Research Acknowledgments
My work is currently supported by the National Science Foundation, DMS--2204148, and by the The Thomas F. and Kate Miller Jeffress Memorial Trust, Bank of America, Trustee.
Here is a link to my papers on the arXiv. Authorship on all articles is alphabetical.
Featured articles
We generalize a result of Scharlemann to spatial graphs, proving that unknotting number one theta curves are prime.
We prove an equivariant version of the Cosmetic Surgery Conjecture and study cosmetic tangle fillings using an immersed curves interpretation of Khovanov homology.
We prove that L-space knots have no essential Conway spheres using an immersed curve interpretation of Heegaard Floer homology for tangles.
We establish a relationship between Milnor invariants, including the triple linking number, and Heegaard Floer homology.
We give a formula for the Heegaard Floer d-invariants of integral surgeries on two-component L–space links of linking number zero in terms of the h-function, generalizing a formula of Ni and Wu. We relate the h-function to the Casson invariant, Sato-Levine invariant, study crossing changs, and give genus bounds.
We prove that the Gordian graph and its relatives are not hyperbolic, prove the concordance knot graph is homogeneous, and show the existence of `torus knot-free balls' of arbitrary radius in the Gordian graph.
We determine when the lens space L(n, 1) is obtained from L(3, 1) by integral surgery by studying Floer d-invariants , and classify bandings relating trefoil and other T(2, n) knots and links.
We prove the nugatory crossing conjecture for alternating and QA knots satisfying a homology condition.
More articles
Tye Lidman and Allison H. Moore. Adjacency in three-manifolds and Brunnian links. arXiv:2308.06211 [math.GT], 2023. arXiv, PDF
Matthew Elpers, Rayan Ibrahim, and Allison H. Moore. Determinants of simple theta curves and graphs with involutive symmetry. arXiv:2211.00626 [math.GT], 2022. arXiv, PDF
Christopher Flippen, Allison H. Moore, and Essak Seddiq. Quotients of the Gordian and H(2)-Gordian graphs.
Journal of Knot Theory and Its Ramifications 30 (2021), no. 5, Paper No. 2150037, 23 pp.
JKTR, arXiv, PDF
Allison H. Moore and Mariel Vazquez. A note on band surgery and the signature of a knot. Bulletin of the London Mathematical Society, 52(6):1191--1208, (2020). BLMS,
arXiv, PDF
Allison H. Moore and Mariel H. Vazquez. Recent advances on the non-coherent band surgery model for site-specific recombination. In Topology and geometry of biopolymers, volume 746 of Contemporary Mathematics, pages 101–125. Amer. Math. Soc., Providence, RI, (2020).
Contemporary Math, arXiv, PDF
Kenneth L. Baker and Allison H. Moore. Montesinos knots, Hopf plumbings, and L-space surgeries. Journal of the Mathematical Society of Japan, 70(1):95--110, (2018).
J. Math Japan, arXiv, PDF
Tye Lidman and Allison H. Moore. Pretzel knots with L-space surgeries. Michigan Mathematical Journal, 65(1):105–130, (2016).
Michigan, arXiv, PDF
Allison H. Moore. Symmetric unions without cosmetic crossing changes. Advances in the Mathematical Sciences: Research from the 2015 Association for Women in Mathematics Symposium, volume 6, pages 103–116. Springer International Publishing, Cham, (2016).
Advances, arXiv, PDF
Allison H. Moore and Laura Starkston. Genus-two mutant knots with the same dimension in knot Floer and Khovanov homologies. Algebraic & Geometric Topology, 15(1):43–63, (2015).
AGT, arXiv, PDF
Thesis
Allison H. Moore. Behavior of knot Floer homology under Conway and genus two mutation. PhD Dissertation, The University of Texas at Austin, May 2013.
Dissertation repository
Software
Braid Generator: Software to generate random braid representatives of a fixed knot type via Markov chain
M. Nasrollahi, S. Witte and A. H. Moore, June 2019
Project now publicly available on Github! and at PyPi.org.
Undergraduate Researchers
Research opportunities for undergrads
I currently have several projects in knot theory (both theoretical, applied, and computational) with opportunities for undergraduate research collaborators. Students with a genuine interest in topology and geometry and some background in linear algebra or coding will be most successful. Prior knowledge of knot theory is not required, and essential skills can be learned along the way. There are some funds to pay students an hourly wage. Students from underrepresented groups are especially encouraged to apply.
Currently enrolled VCU undergrads or grad students who are curious about research, a summer project, or interested in independent study, please get in touch with me by email (moorea14 at VCU dot edu).
Past student projects:
I've had the privilege of working with many excellent undergraduate researchers. Here is a subset of them (and do please contact me if you don't see your name here and would like a shout-out!):
Determinants of simple theta curves and graphs with involutive symmetry. VCU, F21-S23
Link to arXiv.
Quotients of the Gordian graph of knots. Article at journal and on the arXiv.
Topological Molecular Biology
At UC Davis, I held a joint appointment in the department of microbiology and molecular genetics, as a postdoc with the Topological Molecular Biology lab. Part of my research at UC Davis was funded by the NSF, as part of the project "The dynamic genome: studying the interplay between local strand-passage and reconnection" under the direction of PI Mariel Vazquez. The long term goal of this project was to understand the global topological changes effected by recombinases and type II topoisomerases, with a special emphasis on understanding the effects of local reconnection and local crossing changes on the 3D organization of the genome.