publications | Allison H. Moore

2024

Submitted
Signature, slicing foams, and crossing changes of Klein graphs

Kenneth Baker, Allison H. Moore, Danielle O’Donnol, and Scott Taylor

Submitted, 2024

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A totally oriented Klein graph is a trivalent spatial graph in the 3-sphere with a 3-coloring of its edges and an orientation on each bicolored link. A totally oriented Klein foam is a 3-colored 2-complex in the 4-ball whose boundary is a Klein foam and whose bicolored surfaces are oriented. We extend Gille-Robert’s signature for 3-Hamiltonian Klein graphs to all totally oriented Klein graphs and develop an analogy of Murasugi’s bounds relating the signature, slice genus and unknotting number of knots. In particular, we show that the signature of a totally oriented Klein graph produces a lower bound on the negative orbifold Euler characteristic of certain totally oriented Klein foams bounded by Γ. When Γ is abstractly planar, these negative Euler characteristics, in turn, produce a lower bound on a certain natural unknotting number for Γ. Mutatis mutandi, we produce lower bounds on the corresponding Gordian distance between two totally oriented Klein graphs that can be related by a sequence of crossing changes. We also give examples of theta-curves for which our lower bounds on unknotting number improve on previously known bounds.
@article{BMOT, author = {Baker, Kenneth and Moore, Allison H. and O'Donnol, Danielle and Taylor, Scott}, title = {Signature, slicing foams, and crossing changes of Klein graphs}, url = {https://arxiv.org/abs/2405.15044}, year = {2024}, journal = {Submitted}, }
Math. Annalen
Cosmetic operations and Khovanov multicurves

Artem Kotelskiy, Tye Lidman, Allison H. Moore, Liam Watson, and Claudius Zibrowius

Mathematische Annalen, 2024

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We prove an equivariant version of the Cosmetic Surgery Conjecture for strongly invertible knots. Our proof combines a recent result of Hanselman with the Khovanov multicurve invariants \widetildeKh and \widetildeBN. We apply the same techniques to reprove a result of Wang about the Cosmetic Crossing Conjecture and split links. Along the way, we show that \widetildeKh and \widetildeBN detect if a Conway tangle is split.
@article{KLMWZ, author = {Kotelskiy, Artem and Lidman, Tye and Moore, Allison H. and Watson, Liam and Zibrowius, Claudius}, title = {Cosmetic operations and {K}hovanov multicurves}, journal = {Mathematische Annalen}, volume = {389}, year = {2024}, number = {3}, pages = {2903--2930}, doi = {10.1007/s00208-023-02697-5}, url = {https://doi.org/10.1007/s00208-023-02697-5}, }
Submitted
Higher Rank Series and Root Puzzles for Plumbed 3-Manifolds

Allison Moore, and Nicola Tarasca

Submitted, 2024

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We construct plumbed three-manifold invariants in the form of Laurent series twisted by root lattices. Specifically, given a triple consisting of a weakly negative definite plumbing tree, a root lattice, and a generalized Spin^c-structure, we construct a family of Laurent series which are invariant up to Neumann moves between plumbing trees and the action of the Weyl group of the root lattice. We show that there are infinitely many such series for irreducible root lattices of rank at least 2, with each series depending on a solution to a combinatorial puzzle defined on the root lattice. Our invariants recover certain related series recently defined by Gukov-Pei-Putrov-Vafa, Gukov-Manolescu, Park, and Ri as special cases. Explicit computations are given for Brieskorn homology spheres, for which the series may be expressed as modified higher rank false theta functions.
@article{MooreTarasca2024, title = {Higher Rank Series and Root Puzzles for Plumbed 3-Manifolds}, author = {Moore, Allison and Tarasca, Nicola}, journal = {Submitted}, year = {2024}, }

2023

Accepted
Determinants of simple theta curves and symmetric graphs

Matthew Elpers, Rayan Ibrahim, and Allison H. Moore

Accepted for publication in the Journal of Knot Theory and its Ramifications, 2023

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A theta curve is a spatial embedding of the theta-graph in the three-sphere, taken up to ambient isotopy. We define the determinant of a theta curve as an integer-valued invariant arising from the first homology of its Klein cover. When a theta curve is simple, containing a constituent unknot, we prove that the determinant of the theta curve is the product of the determinants of the constituent knots. Our proofs are combinatorial, relying on Kirchhoff’s Matrix Tree Theorem and spanning tree enumeration results for symmetric, signed, planar graphs.
@article{EIM, author = {Elpers, Matthew and Ibrahim, Rayan and Moore, Allison H.}, journal = {Accepted for publication in the {Journal of Knot Theory and its Ramifications}}, url = {https://arxiv.org/abs/2211.00626}, title = {Determinants of simple theta curves and symmetric graphs}, year = {2023}, }
IMRN
Triple linking numbers and Heegaard Floer homology

Eugene Gorsky, Tye Lidman, Beibei Liu, and Allison H. Moore

International Mathematics Research Notices, 2023

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We establish some new relationships between Milnor invariants and Heegaard Floer homology. This includes a formula for the Milnor triple linking number from the link Floer complex, detection results for the Whitehead link and Borromean rings, and a structural property of the d-invariants of surgeries on certain algebraically split links.
@article{GLLM, author = {Gorsky, Eugene and Lidman, Tye and Liu, Beibei and Moore, Allison H.}, title = {Triple linking numbers and {H}eegaard {F}loer homology}, journal = {International Mathematics Research Notices}, year = {2023}, number = {6}, pages = {4501--4554}, doi = {10.1093/imrn/rnab368}, url = {https://doi.org/10.1093/imrn/rnab368}, }
Submitted
Adjacency of three-manifolds and Brunnian links

Tye Lidman, and Allison H. Moore

2023

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We introduce the notion of adjacency in three-manifolds. A three-manifold Y is n-adjacent to another three-manifold Z if there exists an n-component link in Y and surgery slopes for that link such that performing Dehn surgery along any nonempty sublink yields Z. We characterize adjacencies from three-manifolds to the three-sphere, providing an analogy to Askitas and Kalfagianni’s results on n-adjacency in knots.
@misc{LM:Adjacency, author = {Lidman, Tye and Moore, Allison H.}, howpublished = { arXiv:2308.06211 [math.GT] }, journal = {Submitted}, title = {Adjacency of three-manifolds and Brunnian links}, url = {https://arxiv.org/abs/2308.06211}, year = {2023}, }

2022

Math. Zeitschrift
Knot graphs and Gromov hyperbolicity

Stanislav Jabuka, Beibei Liu, and Allison H. Moore

Mathematische Zeitschrift, 2022

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We define a broad class of graphs that generalize the Gordian graph of knots. These knot graphs take into account unknotting operations, the concordance relation, and equivalence relations generated by knot invariants. We prove that overwhelmingly, the knot graphs are not Gromov hyperbolic, with the exception of a particular family of quotient knot graphs. We also investigate the property of homogeneity, and prove that the concordance knot graph is homogeneous. Finally, we prove that that for any n, there exists a knot K such that the ball of radius n in the Gordian graph centered at K contains no connected sum of torus knots.
@article{JLM, author = {Jabuka, Stanislav and Liu, Beibei and Moore, Allison H.}, date = {2022/01/31}, doi = {10.1007/s00209-021-02918-0}, journal = {Mathematische Zeitschrift}, title = {Knot graphs and Gromov hyperbolicity}, url = {https://doi.org/10.1007/s00209-021-02918-0}, year = {2022}, }

Geom. Topol.

L-space knots have no essential Conway spheres

Tye Lidman, Allison H. Moore, and Claudius Zibrowius

Geometry & Topology, 2022

Abs DOI arXiv Bib

@article{LMZ,
  author = {Lidman, Tye and Moore, Allison H. and Zibrowius, Claudius},
  title = {{L}-space knots have no essential {C}onway spheres},
  journal = {Geometry {\&} Topology},
  volume = {26},
  year = {2022},
  number = {5},
  pages = {2065--2102},
  issn = {1465-3060,1364-0380},
  mrclass = {57K18 (57K30 57R58)},
  mrnumber = {4520302},
  mrreviewer = {Thilo\ Kuessner},
  doi = {10.2140/gt.2022.26.2065},
  url = {https://doi.org/10.2140/gt.2022.26.2065},
}

2021

Submitted
Primality of theta-curves with proper rational tangle unknotting number one

Kenneth Baker, Dorothy Buck, Allison H. Moore, Danielle O’Donnol, and Scott Taylor

Accepted for publication in Transactions of the American Mathematical Society, 2021

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We show that if a composite theta-curve has (proper rational) unknotting number one, then it is the order 2 sum of a (proper rational) unknotting number one knot and a trivial theta-curve. We also prove similar results for 2-strand tangles and knotoids.
@article{BBMOT, author = {Baker, Kenneth and Buck, Dorothy and Moore, Allison H. and O'Donnol, Danielle and Taylor, Scott}, journal = {Accepted for publication in {Transactions of the American Mathematical Society}}, title = {Primality of theta-curves with proper rational tangle unknotting number one}, url = {https://arxiv.org/abs/2201.08213}, year = {2021}, }
JKTR
Quotients of the Gordian and H(2)-Gordian graphs

Christopher Flippen, Allison H. Moore, and Essak Seddiq

Journal of Knot Theory and its Ramifications, 2021

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The Gordian graph and H(2)-Gordian graphs of knots are abstract graphs whose vertex sets represent isotopy classes of unoriented knots, and whose edge sets record whether pairs of knots are related by crossing changes or H(2)-moves, respectively. We investigate quotients of these graphs under equivalence relations defined by several knot invariants including the determinant, the span of the Jones polynomial, and an invariant related to tricolorability. We show, in all cases considered, that the quotient graphs are Gromov hyperbolic. We then prove a collection of results about the graph isomorphism type of the quotient graphs. In particular, we find that the H(2)-Gordian graph of links modulo the relation induced by the span of the Jones polynomial is isomorphic with the complete graph on infinitely many vertices.
@article{FMS, author = {Flippen, Christopher and Moore, Allison H. and Seddiq, Essak}, title = {Quotients of the {G}ordian and {H(2)}-{G}ordian graphs}, journal = {Journal of Knot Theory and its Ramifications}, volume = {30}, year = {2021}, number = {5}, pages = {Paper No. 2150037, 23}, doi = {10.1142/S0218216521500371}, url = {https://doi.org/10.1142/S0218216521500371}, }

2020

Quantum Topology
Surgery on links of linking number zero and the Heegaard Floer d-invariant

Eugene Gorsky, Beibei Liu, and Allison H. Moore

Quantum Topology, 2020

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We study Heegaard Floer homology and various related invariants (such as the h-function) for two-component L-space links with linking number zero. For such links, we explicitly describe the relationship between the h-function, the Sato-Levine invariant and the Casson invariant. 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. As a consequence, for such links with unknotted components, we characterize L-space surgery slopes in terms of the nu-+invariants of the knots obtained from blowing down the components. We give a proof of a skein inequality for the d-invariants of +1 surgeries along linking number zero links that differ by a crossing change. We also describe bounds on the smooth four-genus of links in terms of the h-function, expanding on previous work of the second author, and use these bounds to calculate the four-genus in several examples of links.
@article{GLM, author = {Gorsky, Eugene and Liu, Beibei and Moore, Allison H.}, doi = {10.4171/QT/137}, journal = {Quantum Topology}, issn = {1663-487X}, number = {2}, pages = {323--378}, title = {Surgery on links of linking number zero and the {H}eegaard {F}loer d-invariant}, url = {https://doi-org.proxy.library.vcu.edu/10.4171/QT/137}, volume = {11}, year = {2020}, }
Contemp. Math.
Recent advances on the non-coherent band surgery model for site-specific recombination

Allison H. Moore, and Mariel H. Vazquez

Contemporary Mathematics: Topology and geometry of biopolymers, 2020

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Site-specific recombination is an enzymatic process where two sites of precise sequence and orientation along a circle come together, are cleaved, and the ends are recombined. Site- specific recombination on a knotted substrate produces another knot or a two-component link depending on the relative orientation of the sites prior to recombination. Mathematically, site- specific recombination is modeled as coherent (knot to link) or non-coherent (knot to knot) banding. We here survey recent developments in the study of non-coherent bandings on knots and discuss biological implications.
@article{MV:survey, author = {Moore, Allison H. and Vazquez, Mariel H.}, journal = {Contemporary Mathematics: Topology and geometry of biopolymers}, doi = {10.1090/conm/746/15004}, pages = {101--125}, publisher = {Amer. Math. Soc., Providence, RI}, title = {Recent advances on the non-coherent band surgery model for site-specific recombination}, url = {https://doi-org.proxy.library.vcu.edu/10.1090/conm/746/15004}, volume = {746}, year = {2020}, }
BLMS
A note on band surgery and the signature of a knot

Allison H. Moore, and Mariel Vazquez

Bulletin of the London Mathematical Society, 2020

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Band surgery is an operation which transforms a knot or link in the three-sphere into another knot or link. We prove that if two quasi-alternating knots K and K′ of the same square-free determinant are related by a band surgery, then the absolute value of the difference in their signatures is either 0 or 8. This obstruction follows from a more general theorem about the difference in the Heegaard Floer d-invariants for two L-spaces of the same square-free first homology order, and that are related by distance one Dehn fillings. These results imply that T(2,5) is the only torus knot T(2,m) with m square-free that admits a chirally cosmetic banding, i.e. a band surgery operation to its mirror image. Band surgery on knots has important applications to the study of reconnection events in polymer chains, particularly the local action of recombination enzymes on circular DNA molecules. In this note we report on preliminary numerical data based on Monte Carlo simulations of non-coherent banding on knots in the simple cubic lattice. For lattice knots of crossing number at most eight, we observe chirally cosmetic bandings for only three knots: 5_1, 8_8 and 8_20. The relative likelihood of these events in the numerical experiments is negligible, thus suggesting that chirality cosmetic bandings are rare.
@article{MV:band, author = {Moore, Allison H. and Vazquez, Mariel}, title = {A note on band surgery and the signature of a knot}, journal = {Bulletin of the London Mathematical Society}, volume = {52}, year = {2020}, number = {6}, pages = {1191--1208}, doi = {10.1112/blms.12397}, url = {https://doi.org/10.1112/blms.12397}, }

2019

Software

Software: BraidGenerator

Allison H. Moore, Milad Nasrollahi, and Shawn Witte

Open Source Software, 2019

Website
AGT
Distance one lens space fillings and band surgery

Tye Lidman, Allison H. Moore, and Mariel Vazquez

Algebraic & Geometric Topology, 2019

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We prove that if the lens space L(n,1) is obtained by a surgery along a knot in the lens space L(3,1) that is distance one from the meridional slope, then n is in −6,±1,±2,3,4,7. This result yields a classification of the coherent and non-coherent band surgeries from the trefoil to T(2,n) torus knots and links. The main result is proved by studying the behavior of the Heegaard Floer d-invariants under integral surgery along knots in L(3,1). The classification of band surgeries between the trefoil and torus knots and links is motivated by local reconnection processes in nature, which are modeled as band surgeries. Of particular interest is the study of recombination on circular DNA molecules.
@article{LMV, author = {Lidman, Tye and Moore, Allison H. and Vazquez, Mariel}, doi = {DOI 10.2140/agt.2019.19.2439}, journal = {Algebraic {\&} Geometric Topology}, number = {5}, pages = {2439--2484}, title = {Distance one lens space fillings and band surgery}, url = {https://arxiv.org/abs/1710.07418}, volume = {19}, year = {2019}, }

2018

J. Math. Soc. Japan
Montesinos knots, Hopf plumbings, and L-space surgeries

Kenneth L. Baker, and Allison H. Moore

Journal of the Mathematical Society of Japan, 2018

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Using Hirasawa-Murasugi’s classification of fibered Montesinos knots we classify the L-space Montesinos knots, providing further evidence towards a conjecture of Lidman-Moore that L-space knots have no essential Conway spheres. In the process, we classify the fibered Montesinos knots whose open books support the tight contact structure on S^3. We also construct L-space knots with arbitrarily large tunnel number and discuss the question of whether L-space knots admit essential tangle decompositions in the context of satellite operations and tunnel number.
@article{BM, author = {Baker, Kenneth L. and Moore, Allison H.}, doi = {10.2969/jmsj/07017484}, journal = {Journal of the Mathematical Society of Japan}, number = {1}, pages = {95--110}, title = {Montesinos knots, {H}opf plumbings, and {L}-space surgeries}, url = {https://doi.org/10.2969/jmsj/07017484}, volume = {70}, year = {2018}, }

2017

Transactions
Cosmetic surgery in L-spaces and nugatory crossings

Tye Lidman, and Allison H. Moore

Transactions of the American Mathematical Society, 2017

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The cosmetic crossing conjecture (also known as the "nugatory crossing conjecture") asserts that the only crossing changes that preserve the oriented isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery characterization of the unknot to prove this conjecture for knots in integer homology spheres whose branched double covers are L-spaces satisfying a homological condition. This includes as a special case all alternating and quasi-alternating knots with square-free determinant. As an application, we prove the cosmetic crossing conjecture holds for all knots with at most nine crossings and provide new examples of knots, including pretzel knots, non-arborescent knots and symmetric unions for which the conjecture holds.
@article{LM:Cosmetic, author = {Lidman, Tye and Moore, Allison H.}, doi = {10.1090/tran/6839}, journal = {Transactions of the American Mathematical Society}, number = {5}, pages = {3639--3654}, title = {Cosmetic surgery in {L}-spaces and nugatory crossings}, url = {http://dx.doi.org/10.1090/tran/6839}, volume = {369}, year = {2017}, }

2016

MMJ
Pretzel knots with L-space surgeries

Tye Lidman, and Allison H. Moore

Michigan Mathematical Journal, 2016

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A rational homology sphere whose Heegaard Floer homology is the same as that of a lens space is called an L-space. We classify pretzel knots with any number of tangles which admit L-space surgeries. This rests on Gabai’s classification of fibered pretzel links.
@article{LM:Pretzels, author = {Lidman, Tye and Moore, Allison H.}, journal = {Michigan Mathematical Journal}, number = {1}, pages = {105--130}, title = {Pretzel knots with {L}-space surgeries}, url = {http://projecteuclid.org/euclid.mmj/1457101813}, volume = {65}, year = {2016}, }
Advances Math. Sci.
Symmetric unions without cosmetic crossing changes

Allison H. Moore

Advances in the mathematical sciences, 2016

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A symmetric union of two knots is a classical construction in knot theory which generalizes connected sum, introduced by Kinoshita and Terasaka in the 1950s. We study this construction for the purpose of finding an infinite family of hyperbolic non- fibered three-bridge knots of constant determinant which satisfy the well-known cosmetic crossing conjecture. This conjecture asserts that the only crossing changes which preserve the isotopy type of a knot are nugatory.
@article{Moore, author = {Moore, Allison H.}, title = {Symmetric unions without cosmetic crossing changes}, journal = {Advances in the mathematical sciences}, series = {Assoc. Women Math. Ser.}, volume = {6}, pages = {103--116}, publisher = {Springer, [Cham]}, year = {2016}, doi = {10.1007/978-3-319-34139-2\_3}, url = {https://doi.org/10.1007/978-3-319-34139-2_3}, }

2015

AGT

Genus-two mutant knots with the same dimension in knot Floer and Khovanov homologies

Allison H. Moore, and Laura Starkston

Algebraic & Geometric Topology, 2015

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We exhibit an infinite family of knots with isomorphic knot Heegaard Floer homology. Each knot in this infinite family admits a nontrivial genus two mutant which shares the same total dimension in both knot Floer homology and Khovanov homology. Each knot is distinguished from its genus two mutant by both knot Floer homology and Khovanov homology as bigraded groups. Additionally, for both knot Heegaard Floer homology and Khovanov homology, the genus two mutation interchanges the groups in delta-gradings k and -k.

2013

Thesis

Thesis: Behavior of Knot Floer Homology Under Conway and Genus Two Mutation

Allison H. Moore

Department of Mathematics, University of Texas at Austin, 2013

Bib

@article{Moore:thesis,
  author = {Moore, Allison H.},
  journal = {Department of Mathematics, University of Texas at Austin},
  title = {Thesis: Behavior of Knot {F}loer Homology Under {C}onway and Genus Two Mutation},
  year = {2013},
}