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The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of –valued functions, the result was later cast in a more general algebraic form, in the language of persistence modules and interleavings. We establish an analogue of this algebraic stability theorem for zigzag persistence modules. To do so, we functorially extend each zigzag persistence module to a two-dimensional persistence module, and establish an algebraic stability theorem for these extensions. One part of our argument yields a stability result for free two-dimensional persistence modules. As an application of our main theorem, we strengthen a result of Bauer et al on the stability of the persistent homology of Reeb graphs. Our main result also yields an alternative proof of the stability theorem for level set persistent homology of Carlsson et al.
To any semigroup presentation and base word may be associated a nonpositively curved cube complex , called a Squier complex, whose underlying graph consists of the words of equal to modulo , where two such words are linked by an edge when one can be transformed into the other by applying a relation of . A group is a diagram group if it is the fundamental group of a Squier complex. We describe hyperplanes in these cube complexes. As a first application, we determine exactly when is a special cube complex, as defined by Haglund and Wise, so that the associated diagram group embeds into a right-angled Artin group. A particular feature of Squier complexes is that the intersections of hyperplanes are “ordered” by a relation . As a strong consequence on the geometry of , we deduce, in finite dimensions, that its universal cover isometrically embeds into a product of finitely many trees with respect to the combinatorial metrics; in particular, we notice that (often) this allows us to embed quasi-isometrically the associated diagram group into a product of finitely many trees, giving information on its asymptotic dimension and its uniform Hilbert space compression. Finally, we exhibit a class of hyperplanes inducing a decomposition of as a graph of spaces, and a fortiori a decomposition of the associated diagram group as a graph of groups, giving a new method to compute presentations of diagram groups. As an application, we associate a semigroup presentation to any finite interval graph , and we prove that the diagram group associated to (for a given base word) is isomorphic to the right-angled Artin group . This result has many consequences on the study of subgroups of diagram groups. In particular, we deduce that, for all , the right-angled Artin group embeds into a diagram group, answering a question of Guba and Sapir.
We show that if a discrete group acts properly and cocompactly on an –dimensional, thick, Euclidean building, then cannot act properly on a contractible –manifold. As an application, if is a torsion-free –arithmetic group over a number field, we compute the minimal dimension of contractible manifold that admits a proper –action. This partially answers a question of Bestvina, Kapovich, and Kleiner.
We study stable commutator length (scl) in free products via surface maps into a wedge of spaces. We prove that scl is piecewise rational linear if it vanishes on each factor of the free product, generalizing a theorem of Danny Calegari. We further prove that the property of isometric embedding with respect to scl is preserved under taking free products. The method of proof gives a way to compute scl in free products which lets us generalize and derive in a new way several well-known formulas. Finally we show independently and in a new approach that scl in free products of cyclic groups behaves in a piecewise quasirational way when the word is fixed but the orders of factors vary, previously proved by Timothy Susse, settling a conjecture of Alden Walker.
We extend the Heegaard Floer homological definition of spectral order for closed contact –manifolds due to Kutluhan, Matić, Van Horn-Morris, and Wand to contact –manifolds with convex boundary. We show that the order of a codimension-zero contact submanifold bounds the order of the ambient manifold from above. As the neighborhood of an overtwisted disk has order zero, we obtain that overtwisted contact structures have order zero. We also prove that the order of a small perturbation of a Giroux –torsion domain has order at most two, hence any contact structure with positive Giroux torsion has order at most two (and, in particular, a vanishing contact invariant).
We prove that if is a nontrivial alternating link embedded (without crossings) in a closed surface , then has a compressing disk whose boundary intersects in no more than two points. Moreover, whenever the surface is incompressible and –incompressible in the link exterior, it can be isotoped to have a standard tube at some crossing of any reduced alternating diagram.
The Drinfeld double of a finite-dimensional Hopf algebra is a quasitriangular Hopf algebra with the canonical element as the universal –matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal –matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang–Baxter equation of the universal –matrix. On the other hand, the Heisenberg double of a finite-dimensional Hopf algebra has the canonical element (the –tensor) satisfying the pentagon relation. In this paper we reconstruct the universal quantum invariant using the Heisenberg double, and extend it to an invariant of equivalence classes of colored ideal triangulations of –manifolds up to colored moves. In this construction, a copy of the –tensor is attached to each tetrahedron, and invariance under the colored Pachnermoves is shown by the pentagon relation of the –tensor.
Let be a monomial ring whose minimal free resolution is rooted. We describe an –algebra structure on . Using this structure, we show that is Golod if and only if the product on vanishes. Furthermore, we give a necessary and sufficient combinatorial condition for to be Golod.
We give a formula for the duality structure of the –manifold obtained by doing zero-framed surgery along a knot in the –sphere, starting from a diagram of the knot. We then use this to give a combinatorial algorithm for computing the twisted Blanchfield pairing of such –manifolds. With the twisting defined by Casson–Gordon-style representations, we use our computation of the twisted Blanchfield pairing to show that some subtle satellites of genus two ribbon knots yield nonslice knots. The construction is subtle in the sense that, once based, the infection curve lies in the second derived subgroup of the knot group.
We build a bridge between geometric group theory and topological dynamical systems by establishing a dictionary between coarse equivalence and continuous orbit equivalence. As an application, we show that group homology and cohomology in a class of coefficients, including all induced and coinduced modules, are coarse invariants. We deduce that being of type (over arbitrary rings) is a coarse invariant, and that being a (Poincaré) duality group over a ring is a coarse invariant among all groups which have finite cohomological dimension over that ring. Our results also imply that every coarse self-embedding of a Poincaré duality group must be a coarse equivalence. These results were only known under suitable finiteness assumptions, and our work shows that they hold in full generality.
We use positive –equivariant symplectic homology to define a sequence of symplectic capacities for star-shaped domains in . These capacities are conjecturally equal to the Ekeland–Hofer capacities, but they satisfy axioms which allow them to be computed in many more examples. In particular, we give combinatorial formulas for the capacities of any “convex toric domain” or “concave toric domain”. As an application, we determine optimal symplectic embeddings of a cube into any convex or concave toric domain. We also extend the capacities to functions of Liouville domains which are almost but not quite symplectic capacities.
We prove that the homotopy prederivator of a cofibration category is equivalent to the homotopy prederivator of its associated quasicategory of frames, as introduced by Szumiło. We use this comparison result to deduce various abstract properties of the obtained prederivators.
The representation of knots by petal diagrams (Adams et al 2012) naturally defines a sequence of distributions on the set of knots. We establish some basic properties of this randomized knot model. We prove that in the random –petal model the probability of obtaining every specific knot type decays to zero as , the number of petals, grows. In addition we improve the bounds relating the crossing number and the petal number of a knot. This implies that the –petal model represents at least exponentially many distinct knots.
Past approaches to showing, in some random models, that individual knot types occur with vanishing probability rely on the prevalence of localized connect summands as the complexity of the knot increases. However, this phenomenon is not clear in other models, including petal diagrams, random grid diagrams and uniform random polygons. Thus we provide a new approach to investigate this question.
We prove that the knot Floer homology of a fibered knot is nontrivial in its next-to-top Alexander grading. Immediate applications include new proofs of Krcatovich’s result that knots with –space surgeries are prime and Hedden and Watson’s result that the rank of knot Floer homology detects the trefoil among knots in the –sphere. We also generalize the latter result, proving a similar theorem for nullhomologous knots in any –manifold. We note that our method of proof inspired Baldwin and Sivek’s recent proof that Khovanov homology detects the trefoil. As part of this work, we also introduce a numerical refinement of the Ozsváth–Szabó contact invariant. This refinement was the inspiration for Hubbard and Saltz’s annular refinement of Plamenevskaya’s transverse link invariant in Khovanov homology.
A closed braid naturally gives rise to a transverse link in the standard contact –space. We study the effect of the dynamical properties of the monodromy of , such as right-veering, on the contact-topological properties of and the values of transverse invariants in Heegaard Floer and Khovanov homologies. Using grid diagrams and the structure of Dehornoy’s braid ordering, we show that is nonzero whenever has fractional Dehn twist coefficient . (For a –braid, we get a sharp result: if and only if the braid is right-veering.)
We define some signature invariants for a class of knotted trivalent graphs using branched covers. We relate them to classical signatures of knots and links. Finally, we explain how to compute these invariants through the example of Kinoshita’s knotted theta graph.
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