Article
Mathematics, Applied
Kouki Sato
Summary: This paper investigates the v(+)-equivalence in the knot concordance group, finding that any genus one knot is v(+)-equivalent to one of the trefoil, its mirror, and the unknot.
SELECTA MATHEMATICA-NEW SERIES
(2023)
Article
Mathematics, Applied
Shida Wang
Summary: Dai, Hom, Stoffregen and Truong defined a family of concordance invariants phi(j). The previously given example of a knot with zero Upsilon invariant but nonzero epsilon invariant by Hom also has nonzero phi invariant. We prove the existence of infinitely many such knots that are linearly independent in the smooth concordance group. Conversely, we construct infinite families of linearly independent knots with zero phi invariant but nonzero Upsilon invariant. We also provide a recursive formula for the phi invariant of torus knots.
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
(2023)
Article
Mathematics
Aaron D. Lauda, Anthony M. Licata, Andrew Manion
Summary: In this study, we demonstrate that the equivariant hypertoric convolution algebras introduced by Braden-Licata-Proudfoot-Webster are affine quasi hereditary in the sense of Kleshchev and compute the Ext groups between standard modules. This implies new homological results about the bordered Floer algebras of Ozsvath-Szabo, including the existence of standard modules over these algebras, in conjunction with the main result of [27]. Furthermore, we establish that the Ext groups between standard modules are isomorphic to the homology of a variant of the Lipshitz-Ozsvath-Thurston bordered strands dg algebras.
ADVANCES IN MATHEMATICS
(2023)
Article
Mathematics
Robert Lipshitz, Sucharit Sarkar
Summary: Building upon the work of Hedden-Ni, this paper demonstrates the use of the module structure on Khovanov homology and untwisted Heegaard Floer homology in detecting split links and branched double covers. Important technical results include the interpretation of the module structure on untwisted Heegaard Floer homology in terms of twisted Heegaard Floer homology, and the fact that the module structure on the reduced Khovanov complex of a link is well defined up to quasiisomorphism.
AMERICAN JOURNAL OF MATHEMATICS
(2022)
Article
Mathematics
Ina Petkova
Summary: Bordered Heegaard Floer homology is an invariant for 3-manifolds that relates a surface to an algebra and a 3-manifold with boundary to a module over that algebra. In this paper, we establish an absolute grading for the relative grading and show that the resulting Z/2-graded module is an invariant of the bordered 3-manifold.
NEW YORK JOURNAL OF MATHEMATICS
(2022)
Article
Mathematics
Jaepil Lee
Summary: An algorithm is introduced to compute the bordered Floer bimodule of a knot and its complement, providing spin(c)-summands of HFK, extendable to arbitrary framing n. This method allows for the computation of module gradings for specific components of HFK.
REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS
(2021)
Article
Mathematics, Applied
Irving Dai, Matthew Hedden, Abhishek Mallick
Summary: Building on the algebraic framework of Hendricks, Manolescu, and Zemke, we introduce and study a set of Floer-theoretic invariants to detect corks. Our invariants obstruct the extension of an involution over any homology ball and utilize the formalism of local equivalence from involutive Heegaard Floer homology, without the use of closed 4-manifold topology or contact topology. We define a modified homology cobordism group Theta(iota)(Z) taking into account involution on each homology sphere, and prove its Z(infinity) -subgroup of strongly nonextendable corks. Using our invariants, we establish new families of corks and prove the strong nonextendability of various known examples. Our main computational tools are a monotonicity theorem and an explicit method of constructing equivariant negative-definite cobordisms via equivariant surgery.
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics
Andras Juhasz, Dylan P. Thurston, Ian Zemke
Summary: All versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural, assigning concrete groups to each based 3-manifold, based link, and balanced sutured manifold, respectively. Isomorphisms are functorially assigned to (based) diffeomorphisms, and it is shown that this assignment is isotopy invariant. By finding a simple generating set for the fundamental group of the space of Heegaard diagrams and proving that Heegaard Floer homology has no monodromy around these generators, sufficient conditions for an arbitrary invariant of multi-pointed Heegaard diagrams to descend to a natural invariant of 3-manifolds, links, or sutured manifolds are given.
MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics, Applied
Andras Juhasz, Ian Zemke
Summary: This article investigates the effect of concordance surgery on the Ozsvath-Szabo 4-manifold invariant using a self-concordance of a knot as a generalization of knot surgery. The computation involves the graded Lefschetz number of the concordance map on the knot Floer homology. The proof utilizes the sutured Floer TQFT, and a perturbed version of sutured Floer homology with a 2-form.
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics
William Chang, Andrew Manion
Summary: In this paper, we give the basic local crossing bimodules in Ozsvath-Szabo's theory of bordered knot Floer homology a structure of 1-morphisms of 2-representations, which categorifies the Uq(gl(1|1)+)-intertwining property of the corresponding maps between ordinary representations. Besides establishing a new connection between bordered knot Floer homology and higher representation theory, similar to the work of Rouquier and Manion, this structure also provides an algebraic reformulation of a compatibility property for Ozsvath and Szabo's bimodules, which is crucial when extending their theory from local crossings to more global tangles and knots.
PACIFIC JOURNAL OF MATHEMATICS
(2023)
Article
Mathematics
Jenniffer Hom, Adam Simon Levine, Tye Lidman
Summary: This paper investigates the relationship between the group of knots in homology spheres that bound homology balls and the smooth knot concordance group in homology cobordisms. Using tools from Heegaard Floer homology, the paper shows that the cokernel of the natural map from the smooth knot concordance group to the group of knots in homology balls is infinitely generated and contains elements of infinite order. In the appendix, a careful proof is provided that any piecewise-linear surface in a smooth 4-manifold can be isotoped to be smooth away from cone points.
DUKE MATHEMATICAL JOURNAL
(2022)
Article
Mathematics, Applied
Motoo Tange
Summary: This paper presents a formula for calculating the Upsilon invariant of any L-space cable knot K-p,K-q, which involves p, gamma(K), and gamma(Tp,q). The integral value of the Upsilon invariant serves as a Q-valued knot concordance invariant. We compute the integral values for L-space iterated cable knots.
TOPOLOGY AND ITS APPLICATIONS
(2023)
Article
Mathematics
Irving Dai, Jennifer Hom, Matthew Stoffregen, Linh Truong
Summary: We prove the existence of an infinite-rank subgroup in the three-dimensional homology cobordism group, using an algebraic variant of the Heegaard Floer package.
DUKE MATHEMATICAL JOURNAL
(2023)
Article
Mathematics
Kristen Hendricks, Matthew Issac, Nicholas McConnell
Summary: This study calculates the involutive knot invariants for a form of pretzel knots P(-2, m, n) where m and n are odd and greater than or equal to 3.
JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS
(2022)
Article
Mathematics
Tye Lidman, Juanita Pinzon-Caicedo, Christopher Scaduto
Summary: In this paper, we investigate L-spaces and their surgeries in Floer homology, and also explore the properties of the contact invariant under different gradings.
INDIANA UNIVERSITY MATHEMATICS JOURNAL
(2022)
Article
Mathematics
Adam Simon Levine, Daniel Ruberman, Saso Strle, Ira M. Gessel
GEOMETRY & TOPOLOGY
(2015)
Article
Mathematics
Adam Simon Levine, Daniel Ruberman, Saso Strle, Ira M. Gessel
GEOMETRY & TOPOLOGY
(2015)
Article
Mathematics
Joshua Evan Greene, Adam Simon Levine
Algebraic and Geometric Topology
(2016)
Article
Mathematics
Matthew Hedden, Adam Simon Levine
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
(2016)
Article
Mathematics
John A. Baldwin, Adam Simon Levine, Sucharit Sarkar
JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS
(2017)
Article
Mathematics
Adam Simon Levine, Daniel Ruberman
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2019)
Article
Mathematics
Daniele Celoria, Marco Golla, Adam Simon Levine
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2020)
Article
Mathematics
Adam Simon Levine, Ian Zemke
BULLETIN OF THE LONDON MATHEMATICAL SOCIETY
(2019)
Article
Mathematics
John A. Baldwin, Nathan Dowlin, Adam Simon Levine, Tye Lidman, Radmila Sazdanovic
Summary: Using Dowlin's spectral sequence from Khovanov homology to knot Floer homology, we are able to prove that reduced Khovanov homology with rational coefficients can detect the figure-eight knot.
BULLETIN OF THE LONDON MATHEMATICAL SOCIETY
(2021)
Article
Mathematics
Onkar Singh Gujral, Adam Simon Levine
Summary: In this paper, the (in)sensitivity of the Khovanov functor to 4-dimensional linking of surfaces is studied. The authors prove that the map on Khovanov homology induced by a cobordism between split links is completely determined by the maps induced by the individual components of the cobordism and does not detect the linking between the components. They also show that a strongly homotopy-ribbon concordance induces an injection on Khovanov homology, which generalizes a previous result. Additionally, it is proved that a non-split link cannot be ribbon concordant to a split link.
JOURNAL OF TOPOLOGY
(2022)
Article
Mathematics
Jenniffer Hom, Adam Simon Levine, Tye Lidman
Summary: This paper investigates the relationship between the group of knots in homology spheres that bound homology balls and the smooth knot concordance group in homology cobordisms. Using tools from Heegaard Floer homology, the paper shows that the cokernel of the natural map from the smooth knot concordance group to the group of knots in homology balls is infinitely generated and contains elements of infinite order. In the appendix, a careful proof is provided that any piecewise-linear surface in a smooth 4-manifold can be isotoped to be smooth away from cone points.
DUKE MATHEMATICAL JOURNAL
(2022)
Article
Mathematics, Applied
Adam Simon Levine, Tye Lidman
FORUM OF MATHEMATICS SIGMA
(2019)
Article
Mathematics, Applied
Adam Simon Levine
FORUM OF MATHEMATICS SIGMA
(2016)
Article
Mathematics
Adam Simon Levine, Sam Lewallen
MATHEMATICAL RESEARCH LETTERS
(2012)
