## Research

My research is in low-dimensional topology and geometry, especially invariants of knots and links in three and four-manifolds. I often use Heegaard Floer homology or Khovanov homology, and many of my results involve local operations on knots like crossing changes, band surgeries and mutations. I have a secondary research interest in the applications of topology to physical systems, for example, how to model and characterize enzymatic actions on circular DNA molecules.

Here is a link to my papers on the arXiv. Authorship on all articles is alphabetical.

## Preprints

Eugene Gorsky, Beibei Liu, Tye Lidman and Allison H. Moore. Heegaard Floer homology and triple linking. arXiv:2006.15484 [math.GT], Submitted 2020.

arXiv, PDF

Tye Lidman, Allison H. Moore and Claudius Zibrowius. L-space knots have no essential Conway spheres. arXiv:2006.03521 [math.GT], 2021. To appear in *Geometry & Topology*.

arXiv, PDF

Stanislav Jabuka, Beibei Liu, and Allison H. Moore. Knot graphs and Gromov hyperbolicity. arXiv:1912.03766 [math.GT], Submitted 2020.

arXiv, PDF

Christopher Flippen, Allison H. Moore, and Essak Seddiq. Quotients of the Gordian and H(2)-Gordian graphs. arXiv:2102.05243 [math.GT], 2021. To appear in the *Journal of Knot Theory and its Ramifications*.

arXiv

## Publications

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

Eugene Gorsky, Beibei Liu and Allison H. Moore. Surgery on links of linking number zero and the Heegaard Floer d-invariant.
*Quantum Topology* 11(2):323--378, 2020.

Quantum Topology,
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

Tye Lidman, Allison H. Moore and Mariel Vazquez. Distance one lens space fillings and band surgeries. *Algebraic & Geometric Topology*; 19(5):2439--2484, 2019.

AGT,
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. Cosmetic surgery in L-spaces and nugatory crossings. *Transactions of the American Mathematical Society*, 369(5):3639--3654, 2017.

Transactions, 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 in Fall 2021 -- Summer 2022

**Project description:**

Knots and links are embeddings of closed loops in three-dimensional space. Knot and link invariants are mathematical objects that capture topological and geometric information about the link and its three-dimensional complement. This project seeks to advance our mathematical understanding of basic operations on knots and links, to explore the relationships between invariants of knots and three-manifolds, and to leverage these developments toward knot-theoretic models of DNA recombination mediated by enzymes. The central hypothesis of the proposal is that knot theory provides a rigorous mathematical framework with which to investigate molecular knotting and entanglement in DNA.

Students with a genuine interest in topology and geometry or 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.

Any VCU undergrads or grad students who are curious about this research project, 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!):

*Journal of Knot Theory and its Ramifications*).

Paper: On the Number of Equivalence Classes of p-Colorings of Symmetric Unions.

## Topological Molecular Biology

At UC Davis, I held a joint appointment in the Topological Molecular Biology lab, in the department of microbiology and molecular genetics. DNA topology refers to the supercoiling, knotting and linking of circular DNA molecules. Replication, reconnection events and other naturally occurring cellular processes cause knotting and linking which can be harmful for the health of the cell. Enzymes such as type II topoisomerases and recombinases act locally to correct these topological problems. Circular DNA can be thought of as a topological knot. Strand-passage events and DNA recombination can be modeled as crossing changes and band-surgeries on knots. Both of these operations on knots are a big focus in my work. 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" with PI M. Vazquez. The long term goal of this project is 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.