Papers

We compute the group of based homotopy classes of self-homotopy equivalences of the non-orientable $4$-manifold $\RP^2\times S^2$. It is an elementary abelian $2$-group of rank $5$. The subgroup of classes whose based lift to the universal cover $S^2\times S^2$ has degree $+1$ has rank $4$ and is isomorphic to the group of free homotopy classes. The group maps onto the group of self-equivalences of the algebraic $2$-type, which has order $8$, with kernel $\pi_4(\RP^2\times S^2)\cong(\Z/2)^2$ generated by pinch-fold maps; that these maps are independent is shown by an adjunction argument. We then construct a non-orientable analogue of the Hambleton–Kreck braid of exact sequences in dimension $4$, for non-orientable manifolds with spin universal cover, stated for finite fundamental group. We apply the braid to $\RP^2\times S^2$ and to $\RP^4$, and for the Hambleton–Hillman manifold with fundamental group $(\Z/2)^2$ we determine the normal $1$-type and show that self-equivalences act on $\pi_2$ through diagonal sign matrices. For $\RP^2\times S^2$ the pinch-fold maps are detected by twisted spin bordism, and, assuming facts from surgery theory stated as hypotheses, exactly one of the three non-trivial pinch-fold maps is homotopic to a homeomorphism.

For p, q >= 3 with gcd(p,q) = 1 we show that the symmetry group of the Turk's head knot Th(p,q), the closure of the braid (sigma_1 sigma_2^{-1} sigma_3 sigma_4^{-1} ... sigma_{p-1}^{+-1})^q, is the dihedral group of order 4q. This proves Conjecture R of Di Prisa and Şavk for q >= 3; the case p = 3 is due to Sakuma and Weeks. The argument is diagrammatic: by the flyping theorem of Menasco and Thistlethwaite, the symmetries of a knot with a prime, reduced alternating diagram without non-trivial flypes are the symmetries of the diagram (we write out a proof for the diagrams at hand), and we show that the standard diagram of Th(p,q) has no non-trivial flypes and exactly 4q symmetries. The chirality of Th(p,q) for even p, which also follows from a theorem of Murasugi, is visible in the group. We compare the argument with the geometric one of Sakuma and Weeks for p = 3.

We determine the Maslov-graded next-to-top term of knot Floer homology for fibered positive links. Its rank in Maslov grading (-1) is the number of nontrivial prime connected-sum factors plus the number of components minus one, and the remaining summand is supported in Maslov grading zero. For a knot, the dimension of this remaining summand is determined by its Alexander polynomial. The proof combines a preferred-crossing construction with reductions of generalized Seifert surfaces to planar pretzel surfaces and a graded skein induction. We also obtain bounds on the fixed points of monodromies for fibered positive knots.

Grid diagrams encode knots and links combinatorially through an (n,n) winding matrix W(G). A fundamental invariant of the grid diagram is its genus g(G), which records essential topological information about the underlying knot. We give an algebraic algorithm that computes g(G) directly from W(G). The method repurposes the Hungarian algorithm ,after a column normalization that makes the matrix non-negative, it applies row-column reductions and minimum-vertex-cover corrections until the zero entries admit a perfect matching, equivalently until the minimum marking number equals n. By Konig's theorem the resulting permutation defines a distinguished grid state, and g(G) is then read from this matrix via the classical grid homology formula (Ozsvath-Stipsicz-Szabo). The novelty of this paper is the algorithm and its strongly polynomial analysis; the underlying invariant is known grid-homology theory. We illustrate the method on the trefoil and additional examples, and discuss extensions to (m,n) matrices and grid-move navigation toward the knot genus g(K).

In \cite{ThurstonStretch} Thurston introduced stretch paths and stretch maps between hyperbolic surfaces. In this paper we extend similar constructions to the space of convex real projective structures on surfaces. Using Fock-Goncharov-like parameters, we introduce stretch paths and construct explicit stretch maps between the projective surface. We show that stretch maps are $e^t$–Lipschitz with respect to a triangular metric we introduce on the projective surface. This allows us to gain control on several length functions on an open set of the space of convex real projective structures (those where there is no “local back–tracking"). As an application we obtain pointwise upper bounds for the Hilbert entropy of some convex real projective structures in terms of Fock-Goncharov-like parameters.

In this article, we demonstrate that there exists an infinite family of lens spaces L(p,q) that cannot be obtained from the 3-sphere by two integral surgeries. The first such lens space in our family is L(177,29).

The knot quandle, introduced independently by Joyce and Matveev, admits a geometric description in terms of nooses. Motivated by this description, we introduce the loop fundamental quandle $LQ(X,A,\ast)$ of a pair $(X,A)$ with a fixed base point $\ast\in X$. Its elements are represented by pairs consisting of a loop in $A$ and a path joining the base point of the loop to the fixed base point $\ast$. More generally, for a loop condition $P$ on $A$, invariant under free homotopy, we define the loop fundamental quandle $LQ(X,A,\ast;P)$. We establish functoriality and homotopy invariance of the loop fundamental quandle and show that it admits a natural augmented quandle structure over $\pi_1(X,\ast)$. For a knot $K$ in $S^3$, applying the construction to its exterior gives the knot loop quandle $LQ(K,\ast;P)$. Appropriate choices of the loop condition recover the knot quandle, its $n$-th power quandles, and the full knot quandle. The construction is inspired by Joyce's noose-theoretic approach, but allows loop data that need not arise as the boundary of a disk satisfying the noose condition.

In this paper, we focus on the examples constructed by Kanenobu and provide new family knots with the same HOMFLYPT polynomial invariants, which are distinguished by the Kauffman polynomial.

We prove that round circles $C_L$ are the unique minimisers of integral Menger curvature among embedded closed $C^1$ curves $\gamma$ of prescribed length $L$ for every $p\geq 2$, i.e. $M_p(C_L) \leq M_p(\gamma)$, with equality if and only if $\gamma$ parametrises a round circle. The proof combines a three-point symmetrisation, an analysis of diagonal truncations, and the periodic Wirtinger inequality. For $p>3$, known self-avoidance and regularity results extend the minimisation result to closed rectifiable curves without assuming embeddedness or smoothness, provided the length of the curve image equals $L$. In particular, $M_p$ detects the unknot and is basic for $p>3$, thereby answering a question posed repeatedly in the literature.

Given a surface of genus $g \geq 2$, its inf spectrum is the ordered multiset of inf lengths of curves on the surface, the infimum being over all hyperbolic metrics. We show that the inf spectrum has exponential growth with coefficients that only depend on g, improving the earlier double exponential upper bound. The proof involves developing a number of tools including the process of "thickening" a surface and the notion of model surfaces. These tools may be of independent interest.

We construct closed symplectic 4-manifolds admitting the structure of hyperbolic $\pi$-orbifold, addressing LeBrun's conjecture. Our simplest example has Euler characteristic $304$, is tessellated by $64$ copies of the right-angled hyperbolic $120$-cell and admits a four-fold branched covering over a product of two surfaces of genus $4$.

For any $n\geq 1$, we construct a continuum of non-conjugate actions by homeomorphisms of $\operatorname{SL}_{n+1}(\mathbb{Q})$ on the $n$-sphere $S^{n}$ that do not extend to a continuous action of $\operatorname{SL}_{n+1}(\mathbb{R})$. We do this by adapting a blow-up construction of Bowden and Mann. We motivate studying these actions and note down some related observations.

Let $N$ be a 3-dimensional integral homology sphere. We prove the existence of a sequence of 4-dimensional topologically negatively curved manifolds $M_k$, $k=1,2,\dots$, satisfying the following: (1) all aspherical manifolds $M_k$ have the same universal cover $U_N$ depending only on $N$; (2) all fundamental groups $\pi_1(M_k)$ are hyperbolic and share the same Gromov boundary $X_N$ depending only on $N$; (3) the groups $\pi_1(M_k)$ represent infinitely many distinct quasi-isometry classes. The last conclusion, which represents the new feature, shows that negatively curved geometries in dimension 4 exhibit substantially richer structure than those in dimension 3.

We give a single counterexample to Conjectures 1–3 as stated in Luo's paper: a minimal two-tetrahedron triangulation of the lens space L(3,1) with a null-homotopic loop edge. The proof uses quadrilateral matching equations, an explicit calculation of shape parameters, and known results on volume maximization.

Let $\Gamma \le \mathrm{SL}(d+1,\mathbb R)$ be a torsion-free hyperbolic group dividing a properly and strictly convex domain $\Omega \subset \mathbb R\mathrm P^d$, and let $\rho: \Gamma \to \mathrm{SL}(d+1,\mathbb R)$ be the inclusion. We study representations of $\Gamma$ into the coaffine group that are deformations of $\rho\oplus 1 \in \mathrm{Hom}(\Gamma, \mathrm{GL}(d+2,\mathbb R))$ and that lie on the boundary of the locus of projective Anosov representations. In this setting, we identify a canonical limit set $\Lambda$ for the action and study its fine geometric features which are, in a suitable sense, stable. We prove the existence of an ending lamination that encodes the failure of the Anosov condition and the non-existence of a Cannon–Thurston map. Under certain conditions, we compute the Hausdorff dimension and show that the corresponding measure is positive.

Within the topological concordance class of the Hopf link, we construct infinitely many classes under the equivalence relation generated by smooth $0$-shake concordance. Each contains infinitely many smooth concordance classes represented by links that are pairwise strongly $0$-shake concordant and whose second components have unbounded smooth slice genus.

For an odd prime $p$, we prove a Smith inequality over $\mathbb{F}_p$ for exact Lagrangian Floer homology under a symplectic $\mathbb{Z}_p$-action, assuming an equivariant stable normal trivialization compatible with the Lagrangians and the required relative Pin data. As an application, we obtain a rank inequality between the symplectic Khovanov homology of a $p$-periodic link and that of its quotient. These results extend the involutive localization of Seidel and Smith. The main construction is a stabilized chain complex. At every invariant intersection point we adjoin the Morse complex of a finite-dimensional normal spectral sphere, whose generators lie in free $\mathbb{Z}_p$-orbits. The differential counts holomorphic strips coupled to sphere half-trajectories by a one-sided weighted asymptotic evaluation. We prove transversality, a uniform spectral cutoff, and the compactification and gluing statements needed for the chain identity. The stabilized complex is quasi-isomorphic to the ordinary Floer complex and contains the complex of the fixed data as a subcomplex with free quotient, so the algebraic Smith inequality applies.

We give a complete classification of tight contact structures on manifolds obtained from any surgery along any torus knot, which provides the first such classification on an infinite family of knots. The main tools are the classification of non-loose torus knots, bordered contact invariants via their immersed curve interpretations, and non-thickenable tori in knot complements. As a byproduct of this classification, we also give the first classification of contact structures on some surface bundles over the circle with fibers of genus larger than $1$. The symplectic fillability of these contact structures is also studied and leads to, among other things, the disproof of a conjecture of Ghiggini and Van Horn-Morris on Stein fillability of some contact structures on a Brieskorn sphere.

The main result is an obstruction to the presence of contact regions for c-symplectic confoliations in any odd ambient dimension. As a first application, we prove a 5-dimensional analogue (under stronger assumptions) of a Reeb stability theorem for confoliations in dimension 3 due to Eliashberg and Thurston. Refining work of the first author with Cardona, we further give two new notions of contact approximations of c-symplectic confoliations in any odd ambient dimension generalizing the 3-dimensional notion of Eliashberg-Thurston. As a second application of our main theorem, we then give an example, in any ambient odd dimension, of a c-symplectic (and in fact strong symplectic) foliation that cannot be contact approximated, in either of these two senses. This relies on the property, of independent interest, that contact structures approximating (in these senses) a weakly c-symplectic confoliation are also weakly fillable. The proofs of all these statements rely on pseudo-holomorphic curve theory. Lastly, we describe several natural classes of examples of these types of contact approximations of c-symplectic confoliations.

We complete the proof that every compact 3-manifold group is linear. Specifically, we construct a faithful representation in GL5(Q[u]) for a finite-index subgroup of every closed non-geometric graph-manifold group.

By a celebrated result of Birman and Series, the set of simple complete geodesics form a nowhere dense subset of a hyperbolic surface. We show that a segment of finite length is in a (small) neighborhood of this set if it can be "extended simply" in both directions for long enough. We deduce that, for large enough genus $g$, there exist surfaces with a Birman-Series set that is roughly $\frac{1}{g^{\sfrac{2}{3}}}$-dense.

In this chapter, we discuss the interaction between low-dimensional dynamics, symbolic dynamics, and geometric topology, focusing on the algebraic characterization and asymptotic behaviour of stretch factors or equivalently pseudo-Anosov dilatations. We first review the algebraic foundations linking Perron–Frobenius theory, multi-variable Euclidean algorithms, and Lind's characterization of topological entropy for shifts of finite type, emphasizing recent constraints on the Perron–Frobenius degree of expansion constants. We then relate these dynamical invariants to the combinatorial world, examining train tracks for surfaces and automorphisms of free groups. Finally, we briefly review the relevance of fibered 3-manifolds, detailing how the Thurston norm, Fried's concavity theorem over fibered cones, and McMullen's Teichmüller polynomial come into play.

We study the (typical) singularities of the distance function from a point in the ambient space to a polytopal complex in Euclidean space. By means of the Goresky-MacPherson splitting we generalize the existing linear theory to the quadratic distance function. Critical points occur on `selected' strata of the PL-space, where normal Morse Data determine the change of topology. On the homology level there exist indices with multiplicities, combined in a Morse vector. We focus mainly on PL-manifolds. We give bounds, depending on the dimension of the strata. In low dimensions we describe singularity types. At the end we discuss relations with Discrete Morse Theory.

For an oriented tame knot $K$, let $G_K$ be its group, with positive meridian $\mu_K$ and preferred longitude $\lambda_K$. We prove that the profinite completion $\widehat{G_K}$, marked by the conjugacy class of $\mu_K\lambda_K$, determines $K$ up to oriented equivalence. In fact, every isomorphism preserving this mark preserves the meridian and longitude with the same conjugator. The proof combines Xu's genuineness theorem with profinite JSJ theory and simultaneous boundary realization, and applies to satellite and composite knots. Consequently, Eisermann's colouring polynomials over all finite groups determine oriented knots.

We construct an action of the positive half of the Witt algebra on colored $\mathfrak{gl}_N$-Khovanov-Rozansky homology and prove that this action is compatible with the functoriality of colored link homology. To establish this result, we use Bar-Natan's framework of canopolies to formulate and prove the relevant statements at the level of tangles.

We provide a presentation of the braid group in which the generating set is the set of all positive half-twists, the relations are balanced positive relations, and every relation is conjugate to one in a finite list. We then prove that for any such presentation, its positive monoid is not isomorphic to the quasipositive braid monoid. Finally, we show that the quasipositive braid monoid has no proper Garside family. These results settle a problem of Calegari from the K3 problems list.

We construct closed Levi non-degenerate CR manifolds with non-vanishing Chern monomials. For every $n \geq 2$ and $\lceil n / 4 \rceil \leq m \leq \lfloor n / 2 \rfloor$, there exists a $(2 n + 1)$-dimensional example of signature $(\lfloor n / 2 \rfloor, \lceil n / 2 \rceil)$ on which all Chern monomials with positive indices at most $m$ and total weight at most $n - m$ are non-zero in cohomology with real coefficients. Taking $m = \lceil n / 4 \rceil$ gives examples with $c_{1}^{\lfloor 3 n / 4 \rfloor} \neq 0$. For $n = 4$ or $n \geq 6$, the latter examples show that the Chern-class vanishing theorem for strictly pseudoconvex CR manifolds does not extend to the Levi non-degenerate case. In these dimensions, their underlying manifolds admit no strictly pseudoconvex CR structures. The construction uses circle bundles over products of integral pseudo-Kähler manifolds.

We construct a polar action of $\mathrm{SU}(3)$ on a closed simply connected $11$-manifold with a compact three-dimensional section that is not injectively immersed. The section is injective over the regular part and has two distinct local branches along a singular stratum. All isotropy groups are connected, and all orbits are simply connected. This gives a negative answer to the injectivity question raised by Grove and Ziller.

We prove that there is no uniform description of $\mathrm{LG}^{(n)}$ in terms of colored Jones polynomials for all knots, even when allowing infinitely many colors and arbitrary operations. We prove the $n=1$ case, using an existing cabling formula and the separation of the Conway and Kinoshita–Terasaka knots by $\mathrm{LG}^{(2)}$. We develop an odd cabling formula and use a Vandermonde argument to show that for each $n\geq 2$ some satellite of each of these knots is distinguished by $\mathrm{LG}^{(n)}$.

The curve graph of a surface is the graph whose vertices are isotopy classes of essential, closed curves on the surface. Edges join vertices with disjoint representatives. The curve graph plays a central role in the theory of mapping class groups of surfaces. Gaster, Greene, and Vlamis have shown that the subgraph of the curve graph induced by curves in a single primitive homology class is uniquely and finitely colorable. In the analogy between the theory of mapping class groups of surfaces and outer automorphisms of free groups, the sphere graph of $M_r$, a connect sum of $r$ copies of $S_1 \times S_2$ plays an analogous role in the theory. Motivated by Gaster, Green, and Vlamis' result, we investigate $\mathcal{S}_v(M_r)$, the subgraph of the sphere graph of $M_r$ induced by spheres representing a single primitive homology class $[v] \in H_2(M_r;\mathbb{Z})$. We show that this graph is uniquely $r-1$ colorable. As a corollary, with $r\ge 3$ we obtain a nontrivial homomorphism from the Torelli subgroup $IO_r \le \operatorname{Out}(F_r)$ to the symmetric group on $r-1$ symbols; this parallels Gaster, Greene, and Vlamis' connection between their coloring and the Chillingworth homomorphism. In the course of our proof we find new smaller generating sets for mapping class groups of $M_r$ with spheres deleted.