At UnKnot Conference III (Granville, OH). Dr. Jennifer McLoud-Mann (left), Selina Foster (mid-left), Malachi Alexander (mid-right), Gianni Krakoff (right)

Mentor: Dr. Jennifer McLoud-Mann
Malachi Alexander, Selina Foster, Gianni Krakoff

Project Title: Hexagonal Mosaic Knots
Abstract: In 2008, Lomonaco and Kauffman introduced square mosaic knots in their paper Quantum knots and mosaics. Lomonaco and Kauffman proposed a new invariant, mosaic number, in which knots could be created on a set of n by n tiles and established definitions, notation and the fundamental theory of knot mosaics. In this paper, we further develop mosaic knot theory by extending the tiles, terms and theorems to hexagonal mosaic knots. We investigate analogous arguments from square mosaic knots to hexagonal mosaic knots including Reidemeister moves, mosaic number and saturated diagrams. In this paper, we establish a new invariant, hextile number, which represents the minimal number of tiles necessary to create a knot diagram and establish hextile numbers for all knots with crossing numbers less than seven. In addition, we introduce new concepts such as edge-to-curve graphs and algebraic saturation that are applicable to general knot mosaic theory. Our results allow us to create more complicated knot diagrams on tiles while still maintaining an underlying discrete structure.

As part of this research experience, I visited the University of Washington Bothell in Washington, Bothell for eight weeks during the summer of 2016. During the visit, I worked under the mentorship of Dr. Jennifer McLoud-Mann with two other undergraduates, Selina Foster and Gianni Krakoff, as part of the Knot Theory group of the program.

At University of Washington Bothell (Bothell, WA).

We have presented on the results of our research at Joint Mathematics Meeting in Atlanta, Georgia as a poster and as an oral talk. I have also given a poster presentation and oral presentation independently at California State University, Monterey Bay (CSUMB) at my department research symposium and campus research showcase, respectively.

At Joint Mathematics Meeting 2017 (Atlanta, GA).

For an electronic copy of the poster presentation and oral presentation slides, click the following links:
Hexagonal Mosaic Knots Poster (JMM 2017)
Hexagonal Mosaic Knots Talk (JMM 2017)
Hexagonal Mosaic Knots Talk (Spring Showcase at CSUMB)

Additional Resources:
UW Bothell Mathematics REU Website
UnKnot Conference III Article