Journal: Algebraic & Geometric Topology

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Mathematical Sciences Publishers

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ISSN

1472-2747
1472-2739

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2016 - 201932020 - 20258
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Publications 1 - 10 of 11
  • The upsilon invariant at 1 of 3-braid knots
    Item type: Journal Article
    Truöl, Paula (2023)
    Algebraic & Geometric Topology
    We provide explicit formulas for the integer-valued smooth concordance invariant v(K) = TK (1) for every 3-braid knot K. We determine this invariant, which was defined by Ozsvath, Stipsicz and Szabo (2017), by constructing cobordisms between 3-braid knots and (connected sums of) torus knots. As an application, we show that for positive 3-braid knots K several alternating distances all equal the sum g(K) + v(K), where g(K) denotes the 3-genus of K. In particular, we compute the alternation number, the dealternating number and the Turaev genus for all positive 3-braid knots. We also provide upper and lower bounds on the alternation number and dealternating number of every 3-braid knot which differ by 1.
  • A model for configuration spaces of points
    Item type: Journal Article
    Campos, Ricardo; Willwacher, Thomas (2023)
    Algebraic & Geometric Topology
    We construct a real combinatorial model for the configuration spaces of points of compact smooth oriented manifolds without boundary. We use these models to show that the real homotopy type of configuration spaces of a simply connected such manifold only depends on the real homotopy type of the manifold. Moreover, we show that for framed D-dimensional manifolds these models capture a natural right homotopy action of the little D-disks operad.
  • An embedding of the Morse boundary in the Martin boundary
    Item type: Journal Article
    Cordes, Matthew; Dussaule, Matthew; Gekhtman, Ilya (2022)
    Algebraic & Geometric Topology
    We construct a one-to-one continuous map from the Morse boundary of a hierarchically hyperbolic group to its Martin boundary. This construction is based on deviation inequalities generalizing Ancona’s work on hyperbolic groups. This provides a possibly new metrizable topology on the Morse boundary of such groups. We also prove that the Morse boundary has measure 0 with respect to the harmonic measure unless the group is hyperbolic.
  • Hamiltonian classification of toric fibres and symmetric probes
    Item type: Journal Article
    Brendel, Joé (2025)
    Algebraic & Geometric Topology
    In a toric symplectic manifold, regular fibres of the moment map are Lagrangian tori which are called toric fibres. We discuss the question of which two toric fibres are equivalent up to a Hamiltonian diffeomorphism of the ambient space. On the construction side of this question, we introduce a new method of constructing equivalences of toric fibres by using a symmetric version of McDuff's probes. On the other hand, we derive some obstructions to such equivalence by using Chekanov's classification of product tori together with a lifting trick from toric geometry. Furthermore, we conjecture that (iterated) symmetric probes yield all possible equivalences and prove this conjecture for Cn, C P 2, C x S2, C2 x T*S1, T*S1 x S2 and monotone S2 x S2. This problem is intimately related to determining the Hamiltonian monodromy group of toric fibres, ie determining which automorphisms of the homology of the toric fibre can be realized by a Hamiltonian diffeomorphism mapping the toric fibre in question to itself. For the above list of examples, we determine the Hamiltonian monodromy group for all toric fibres.
  • The topological slice genus of satellite knots
    Item type: Journal Article
    Feller, Peter; Miller, Allison N.; Pinzón-Caicedo, Juanita (2022)
    Algebraic & Geometric Topology
    We present evidence supporting the conjecture that, in the topological category, the slice genus of a satellite knot P(K) is bounded above by the sum of the slice genera of K and P(U). Our main result establishes this conjecture for a variant of the topological slice genus, the Z–slice genus. Notably, the conjectured upper bound does not involve the algebraic winding number of the pattern P. This stands in stark contrast with the smooth category, where, for example, there are many genus 1 knots whose (n,1)–cables have arbitrarily large smooth 4–genera. As an application, we show that the (n,1)–cable of any knot of 3–genus 1 (eg the figure-eight or trefoil knot) has topological slice genus at most 1, regardless of the value of n ∈ N. Further, we show that the lower bounds on the slice genus coming from the Tristram–Levine and Casson–Gordon signatures cannot be used to disprove the conjecture.
  • Thickness, relative hyperbolicity, and randomness in coxeter groups
    Item type: Journal Article
    Behrstock, Jason; Hagen, Mark F.; Sisto, Alessandro; et al. (2017)
    Algebraic & Geometric Topology
  • Swiss-cheese action on the totalization of action-operads
    Item type: Journal Article
    Ducoulombier, Julien (2016)
    Algebraic & Geometric Topology
  • Braided Thompson groups with and without quasimorphisms
    Item type: Journal Article
    Fournier-Facio, Francesco; Lodha, Yash; Zaremsky, Matthew C.B. (2024)
    Algebraic & Geometric Topology
    We study quasimorphisms and bounded cohomology of a variety of braided versions of Thompson groups. Our first main result is that the Brin--Dehornoy braided Thompson group $bV$ has an infinite-dimensional space of quasimorphisms and thus infinite-dimensional second bounded cohomology. This implies that despite being perfect, $bV$ is not uniformly perfect, in contrast to Thompson's group $V$. We also prove that relatives of $bV$ like the ribbon braided Thompson group $rV$ and the pure braided Thompson group $bF$ similarly have an infinite-dimensional space of quasimorphisms. Our second main result is that, in stark contrast, the close relative of $bV$ denoted $\hat{bV}$, which was introduced concurrently by Brin, has trivial second bounded cohomology. This makes $\hat{bV}$ the first example of a left-orderable group of type $\operatorname{F}_\infty$ that is not locally indicable and has trivial second bounded cohomology. This also makes $\hat{bV}$ an interesting example of a subgroup of the mapping class group of the plane minus a Cantor set that is non-amenable but has trivial second bounded cohomology, behaviour that cannot happen for finite-type mapping class groups.
  • Twisted signatures of fibered knots
    Item type: Journal Article
    Conway, Anthony; Nagel, Matthias (2021)
    Algebraic & Geometric Topology
    This paper concerns twisted signature invariants of knots and 3-manifolds. In the fibered case, we reduce the computation of these invariants to the study of the intersection form and monodromy on the twisted homology of the fiber surface. Along the way, we use rings of power series to obtain new interpretations of the twisted Milnor pairing introduced by Kirk and Livingston. This allows us to relate these pairings to twisted Blanchfield pairings. Finally, we study the resulting signature invariants, all of which are twisted generalizations of the Levine-Tristram signature.
  • On the functoriality of sl₂ tangle homology
    Item type: Journal Article
    Beliakova, Anna; Hogancamp, Matthew; Putyra, Krzysztof K.; et al. (2023)
    Algebraic & Geometric Topology
    We construct an explicit equivalence between the (bi)category of gl₂ webs and foams and the Bar-Natan (bi)category of Temperley–Lieb diagrams and cobordisms. With this equivalence we can fix functoriality of every link homology theory that factors through the Bar-Natan category. To achieve this, we define web versions of arc algebras and their quasihereditary covers, which provide strictly functorial tangle homologies. Furthermore, we construct explicit isomorphisms between these algebras and the original ones based on Temperley–Lieb cup diagrams. The immediate application is a strictly functorial version of the Beliakova–Putyra–Wehrli quantization of the annular link homology.
Publications 1 - 10 of 11