Finding Fibonacci in the Hyperbolic Plane. To appear in Mathematics Magazine. [view]
show abstractWe consider a combinatorial approximation of the hyperbolic plane, and show that the perimeter of a disk with radius in this model is given by
. We use properties of Fibonacci numbers to show that this model satisfies a linear isoperimetric inequality.

Random groups at density act non-trivially on a CAT(0) cube complex. Transactions of the AMS 376 (2023), 1653-1682. [arXiv]
For random groups in the Gromov density model at , we construct walls in the Cayley complex X which give rise to a non-trivial action by isometries on a CAT(0) cube complex. This extends results of Ollivier-Wise and Mackay-Przytycki at densities
and
, respectively. We are able to overcome one of the main combinatorial challenges remaining from the work of Mackay-Przytycki, and we give a construction that plausibly works at any density
.

Graphs with Many Hamiltonian Paths. With Erik Carlson, Willem Fletcher, Chi Nguyen, Xingyi Zhang. [arXiv] Submitted.
show abstractA graph is hamiltonian-connected if every pair of vertices can be connected by a hamiltonian path, and it is hamiltonian if it contains a hamiltonian cycle. Every hamiltonian-connected graph is hamiltonian, however we also construct families of nonhamiltonian graphs with `many’ hamiltonian paths, where ‘many’ is interpreted with respect to the number of pairs of vertices connected by hamiltonian paths. We then consider minimal graphs that are hamiltonian-connected; we show that any order- graph that is hamiltonian-connected graphs must have
edges, and we construct an infinite family of graphs realizing this minimum.

Property (T) in -gonal random groups. Glasgow Mathematical Journal, 64 (3): 734-738, 2022. [arXiv]
The -gonal models of random groups are defined as the quotients of free groups on n generators by cyclically reduced words of length k. As
tends to infinity, this model approaches the Gromov density model. In this paper we show that for any fixed
, if positive
-gonal random groups satisfy Property (T) with overwhelming probability for densities
, then so do
-gonal random groups, for any
. In particular, this shows that for densities above
, groups in
-gonal models satisfy Property (T) with probability 1 as
approaches infinity.

Knot projections with a single multi-crossing. With Colin Adams, Thomas Crawford, Benjamin DeMeo, Michael Landry, Alex Tong Lin, Seojung Park, Saraswathi Venkatesh, Farrah Yhee. Journal of Knot Theory and Its Ramifications. Volume 24 (3) 2015. [arXiv]
show abstractIntroduced recently, an -crossing is a singular point in a projection of a link at which n strands cross such that each strand travels straight through the crossing. We introduce the notion of an übercrossing projection, a knot projection with a single
-crossing. Such a projection is necessarily composed of a collection of loops emanating from the crossing. We prove the surprising fact that all knots have a special type of übercrossing projection, which we call a petal projection, in which no loops contain any others. The rigidity of this form allows all the information about the knot to be concentrated in a permutation corresponding to the levels at which the strands lie within the crossing. These ideas give rise to two new invariants for a knot
: the übercrossing number
, and petal number
. These are the least number of loops in any übercrossing or petal projection of
, respectively. We relate
and
to other knot invariants, and compute
for several classes of knots, including all knots of nine or fewer crossings.

Complementary regions of multi-crossing projections of knots.[arXiv]
show abstractAn increasing sequence of integers is said to be universal for knots if every knot has a reduced regular projection on the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. Adams, Shinjo, and Tanaka have, in a work, shown that and
(where
is a positive integer greater than
), among others, are universal. In a forthcoming paper, Adams introduces the notion of a multi-crossing projection of a knot. An
-crossing projection is a projection of a knot in which each crossing has
strands, rather than
strands as in a regular projection. We then extend the notion of universality to such knots. These results allow us to prove that
is a universal sequence for both n-crossing knot projections, for all
. Adams further proves that all knots have an
-crossing projection for all positive
. Another proof of this fact is included in this paper. This is achieved by constructing
-crossing template knots, which enable us to construct multi-crossing projections with crossings of any multiplicity.

Other Papers
Mathematics of the Fon people. With T Helke. In progress.
Topic Proposal: Cubulated Groups