# Allison H. Moore's research while affiliated with Virginia Commonwealth University and other places

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## Publications (23)

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...

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.

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.

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 $\widetilde{\operatorname{Kh}}$ and $\widetilde{\operatorname{BN}}$. We apply the same techniques to reprove a result of Wang about the Cosmetic Crossing Conjecture...

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...

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...

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 K0 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 gene...

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.

We prove that L-space knots do not have essential Conway spheres with the technology of peculiar modules, a Floer theoretic invariant for tangles.

We define three types of knot graphs, constructed with the help of unknotting operations, the concordance relation and knot invariants. Some of these graphs have been previously studied in the literature, others are defined here for the first time. The main question we pose and answer in a host of cases is whether the knot graphs are Gromov hyperbo...

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, we characterize L-space surgery slopes for such links in terms of the $\tau$-invariant when the components are unknotted. For genera...

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. Ma...

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...

We study lens spaces that are related by distance one Dehn fillings. More precisely, 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 $\{-6, \pm 1, \pm 2, 3, 4, 7\}$. This is proved by studying the behavior of the Heegaard Floer...

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 con...

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 con...

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...

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...

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.

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...

## Citations

... Given a property that L-space knots exhibit, it is natural to ask if almost L-space knots too exhibit that property. For example, L-space knots do not have essential Conway spheres by a result of Lidman-Moore-Zibrowius [LMZ20], so it is natural to ask the following; Question 1.8. Do almost L-space knots have essential Conway spheres? ...

... Many authors have researched various properties of band surgery and related constructions over the past decade. See [40,42,43,45] for more details on the relationship between band surgery and polynomial invariants, [13,39,41,46,77] for the H(2)-Gordian distance, [1,2,6,7,44,62] for calculation of the H(2)-Gordian distance to the unknot, [59,63] for the relationship between band surgery and lens spaces, [14] for fibered links band surgery, [33] for a description of cosmetic surgery, [35,79,80] for details on graphs and complexes associated with band surgery, and [34,64] for applications in biology. ...

Reference: Lernaean knots and band surgery

... Then the generator of HF − (S 3 d 1 ,d 2 (L), s) is H * (A 11 s ) for some s ∈ H(L). For details, see Section 3 and [3]. This will give restrictions to the differentials in the surgery complex, which is related to the H-function. ...

... In turn, this provided strong evidence that the simplification action of specific DNA enzymes is driven by a geometric selection of sites [3]. One further potential use of grid diagrams is to help with the search of band changes and the determination of Gordian distance between knot types [17,25,5,6]. ...

... Many authors have researched various properties of band surgery and related constructions over the past decade. See [40,42,43,45] for more details on the relationship between band surgery and polynomial invariants, [13,39,41,46,77] for the H(2)-Gordian distance, [1,2,6,7,44,62] for calculation of the H(2)-Gordian distance to the unknot, [59,63] for the relationship between band surgery and lens spaces, [14] for fibered links band surgery, [33] for a description of cosmetic surgery, [35,79,80] for details on graphs and complexes associated with band surgery, and [34,64] for applications in biology. ...

Reference: Lernaean knots and band surgery

... The conjecture is usually viewed as a claim about knots and their crossing changes. See for example [Tor99,Kal12,BFKP12,BK16,Moo16,LM17]. Alternatively, the conjecture can viewed as a claim about two-component links and their band surgeries. ...

... The crossing is said to be nugatory if ∂D bounds a disk in S 3 − L, and a crossing change is cosmetic if it preserves the isotopy type of L. Conjecture 1.1 has been affirmed for two-bridge knots [18] and fibered knots [11], and significant partial results exist for genus one knots and satellite knots [1,2,9,10]. Further, Lidman and Moore have verified the conjecture for all knots L ⊂ S 3 such that the branched double-cover Σ(L) is an L-space and L has square-free determinant [13]; their work has been extended by Ito [8]. ...

... • The only L-space Montesinos knots are the P (−2, 3, q), q ≥ 1 pretzel knots [LM16,BM18]. ...

... • The only L-space Montesinos knots are the P (−2, 3, q), q ≥ 1 pretzel knots [LM16,BM18]. ...

... In general, knot invariants tend not to distinguish a knot from its Conway mutant [5]; a very short sampling of related references include [1,6,16,30,32]. References for the application of Heegaard Floer methods to mutation (but not in the setting of concordance or string reversal) include [19,20,26,28]. Recent work that touches upon Conway mutation and concordance includes [21,25,29]. ...