We study surface representatives of homology classes of finite complexes which minimize certain complexity measures, including its genus and Euler characteristic. Our main result is that up to surgery at nullhomotopic curves minimizers are homotopic to cellwise coverings to the 2-skeleton. From this we conclude that the minimizing problem is in general algorithmically undecidable, but can be solved for 2-dimensional CAT(-1)-complexes.
We report on the computation of the integral homology of the mapping class group of genus g surfaces with one boundary curve and m punctures, when 2g + m is smaller than 6. In particular, it includes the genus 2 case with no or one puncture.
To each knot $K\subset S^3$ one can associated its knot Floer homology $\hat{HFK}(K)$, a finitely generated bigraded abelian group. In general, the nonzero ranks of these homology groups lie on a finite number of slope one lines with respect to the bigrading. The width of the homology is, in essence, the largest horizontal distance between two such lines. Also, for each diagram $D$ of $K$ there is an associated Turaev surface, and the Turaev genus is the minimum genus of all Turaev surfaces for $K$. We show that the width of knot Floer homology is bounded by Turaev genus plus one. Skein relati…
Given a genus-g Heegaard splitting of a 3-sphere, the genus-g Goeritz group is defined to be the group of the isotopy classes of orientation preserving homeomorphism of the 3-sphere that preserve the splitting. In this paper, we determine the twisted first (co)homology group of the genus-2 Goeritz group of 3-sphere.
Using the Heegaard Floer homology of Ozsvath and Szabo we investigate obstructions to definite intersection pairings bounded by rational homology spheres. As an application we obtain new lower bounds for the four-ball genus of Montesinos links.
It is known that knot Floer homology detects the genus and Alexander polynomial of a knot. We investigate whether knot Floer homology of $K$ detects more structure of minimal genus Seifert surfaces for $K$. We define an invariant of algebraically slice, genus one knots and provide examples to show that knot Floer homology does not detect this invariant. Finally, we remark that certain metabelian $L^2$-signatures bound this invariant from below.
We consider manifold-knot pairs $(Y,K)$ where $Y$ is a homology sphere that bounds a homology ball. We show that the minimum genus of a PL surface $Σ$ in a homology ball $X$ such that $\partial (X, Σ) = (Y, K)$ can be arbitrarily large. Equivalently, the minimum genus of a surface cobordism in a homology cobordism from $(Y, K)$ to any knot in $S^3$ can be arbitrarily large. The proof relies on Heegaard Floer homology.
We show that perturbing the definition of sl(n) Khovanov-Rozansky link homology gives a lower bound on the slice genus of a knot. As a corollary this yields another proof of Milnor's conjecture on the slice genus of torus knots.
We determine the first homology group with coefficients in $H_1(N;\mathbb{Z})$ for the mapping class group of a non-orientable surface $N$ of genus three with two boundary components.
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 bigraded groups. Additionally, for both knot Heegaard Floer homology and Khovanov homology, the genus two mutation interchanges the groups in $δ$-gradings $k$ and $-k$.
The Turaev genus of a link can be thought of as a way of measuring how non-alternating a link is. A link is Turaev genus zero if and only if it is alternating, and in this viewpoint, links with large Turaev genus are very non-alternating. In this paper, we study Turaev genus one links, a class of links which includes almost alternating links. We prove that the Khovanov homology of a Turaev genus one link is isomorphic to $\mathbb{Z}$ in at least one of its extremal quantum gradings. As an application, we compute or nearly compute the maximal Thurston Bennequin number of a Turaev genus one link…
We study the action of the Veech group of square-tiled surfaces of genus two on homology. This action defines the homology Veech group which is a subgroup of $\textrm{SL}_2(\mathcal{O}_D)$ where $\mathcal{O}_D$ is a quadratic order of square discriminant. Extending a result of Weitze-Schmithüsen we show that also the homology Veech group is a totally non-congruence subgroup with exceptions stemming only from the prime ideals lying above 2. While Weitze-Schmithüsen's result for Veech groups is asymmetric with respect to the spin structure our use of the homology Veech group yields a completely …
We use Lee's work on the Khovanov homology to define a knot invariant s. We show that s(K) is a concordance invariant and that it provides a lower bound for the slice genus of K. As a corollary, we give a purely combinatorial proof of the Milnor conjecture.
Given an element in the first homology of a rational homology 3-sphere $Y$, one can consider the minimal rational genus of all knots in this homology class. This defines a function $Θ$ on $H_1(Y;\mathbb Z)$, which was introduced by Turaev as an analogue of Thurston norm. We will give a lower bound for this function using the correction terms in Heegaard Floer homology. As a corollary, we show that Floer simple knots in L-spaces are genus minimizers in their homology classes, hence answer questions of Turaev and Rasmussen about genus minimizers in lens spaces.
According to seminal work of Kontsevich, the unstable homology of the mapping class group of a surface can be computed via the homology of a certain lie algebra. In a recent paper, S. Morita analyzed the abelianization of this lie algebra, thereby constructing a series of candidates for unstable classes in the homology of the mapping class group. In the current paper, we show that these cycles are all nontrivial, representing degree 4k+1 homology classes in the homology of the mapping class group of a genus one surface with 4k+1 punctures.
We exhibit a finitely generated group $\M$ whose rational homology is isomorphic to the rational stable homology of the mapping class group. It is defined as a mapping class group associated to a surface $\su$ of infinite genus, and contains all the pure mapping class groups of compact surfaces of genus $g$ with $n$ boundary components, for any $g\geq 0$ and $n>0$. We construct a representation of $\M$ into the restricted symplectic group ${\rm Sp_{res}}({\cal H}_r)$ of the real Hilbert space generated by the homology classes of non-separating circles on $\su$, which generalizes the classical …
We compute the Heegaard Floer homology of any rational homology 3-sphere with an open book decomposition of the form (T,ϕ), where T is a genus one surface with one boundary component. In addition, we compute the Heegaard Floer homology of any T^2-bundle over S^1 with first Betti number equal to one, and we compare our results with those of Lebow on the embedded contact homology of such torus bundles. We use these computations to place restrictions on Stein-filllings of the contact structures compatible with such open books, to narrow down somewhat the class of 3-braid knots with finite concord…
In an earlier work, we introduced a family of t-modified knot Floer homologies, defined by modifying the construction of knot Floer homology HFK-minus. The resulting groups were then used to define concordance homomorphisms indexed by t in [0,2]. In the present work we elaborate on the special case t=1, and call the corresponding modified knot Floer homology the unoriented knot Floer homology. Using elementary methods (based on grid diagrams and normal forms for surface cobordisms), we show that the resulting concordance homomorphism gives a lower bound for the smooth 4-dimensional crosscap nu…
Suppose C is a singular curve in CP^2 and it is topologically an embedded surface of genus g; such curves are called cuspidal. The singularities of C are cones on knots K_i. We apply Heegaard Floer theory to find new constraints on the sets of knots {K_i} that can arise as the links of singularities of cuspidal curves. We combine algebro-geometric constraints with ours to solve the existence problem for curves with genus one, d>33, that possess exactly one singularity which has exactly one Puiseux pair (p;q). The realized triples (p,d,q) are expressed as successive even terms in the Fibonacci …
We prove that, like the Seiberg-Witten monopole homology, the Heegaard Floer homology for a three-manifold determines its Thurston norm. As a consequence, we show that knot Floer homology detects the genus of a knot. This leads to new proofs of certain results previously obtained using Seiberg-Witten monopole Floer homology (in collaboration with Kronheimer and Mrowka). It also leads to a purely Morse-theoretic interpretation of the genus of a knot. The method of proof shows that the canonical element of Heegaard Floer homology associated to a weakly symplectically fillable contact structure i…
Let X be a simply-connected closed oriented 4-manifold and A an embedded surface of genus g and negative self-intersection -N. We show that for fixed genus g there is an upper bound on N if the homology class of A is divisible or characteristic. In particular, for genus zero, there is a lower bound on the self-intersection of embedded spheres in these kinds of homology classes. This question is related to a problem from the Kirby list.
Given a one-dimensional homology class in a lens space, a question related to the Berge conjecture on lens space surgeries is to determine all knots realizing the minimal rational genus of all knots in this homology class. It is known that simple knots are rational genus minimizers. In this paper, we construct many non-simple genus minimizers. This negatively answers a question of Rasmussen.
Let K be a knot in S^3 of genus g and let n>0. We show that if rk HFK(K,g) < 2^{n+1} (where HFK denotes knot Floer homology), in particular if K is an alternating knot such that the leading coefficient a_g of its Alexander polynomial satisfies |a_g| <2^{n+1}, then K has at most n pairwise disjoint non-isotopic genus g Seifert surfaces. For n=1 this implies that K has a unique minimal genus Seifert surface up to isotopy.
Ozsvath and Szabo conjectured that knot Floer homology detects fibred knots. We propose a strategy to approach this conjecture based on Gabai's theory of sutured manifold decomposition and contact topology. We implement this strategy for genus-one knots, obtaining as a corollary that, if rational surgery on a knot $K$ gives the Poincare homology sphere $Σ(2,3,5)$, then $K$ is the left-handed trefoil knot.
We use the knot filtration on the Heegaard Floer complex to define an integer invariant tau(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlike the signature, tau gives sharp bounds on the four-ball genera of torus knots. As another illustration, we calculate the invariant for several ten-crossing knots.
We prove that the map on knot Floer homology induced by a ribbon concordance is injective. As a consequence, we prove that the Seifert genus is monotonic under ribbon concordance. We also generalize a theorem of Gabai about the super-additivity of the Seifert genus under band connected sum. Our result gives evidence for a conjecture of Gordon that ribbon concordance is a partial order on the set of knots.
We show that if M is a complete, finite-volume, hyperbolic 3-manifold having exactly one cusp, and if H_1(M;Z_2) has dimension at least 6, then M has volume greater than 5.06. We also show that if M is a closed, orientable hyperbolic 3-manifold such that H_1(M;Z_2) has dimension at least 4, and if the image of the cup product map in H^2(M;Z_2) has dimension at most 1, then M has volume greater than 3.08. The proofs of these geometric results involve new topological results relating the Heegaard genus of a closed Haken manifold M to the Euler characteristic of the kishkes (i.e guts) of the comp…
We modify the construction of knot Floer homology to produce a one-parameter family of homologies for knots in the three-sphere. These invariants can be used to give homomorphisms from the smooth concordance group to the integers, giving bounds on the four-ball genus and the concordance genus of knots. We give some applications of these homomorphisms.
We investigate constraints on embeddings of a non-orientable surface in a $4$-manifold with the homology of $M \times I$, where $M$ is a rational homology $3$-sphere. The constraints take the form of inequalities involving the genus and normal Euler class of the surface, and either the Ozsváth--Sazbó $d$-invariants or Atiyah--Singer $ρ$-invariants of $M$. One consequence is that the minimal genus of a smoothly embedded surface in $L(2p,q) \times I$ is the same as the minimal genus of a surface in $L(2p,q)$. We also consider embeddings of non-orientable surfaces in closed $4$-manifolds.
We define several concordance invariants using knot Floer homology which give improvements over known slice genus and clasp number bounds from Heegaard Floer homology. We also prove that the involutive correction terms of Hendricks and Manolescu give both a slice genus and a clasp number bound.
We extend the techniques in a previous paper to calculate the Heegaard Floer homology groups for fibered 3-manifolds M whose monodromy is a power of a Dehn twist about a genus-1 separating circle on a surface of genus g > 1. We only consider non-torsion Spin^c-structures on M.
In this article we give a survey of homology computations for moduli spaces $\mathfrak{M}_{g,1}^m$ of Riemann surfaces with genus $g\geqslant 0$, one boundary curve, and $m\geqslant 0$ punctures. While rationally and stably this question has a satisfying answer by the Madsen-Weiss theorem, the unstable homology remains notoriously complicated. We discuss calculations with integral, mod-2, and rational coefficients. Furthermore, we determine, in most cases, explicit generators using homology operations.
In an earlier paper, we introduced a collection of graded Abelian groups $\HFKa(Y,K)$ associated to knots in a three-manifold. The aim of the present paper is to investigate these groups for several specific families of knots, including the Kinoshita-Terasaka knots and their ``Conway mutants''. These results show that $\HFKa$ contains more information than the Alexander polynomial and the signature of these knots; and they also illustrate the fact that $\HFKa$ detects mutation. We also calculate $\HFKa$ for certain pretzel knots, and knots with small crossing number ($n\leq 9$). Our calculatio…
In this article, we generalize the results discussed in [arXiv:1004.3762] by introducing a genus to generic fibers of Lefschetz fibrations. That is, we give families of relations in the mapping class groups of genus-1 surfaces with boundaries that represent rational homology disk smoothings of weighted homogeneous surface singularities whose resolution graphs are $3$-legged with a bad central vertex.
In this paper, we study the Khovanov homology of an alternating virtual link $L$ and show that it is supported on $g+2$ diagonal lines, where $g$ equals the virtual genus of $L$. Specifically, we show that $Kh^{i,j}(L)$ is supported on the lines $j=2i-σ_ξ+2k-1$ for $0\leq k\leq g+1$ where $σ_{ξ^*}(L)+2g= σ_ξ(L)$ are the signatures of $L$ for a checkerboard coloring $ξ$ and its dual $ξ^*$. Of course, for classical links, the two signatures are equal and this recovers Lee's $H$-thinness result for $Kh^{*,*}(L)$. Our result applies more generally to give an upper bound for the homological width o…
Let $H_{g}$ be a 3-dimensional handlebody of genus $g$. We determine the twisted first homology group of the mapping class group of $H_{g}$ with coefficients in the first integral homology group of the boundary surface $\partial H_{g}$ for $g\geq 2$.
We define an integer-valued invariant of special cube complexes called the genus, and prove that having genus one characterizes special cube complexes with abelian fundamental group. Using the genus, we obtain a new proof that the fundamental group of a special cube complex is either free abelian or surjects onto a non-cyclic free group. We also investigate automorphisms of special cube complexes, and give a new geometric proof that the Torelli subgroup for a right-angled Artin group is torsion-free.
We say a null-homologous knot $K$ in a $3$--manifold $Y$ has Property G, if the properties about the Thurston norm and fiberedness of the complement of $K$ is preserved under the zero surgery on $K$. In this paper, we will show that, if the smooth $4$--genus of $K\times\{0\}$ (in a certain homology class) in $(Y\times[0,1])\#N\overline{\mathbb CP^2}$, where $Y$ is a rational homology sphere, is smaller than the Seifert genus of $K$, then $K$ has Property G. When the smooth $4$--genus is $0$, $Y$ can be taken to be any closed, oriented $3$--manifold.
Genus 2 mutation is the process of cutting a 3-manifold along an embedded closed genus 2 surface, twisting by the hyper-elliptic involution, and gluing back. This paper compares genus 2 mutation with the better-known Conway mutation in the context of knots in the 3-sphere. Despite the fact that any Conway mutation can be achieved by a sequence of at most two genus 2 mutations, the invariants that are preserved by genus 2 mutation are a proper subset of those preserved by Conway mutation. In particular, while the Alexander and Jones polynomials are preserved by genus 2 mutation, the HOMFLY-PT p…
We sketch a construction of Legendrian Symplectic Field Theory (SFT) for conormal tori of knots and links. Using large $N$ duality and Witten's connection between open Gromov-Witten invariants and Chern-Simons gauge theory, we relate the SFT of a link conormal to the colored HOMFLY-PT polynomials of the link. We present an argument that the HOMFLY-PT wave function is determined from SFT by induction on Euler characteristic, and also show how to, more directly, extract its recursion relation by elimination theory applied to finitely many noncommutative equations. The latter can be viewed as the…
We determine when certain state cycles represent nontrivial Khovanov homology classes by analyzing features of the state graph. Using this method, we are able to produce hyperbolic knots with arbitrarily many diagonals containing nontrivial state cycle homology classes. This gives lower bounds on the Khovanov width of knots whose complexity precludes computation of the full homology.
An estimate for the genus function in circle bundles over irreducible 3-manifolds is proven. This estimate is in many cases an equality and it relates the minimal genus of the surfaces representing a given homology class with the self-intersection of the class and the Thurston norm of the underlying 3-manifold.
We give bounds on the gap functions of the singularities of a cuspidal plane curve of arbitrary genus, generalising recent work of Borodzik and Livingston. We apply these inequalities to unicuspidal curves whose singularity has one Puiseux pair: we prove two identities tying the parameters of the singularity, the genus, and the degree of the curve; we improve on some degree-multiplicity asymptotic inequalities; finally, we prove some finiteness results, we construct infinite families of examples, and in some cases we give an almost complete classification.
We define two concordance invariants of knots using framed instanton homology. These invariants $ν^\sharp$ and $τ^\sharp$ provide bounds on slice genus and maximum self-linking number, and the latter is a concordance homomorphism which agrees in all known cases with the $τ$ invariant in Heegaard Floer homology. We use $ν^\sharp$ and $τ^\sharp$ to compute the framed instanton homology of all nonzero rational Dehn surgeries on: 20 of the 35 nontrivial prime knots through 8 crossings, infinite families of twist and pretzel knots, and instanton L-space knots; and of 19 of the first 20 closed hyper…
We apply Heegaard-Floer homology theory to establish generalized slicing Bennequin inequalities closely related to a recent result of T. Mrowka and Y. Rollin proved using Seiberg-Witten monopoles.
An important difference between high dimensional smooth manifolds and smooth 4-manifolds that in a 4-manifold it is not always possible to represent every middle dimensional homology class with a smoothly embedded sphere. This is true even among the simplest 4-manifolds: $X_0(K)$ obtained by attaching an $0$-framed 2-handle to the 4-ball along a knot $K$ in $S^3$. The $0$-shake genus of $K$ records the minimal genus among all smooth embedded surfaces representing a generator of the second homology of $X_0(K)$ and is clearly bounded above by the slice genus of $K$. We prove that slice genus is …
We construct elements of the third quandle homology groups of knot quandles, which are called the shadow fundamental classes. They play the same roles for the shadow quandle cocycle invariants of knots as the fundamental classes of knot quandles does for the quandle cocycle invariants. As an application of the shadow fundamental classes, we show the relation between the shadow quandle cocycle invariants and the based shadow quandle cocycle invariants. Moreover, we will show, for a prime knot, that any third quandle homology classes are considered as images of the shadow fundamental classes of …
