R. McConkey, C. St. Clair, T. Wells, and C. Zhang, Deeply Slice Knot Detection Via Immersed Curves, in preparation
R. McConkey, L. Seaton, Satellite Operations and Theta, Submitted (Arxiv 2509.22939)
R. McConkey, Linear Bounds of Crosscap Numbers of Knots, Algebraic and Geometric Topology, Vol 25, No 7, 4009-4035 (2025).
E. Kalfagianni and R. McConkey, Crossing Numbers of Cable Knots, Bulletin of London Math. Society, Vol 56, No 24, 3400-3411 (2024)
My recent work has been focused on two main topics. One of which is the Theta invariant discovered by Bar Natan and Van der Veen. I've been working on this project with Luke Seaton at Michigan State University. Theta is a believed to be a very powerful knot invariant that can be quickly computed by a computer. The resulting multivariable polynomial can be beautifully represented by a hexagonal QR-code. Luke and I have a paper on Arxiv (submitted) where we prove that Theta is additive over a connected sum and give a conjecture for the formula of theta for the Whitehead double of a knot. We have shown that our conjecture holds true for the first 2977 prime knots. We have been accepted to attend SLMath's PROOF program this summer (2026) to further our work and will be attending for two weeks.
The other main area for my research as of late has been with knots in thickened surfaces. Classical knot theory takes place in S^3 or R^3, but thickened surfaces take the knots to other spaces that can have more interesting topology. A big question is what can we generalize from the classical setting to the thickened surfaces setting. For me my focus is largely to do with the generalizations of the Jones polynomial in this setting. Currently I have paper in preparation with E. Buchanan.
I also have a paper in preparation with C. St. Clair (Saginaw Valley State), T. Wells (LSU), C. Zhang (Stony Brook) in the realm of bordered Floer Homology. While Floer homology is not my area of study historically, this was a fun opportunity to learn a bit more about a popular topic in the field of knot theory.
My past work has been on the relationship between fundamental link invariants and quantum link invariants. Fundamental link invariants include invariants such as crossing number, uncrossing number, knot genus, and crosscap number. Whereas quantum link invariants include examples such as the Colored Jones Polynomial. This work resulted in two papers which are both published. The links are below.
Crossing Numbers of Cable Knots, with E. Kalfagianni, Bulletin of London Math Society Vol 56, No 24 3400- 3411 (2024)
Linear Bounds of Crosscap Numbers of Knots, Algebraic and Geometric Topology Vol 25, No 27 4009-4035 (2025)
E. Kalfagianni and R. McConkey, Crossing Numbers of Cable Knots, Bulletin of
London Math. Society, Vol 56, No 24, 3400-3411 (2024)