The topology of symplectic 4-manifolds is related to that of singular plane curves via the concept of branched covers. Thus, various classification problems concerning symplectic 4-manifolds can be reformulated as questions about singular plane curves. Moreover, using braid monodromy, these can in turn be reformulated in the language of braid group factorizations. While the results mentioned in this paper are not new, we hope that they will stimulate interest in these questions, which remain essentially wide open.
There are some similarities between cohomology of SU(2)-representation varieties of the fundamental group of some link complements and the Khovanov homology of the links. We start here a program to explain a possible source of these similarities. We introduce a symplectic manifold ${\mathscr M}$ with an action of the braid group $B_{2n}$ preserving the symplectic structure. The action allows to associate a Lagrangian submanifold of ${\mathscr M}$ to every braid. The representation variety of a link can then be described as the intersection of such Lagrangian submanifolds, given a braid present…
By evaluating the Burau representation at t=-1, we obtain a symplectic representation of the braid group. We define the congruence subgroups of the braid group to be the preimages of the principal congruence subgroups of the symplectic group. Our main result is that the level four congruence subgroup of the braid group is equal to the group generated by squares of Dehn twists and is also equal to the group generated by squares of pure braids. We also compute the image of the point pushing subgroup under the symplectic representation.
In this paper we prove certain Hurwitz equivalence properties in the braid group. Our main result is that every two factorizations of $Δ_n ^2$ where the elements of the factorization are semi-frame are Hurwitz equivalent. The results of this paper are generalization of the results in \cite{B4}. We use a new presentation of the braid group, called the Birman-Ko-Lee presentation, to define the semi-frame structure. The main result of this paper can be applied to compute the BMT invariant of surfaces. The BMT is the class of Hurwitz equivalent factorizations of the central element of the braid gr…
We prove that the symplectic group $Sp(2n,\mathbb Z)$ and the mapping class group $Mod_{S}$ of a compact surface $S$ satisfy the $R_{\infty}$ property. We also show that $B_n(S)$, the full braid group on $n$-strings of a surface $S$, satisfies the $R_{\infty}$ property in the cases where $S$ is either the compact disk $D$, or the sphere $S^2$. This means that for any automorphism $ϕ$ of $G$, where $G$ is one of the above groups, the number of twisted $ϕ$-conjugacy classes is infinite.
Every compact symplectic 4-manifold can be realized as a branched cover of the complex projective plane branched along a symplectic curve with cusp and node singularities; the covering map is induced by a triple of sections of a "very ample" line bundle. In this paper, we give an explicit formula describing the behavior of the braid monodromy invariants of the branch curve upon degree doubling of the linear system (from a very ample bundle $L^k$ to $L^{2k}$). As a consequence, we derive a similar degree doubling formula for the mapping class group monodromy of Donaldson's symplectic Lefschetz …
We investigate the local contribution of the braid monodromy factorization in the context of the links obtained by the closure of these braids. We consider plane curves which are arrangements of lines and conics as well as some algebraic surfaces, where some of the former occur as local configurations in degenerated and regenerated surfaces in the latter. In particular we focus on degenerations which involve intersection points of multiplicity two and three. We demonstrate when the same links arise even when the local arrangements are different.
We show that any knot which is smoothly the closure of a 3-braid cannot be Lagrangian concordant to and from the maximum Thurston-Bennequin Legendrian unknot except the unknot itself. Our obstruction comes from drawing the Weinstein handlebody diagrams of particular symplectic fillings of cyclic branched double covers of knots in $S^3$. We use the Legendrian contact homology differential graded algebra of the links in these diagrams to compute the symplectic homology of these fillings to derive a contradiction. As a corollary, we find an infinite family of contact manifolds which are rational …
We develop a new algebraic theory of positive braids and conjugacy classes in the braid group $B_3$. We use our theory to establish a complete classification of isotopy classes of degree three symplectic curves in $\mathbb{CP}^2$ with only $A_n$-singularities for $n\geq 1$ (an $A_n$-singularity is locally modelled by the equation $z^2 = w^n$) independent of Gromov's theory of pseudoholomorphic curves. We show that if $C$ and $C'$ are degree three symplectic curves in $\mathbb{CP}^2$ with the same numbers of $A_n$-singularities for each $n\geq 1$, then $C$ is isotopic to $C'$. Furthermore, our …
After reviewing recent results on symplectic Lefschetz pencils and symplectic branched covers of CP^2, we describe a new construction of maps from symplectic manifolds of any dimension to CP^2 and the associated monodromy invariants. We also show that a dimensional induction process makes it possible to describe any compact symplectic manifold by a series of words in braid groups and a word in a symmetric group.
We show that there are contact 3-manifolds of support genus one which admit infinitely many Stein fillings, but do not admit arbitrarily large ones. These Stein fillings arise from genus-1 allowable Lefschetz fibrations with distinct homology groups, all filling a fixed minimal genus open book supporting the boundary contact 3-manifold. In contrast, we observe that there are only finitely many possibilities for the homology groups of Stein fillings of a given contact 3-manifold with support genus zero. We also show that there are 4-strand braids which admit infinitely many distinct Hurwitz cla…
We study fillings of contact structures supported by planar open books by analyzing positive factorizations of their monodromy. Our method is based on Wendl's theorem on symplectic fillings of planar open books. We prove that every virtually overtwisted contact structure on L(p,1) has a unique filling, and describe fillable and non-fillable tight contact structures on certain Seifert fibered spaces.
We construct a family of pairs of non-isotopic symplectic surfaces in the standard symplectic $4$-disk such that they are bounded by the same transverse knot in the standard contact $3$-sphere and fundamental groups of their complements are isomorphic. In the appendix, we prove explicitly that one can obtain a symplectic surface in the standard symplectic $4$-disk from a braided surface in a bidisk.
We prove that every symplectic 4-manifold admits a trisection that is compatible with the symplectic structure in the sense that the symplectic form induces a Weinstein structure on each of the three sectors of the trisection. Along the way, we show that a (potentially singular) symplectic braided surface in $\mathbb{CP}^2$ can be symplectically isotoped into bridge position.
We initiate a study of positive multisections of Lefschetz fibrations via positive factorizations in framed mapping class groups of surfaces. Using our methods, one can effectively capture various interesting symplectic surfaces in symplectic 4-manifolds as multisections, such as Seiberg-Witten basic classes and exceptional classes, or branched loci of compact Stein surfaces as branched coverings of the 4-ball. Various problems regarding the topology of symplectic 4-manifolds, such as the smooth classification of symplectic Calabi-Yau 4-manifolds, can be translated to combinatorial problems in…
We study Question 7.9 in the paper "Monoids in the mapping class group" by Etnyre and Van Horn-Morris; whether a symmetric mapping class admitting a positive factorization is a lift of a quasi-positive braid. We answer affirmatively for mapping classes satisfying certain cyclic conditions.
Continuing the quest for exclusive Racah matrices, which are needed for evaluation of colored arborescent-knot polynomials in Chern-Simons theory, we suggest to extract them from a new kind of a double-evolution -- that of the antiparallel double-braids, which is a simple two-parametric family of two-bridge knots, generalizing the one-parametric family of twist knots. In the case of rectangular representations $R=[r^s]$ we found an evidence that the corresponding differential expansion miraculously factorizes and can be obtained from that for the twist knots. This reduces the problem of rectan…
Let $M$ be a closed, oriented, connected 3--manifold and $(B,π)$ an open book decomposition on $M$ with page $Σ$ and monodromy $φ$. It is easy to see that the first Betti number of $Σ$ is bounded below by the number of $S^2\times S^1$--factors in the prime factorization of $M$. Our main result is that equality is realized if and only if $φ$ is trivial and $M$ is a connected sum of $S^2\times S^1$'s. We also give some applications of our main result, such as a new proof of the result by Birman and Menasco that if the closure of a braid with $n$ strands is the unlink with $n$ components then the…
We construct an action of the braid group B_{2g+2} on the free group F_{2g} extending an action of B_4 on F_2 introduced earlier by Reutenauer and the author. Our action induces a homomorphism from B_{2g+2} into the symplectic modular group Sp_{2g}(Z). In the special case g=2 we show that the latter homomorphism is surjective and determine its kernel, thus obtaining a braid-like presentation of Sp_4(Z).
We prove that any symplectic 4-manifold which is not a rational or ruled surface, after sufficiently many blow-ups, admits an arbitrary number of nonisomorphic Lefschetz fibrations of the same genus which cannot be obtained from one another via Luttinger surgeries. This generalizes results of Park and Yun who constructed pairs of nonisomorphic Lefschetz fibrations on knot surgered elliptic surfaces. In turn, we prove that there are monodromy factorizations of Lefschetz pencils which have the same characteristic numbers but cannot be obtained from each other via partial conjugations by Dehn twi…
We show that the smallest non-cyclic quotients of braid groups are symmetric groups, proving a conjecture of Margalit. Moreover we recover results of Artin and Lin about the classification of homomorphisms from braid groups on n strands to symmetric groups on k letters, where k is at most n. Unlike the original proofs, our method does not use the Bertrand-Chebyshev theorem, answering a question of Artin. Similarly for mapping class group of closed orientable surfaces, the smallest non-cyclic quotient is given by the mod two reduction of the symplectic representation. We provide an elementary p…
In this article, we study the maximal length of positive Dehn twist factorizations of surface mapping classes. In connection to fundamental questions regarding the uniform topology of symplectic 4-manifolds and Stein fillings of contact 3-manifolds coming from the topology of supporting Lefschetz pencils and open books, we completely determine which boundary multitwists admit arbitrarily long positive Dehn twist factorizations along nonseparating curves, and which mapping class groups contain elements admitting such factorizations. Moreover, for every pair of positive integers g,n, we tell whe…
We consider the canonical contact structures on links of rational surface singularities with reduced fundamental cycle. These singularities can be characterized by their dual resolution graphs: the graph is a tree, and the weight of each vertex is no greater than its negative valency. In a joint work with Starkston, we previously showed that if the weight of each vertex in the graph is at most -5, the contact structure has a unique symplectic filling (up to symplectic deformation and blow-up). The proof was based on a symplectic analog of de Jong-van Straten's description of smoothings of thes…
We prove that there exists no a priori bound on the Euler characteristic of a closed symplectic 4-manifold coming solely from the genus of a compatible Lefschetz pencil on it, nor is there a similar bound for Stein fillings of a contact 3-manifold coming from the genus of a compatible open book --- except possibly for a few low genera cases. To obtain our results, we produce the first examples of factorizations of a boundary parallel Dehn twist as arbitrarily long products of positive Dehn twists along non-separating curves on a fixed surface with boundary. This solves an open problem posed by…
To a presentation of an oriented link as the closure of a braid we assign a complex of bigraded vector spaces. The Euler characteristic of this complex (and of its triply-graded cohomology groups) is the HOMFLYPT polynomial of the link. We show that the dimension of each cohomology group is a link invariant.
In this paper we construct a homomorphism of the affine braid group $Br_n^{aff}$ in the convolution algebra of the equivariant matrix factorizations on the space $\overline{\mathcal{X}}_2=\mathfrak{b}_n\times GL_n\times\mathfrak{n}_n$ considered in the earlier paper of the authors. We explain that the pull-back on the stable part of the space $\overline{\mathcal{X}_2}$ intertwines with the natural homomorphism from the affine braid group $Br_n^{aff}$ to the finite braid group $Br_n$. This observation allows us derive a relation between the knot homology of the closure of $β\in Br_n$ and the kn…
Questions of geography of various classes of $4$-manifolds have been a central motivating question in $4$-manifold topology. Baykur and Korkmaz asked which small, simply connected, minimal $4$-manifolds admit a genus $2$ Lefschetz fibration. They were able to classify all the possible homeomorphism types and realize all but one with the exception of a genus $2$ Lefschetz fibration on a symplectic $4$-manifold homeomorphic, but not diffeomorphic to $3 \mathbb{CP}^2 \# 11\overline{\mathbb{CP}}^2$. We give a positive factorization of type $(10,10)$ that corresponds to such a genus $2$ Lefschetz f…
We call a symplectic rational surface $(X,ω)$ \textit{positive} if $c_1(X)\cdot[ω]>0$. The positivity condition of a rational surface is equivalent to the existence of a divisor $D\subset X$, such that $(X, D)$ is a log Calabi-Yau surface. The cohomology class of a symplectic form can be endowed with a \textit{type} using the root system associated to its Lagrangian spherical classes. In this paper, we prove that the symplectic Torelli group of a positive rational surface is trivial if it is of type $\mathbb{A}$, and is a sphere braid group if it is of type $\mathbb{D}$. As an application, we …
In this paper we prove certain Hurwitz equivalence properties in $B_n$. Our main result is that every two Artin's factorizations of $Δ_n ^2$ of the form $H_{i_1} ... H_{i_{n(n-1)}}, \quad F_{j_1} ... F_{j_{n(n-1)}}$ (with $i_k, j_k \in \{1,...,n-1 \}$), where $\{H_1,...,H_{n-1} \}, \{F_1,...,F_{n-1} \}$ are frames, are Hurwitz equivalent. This theorem is a generalization of the theorem we have proved in \cite{Tz}, using an algebraic approach unlike the proof in \cite{Tz} which is geometric. The application of the result is applied to the classification of algebraic surfaces up to deformation. …
We explicitly produce symplectic genus-3 Lefschetz pencils (with base points), whose total spaces are homeomorphic but not diffeomorphic to rational surfaces CP^2 # p (-CP^2) for p= 7, 8, 9. We then give a new construction of an infinite family of symplectic Calabi-Yau surfaces with first Betti number b_1=2,3, along with a surface with b_1=4 homeomorphic to the 4-torus. These are presented as the total spaces of symplectic genus-3 Lefschetz pencils we construct via new positive factorizations in the mapping class group of a genus-3 surface. Our techniques in addition allow us to answer in the …
We study the problem of computing the homology of the configuration spaces of a finite cell complex $X$. We proceed by viewing $X$, together with its subdivisions, as a subdivisional space--a kind of diagram object in a category of cell complexes. After developing a version of Morse theory for subdivisional spaces, we decompose $X$ and show that the homology of the configuration spaces of $X$ is computed by the derived tensor product of the Morse complexes of the pieces of the decomposition, an analogue of the monoidal excision property of factorization homology. Applying this theory to the co…
Purpose of the Conference article, intended for a wider audience, is to introduce concepts and techniques used by Bronislaw Wajnryb and the author in order to show the diffeomorphism of certain elementary algebraic surfaces, called ABC surfaces, which are not deformation equivalent. In the first part are recalled the concepts of connected sum, surgeries, Poincare' group, Braids, Riemann surfaces and Dehn's Mapping class group. Then the topics become more advanced, including Lefschetz fibrations, ABC surfaces, symplectic fibrations and braid monodromy.
