory

3:30 p.m. - 3:50 p.m. Jennifer Townsend, Bellevue College

Phrasing knot theory questions as games can provide inherent motivation for research, encourage creative construction of (counter)examples, and lead to unique perspectives and approaches. It also opens doors into new and approachable questions. We examine a few such questions, and discuss how a game-based approach can benefit students new to mathematical research.

Unknotting Knots

4:00 p.m. - 4:20 p.m.
Allison Henrich, Seattle University

There are several ways of unknotting knots. The classical unknotting operation is the crossing change, but there are others. The delta and sharp moves are two interesting examples that were discovered in the 1980’s by Murakami and Nakanishi. More recently, Ayaka Shimizu discovered another unknotting operation called the region crossing change. This new move has led us to ask classical unknotting questions in the region crossing change setting. We’ll consider several of the questions that have been studied by undergraduate researchers in the last couple of years as well as related questions that have yet to be explored.