The lens space $L_{p,q}$ is the orbit space of a $\mathbb{Z}_{p}$-action on the three sphere. We investigate polynomials of two complex variables that are invariant under this action, and thus define links in $L_{p,q}$. We study properties of these links, and their relationship with the classical algebraic links. We prove that all algebraic links in lens spaces are fibered, and obtain results about their Seifert genus. We find some examples of algebraic knots in lens spaces, whose lift in the $3$-sphere is a torus link.
We determine lens surgeries (i.e.\ Dehn surgery yielding a lens space) along the $n$-twisted Whitehead link. To do so, we first give necessary conditions to yield a lens space from the Alexander polynomial of the link as: (1) $n=1$ (i.e. the Whitehead link), and (2) one of surgery coefficients is 1, 2 or 3. Our interests are not only lens surgery itself but also how to apply the Alexander polynomial for this kind of problems.
We show how the Alexander polynomial of links in lens spaces is related to the classical Alexander polynomial of a link in the 3-sphere, obtained by cutting out the exceptional lens space fibre. It follows from these relationship that a certain normalization of the Alexander polynomial satisfies a skein relation in lens spaces.
Following the approach to grid homology of links in $S^3$, we prove combinatorially that the grid homology of links in lens spaces defined by Baker, Grigsby, and Hedden is a link invariant. Further, using the sign assignment defined by Celoria, we prove that the generalization of grid homology to integral coefficients is a link invariant.
We prove the existence of a polynomial invariant that satisfies the HOMFLY skein relation for links in a lens space. In the process we also develop a skein theory of toroidal grid diagrams in a lens space.
We give a concise proof of a classification of lens spaces up to orientation-preserving homeomorphisms. The chief ingredient in our proof is a study of the Alexander polynomial of ` symmetric' links in $S^3$.
An important geometric invariant of links in lens spaces is the lift in the 3-sphere of a link $L$ in $L(p,q)$, that is the counterimage $\widetilde L$ of $L$ under the universal covering of $L(p,q)$. If lens spaces are defined as a lens with suitable boundary identifications, then a link in $L(p,q)$ can be represented by a disk diagram, that is to say, a regular projection of the link on a disk. Starting from a disk diagram of $L$, we obtain a diagram of the lift $\widetilde L$ in the 3-sphere. With this construction we are able to find different knots and links in $L(p,q)$ having equivalent …
We construct a virtual quandle for links in lens spaces $L(p,q)$, with $q=1$. This invariant has two valuable advantages over an ordinary fundamental quandle for links in lens spaces: the virtual quandle is an essential invariant and the presentation of the virtual quandle can be easily written from the band diagram of a link.
We take advantage of the correspondence between fibered links, open book decompositions and contact structures on a closed connected 3-dimensional manifold to determine a mixed link diagram presentation for a particular fibered link $L$ in the lens space $L(p,q)$. Moreover, we construct a diagram for the lift of $L$ in the 3-sphere $S^3$ in order to show that, for $q=1$, $L$ is compatible with the standard contact structure of $L(p,1)$ when $p \leq -1$.
In this paper we study the relation between two diagrammatic representations of links in lens spaces: the disk diagram and the grid diagram and we find how to pass from one to the other. We also investigate whether the HOMFLY-PT invariant and the Link Floer Homology are essential invariants, that is, we try to understand if these invariants are able to distinguish links in $L(p,q)$ covered by the same link in $\mathbf{S}^3$. In order to do so, we generalize the combinatorial definition of Knot Floer Homology in lens spaces to the case of links and we analyze how both the invariants change when…
We study lens space surgeries along two different families of 2-component links, denoted by $A_{m,n}$ and $B_{p,q}$, related with the rational homology 4-ball used in J.\ Park's (generalized) rational blow down. We determine which coefficient $r$ of the knotted component of the link yields a lens space by Dehn surgery. The link $A_{m,n}$ yields a lens space only by the known surgery with $r=mn$ and unexpectedly with $r=7$ for $(m,n)=(2,3)$. On the other hand, $B_{p,q}$ yields a lens space by infinitely many $r$. Our main tool for the proof is the Reidemeister-Turaev torsions, i.e.\ Reidemeiste…
In this paper we study some aspects of knots and links in lens spaces. Namely, if we consider lens spaces as quotient of the unit ball $B^{3}$ with suitable identification of boundary points, then we can project the links on the equatorial disk of $B^{3}$, obtaining a regular diagram for them. In this contest, we obtain a complete finite set of Reidemeister type moves establishing equivalence, up to ambient isotopy, a Wirtinger type presentation for the fundamental group of the complement of the link and a diagrammatic method giving the first homology group. We also compute Alexander polynomia…
Links in lens spaces may be defined to be equivalent by ambient isotopy or by diffeomorphism of pairs. In the first case, for all the combinatorial representations of links, there is a set of Reidemeister-type moves on diagrams connecting isotopy equivalent links. In this paper we provide a set of moves on disk, band and grid diagrams that connects diffeo equivalent links: there are up to four isotopy equivalent links in each diffeo equivalence class. Moreover, we investigate how the diffeo equivalence relates to the lift of the link in the $3$-sphere: in the particular case of oriented primit…
In this paper the properties of the Kauffman bracket skein module of $L(p,q)$ are investigated. Links in lens spaces are represented both through band and disk diagrams. The possibility to transform between the diagrams enables us to compute the Kauffman bracket skein module on an interesting class of examples consisting of inequivalent links with equivalent lifts in the $3$-sphere. The computation show that the Kauffman bracket skein module is an essential invariant, that is, it may take different values on links with equivalent lifts. We also show how the invariant is related to the Kauffman…
We give criteria for an invariant of lens space links to bound the maximal self-linking number in certain tight contact lens spaces. As a corollary we extend the Franks-Williams-Morton inequality to the setting of lens spaces.
We present a reduced Burau-like representation for the mixed braid group on one strand representing links in lens spaces and show how to calculate the Alexander polynomial of a link directly from the mixed braid.
Grid diagrams encode useful geometric information about knots in S^3. In particular, they can be used to combinatorially define the knot Floer homology of a knot K in S^3, and they have a straightforward connection to Legendrian representatives of K in (S^3, ξ_\st), where ξ_\st is the standard, tight contact structure. The definition of a grid diagram was extended to include a description for links in all lens spaces, resulting in a combinatorial description of the knot Floer homology of a knot K in L(p, q) for all p > 0. In the present article, we explore the connection between lens space gri…
In this survey we summarize results regarding the Kauffman bracket, HOMFLYPT, Kauffman 2-variable and Dubrovnik skein modules, and the Alexander polynomial of links in lens spaces, which we represent as mixed link diagrams. These invariants generalize the corresponding knot polynomials in the classical case. We compare the invariants by means of the ability to distinguish between some difficult cases of knots with certain symmetries.
We study the concept of the fourth skein module of 3-manifolds, that is a skein module based on the skein relation b_0L_0 + b_1L_1 + b_2L_2 + b_3L_3 = 0 and a framing relation L^{(1)} = aL, where a, b_0, b_3 are invertible. We introdule the concept of n-algebraic tangles (and links) and analyze the skein module for 3-algebraic links. As a byproduct we prove the Montesinos-Nakanishi 3-move conjecture for 3-algebraic links (including 3-bridge links).
Let $F$ be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle $(B,T)$. Then $F$ separates the strings of $T$ in $B$ and the boundary slope of $F$ is uniquely determined by $(B,T)$ and hence we can define the slope of the algebraic tangle. In addition to the Conway's tangle sum, we define a natural product of two tangles. The slopes and binary operation on algebraic tangles lead an algebraic structure which is isomorphic to the rational numbers. We introduce a new knot and link class, algebraically alternating …
Let $M_λ$ be the $λ$-component Milnor link. For $λ\ge 3$, we determine completely when a finite slope surgery along $M_λ$ yields a lens space including $S^3$ and $S^1\times S^2$, where {\it finite slope surgery} implies that a surgery coefficient of every component is not $\infty$. For $λ=3$ (i.e.\ the Borromean rings), there are three infinite sequences of finite slope surgeries yielding lens spaces. For $λ\ge 4$, any finite slope surgery does not yield a lens space. As a corollary, $M_λ$ for $λ\ge 3$ does not yield both $S^3$ and $S^1\times S^2$ by any finite slope surgery. We generalize the…
We compute the invariants for a class of knots and links in arbitrary representations in $S^3/\mathbb{Z}_p$ in the large $k$ (level), large $N$ (rank) limit, keeping $N/(k+N)=λ$ fixed, in $U(N)$ and $Sp(N)$ Chern-Simons theories. Using the relation between the saddle point description and collective field theory, we first find that the invariants for the Hopf link and unknot are given by the on shell collective field theory action. We next show that the results of these two invariants can be used to compute the invariants of other torus knots and links. We also discuss the large $N$ phase stru…
Divide knots and links, defined by A'Campo in the singularity theory of complex curves, is a method to present knots or links by real plane curves. The present paper is a continuation of the author's previous result that every knot in the major subfamilies of Berge's lens space surgery (i.e., knots yielding a lens space by Dehn surgery) is presented by an L-shaped curve as a divide knot. In the present paper, L-shaped curves are generalized and it is shown that every knot in the minor subfamilies, called sporadic examples, of Berge's lens space surgery is presented by a generalized L-shaped cu…
We determine the non-null homologous knots in lens spaces whose exteriors contain properly embedded once-punctured tori. All such knots arise as surgeries on the Whitehead link and are grid number 1 in their lens spaces. As a corollary, we classify once-punctured torus bundles that admit a lens space filling.
The paper gives topological as well as rigid isotopy classification of smooth irreducible algebraic curves in the real projective 3-space for the case when the degree of the curve is at most six and its genus is at most one.
It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links. The Morton-Franks-Williams inequality gives a lower bound for braid index. And sharpness of the inequality on a knot type implies the truth of the conjecture for the knot type. We prove that there are infinitely many examples of knots and links for which the inequality is not sharp but the conjecture is still true. We also show that if the conjecture is true for K and L, then it is also true for the (p,q)-cable…
We study the behavior of the Witten-Reshetikhin-Turaev SU(2) invariants of links in L(p,q) as a function of the level r-2. They are given by 1 over the square root of r times one of p Laurent polynomials evaluated at e to the 2 pi i divided by 4pr. The congruence class of r modulo p determines which polynomial is applicable. If p is zero modulo four, the meridian of L(p,q) is non-trivial in the skein module but has trivial Witten-Reshetikhin-Turaev SU(2) invariants. On the other hand, we show that one may recover the element in the Kauffman bracket skein module of L(p,q) represented by a link …
A linkage is a finite graph with lengths assigned to each edge. A planar realization is a map to the plane which preserves edge lengths. It can be thought of as a mechanical device formed from stiff rods and rotating joints. We look at the configuration space of all planar realizations of a linkage (following work of Kapovich-Millson). We also look at configuration spaces of cabled linkages, where some edges are flexible cables. These configuration spaces are classified up to analytic isomorphism.
In this survey paper we present results about link diagrams in Seifert manifolds using arrow diagrams, starting with link diagrams in $F\times S^1$ and $N\hat{\times}S^1$, where $F$ is an orientable and $N$ an unorientable surface. Reidemeister moves for such arrow diagrams make the study of link invariants possible. Transitions between arrow diagrams and alternative diagrams are presented. We recall results about %the knot group presentation for lens spaces and the Kauffman bracket and HOMFLYPT skein modules of some Seifert manifolds using arrow diagrams, namely lens spaces, a product of a di…
We construct a family of hyperbolic link complements by gluing tangles along totally geodesic four-punctured spheres, then investigate the commensurability relation among its members. Those with different volume are incommensurable, distinguished by their scissors congruence classes. Mutation produces arbitrarily large finite subfamilies of nonisometric manifolds with the same volume and scissors congruence class. Depending on the choice of mutation, these manifolds may be commensurable or incommensurable, distinguished in the latter case by cusp parameters. All have trace field Q(i,\sqrt{2}),…
We compute the Heegaard-Floer link homology of algebraic links in terms of the multivariate Hilbert function of the corresponding plane curve singularities. The main result of the paper identifies four homologies: (a) the Heegaard-Floer link homology of the local embedded link of the germ, (b) the lattice homology associated with the Hilbert function, (c) the homologies of the projectivized complements of local hyperplane arrangements cut out from the local algebra, and (d) a certain variant of the Orlik-Solomon algebra of these local arrangements. In particular, the Poincaré polynomials of al…
The present paper studies the structure of characteristic varieties of fundamental groups of graph manifolds. As a consequence, a simple proof of Papadima's question is provided on the characterization of algebraic links that have quasi-projective fundamental groups. The type of quasi-projective obstructions used here are in the spirit of Papadima's original work.
We prove that if the lens space $L(n, 1)$ is obtained by a surgery along a knot in the lens space $L(3,1)$ that is distance one from the meridional slope, then $n$ is in $\{-6, \pm 1, \pm 2, 3, 4, 7\}$. This result yields a classification of the coherent and non-coherent band surgeries from the trefoil to $T(2, n)$ torus knots and links. The main result is proved by studying the behavior of the Heegaard Floer $d$-invariants under integral surgery along knots in $L(3,1)$. The classification of band surgeries between the trefoil and torus knots and links is motivated by local reconnection proces…
Symplectic fillings of standard tight contact structures on lens spaces are understood and classified. The situation is different if one considers non-standard tight structures (i.e. those that are virtually overtwisted), for which a classification scheme is still missing. In this work we use different approaches and employ various techniques to improve our knowledge of symplectic fillings of virtually overtwisted contact structures. We study curves configurations on surfaces to solve the problem in the case of a specific family of lens spaces. Then we give general constraints on the topology …
We define combinatorial invariants of Legendrian and transverse links in universally tight lens spaces using grid diagrams, generalizing [OST08] and prove that they are equivalent to the invariants defined in [BVVV13] and [LOSS09]. We use these combinatorial invariants to characterize index one grid diagrams for knots in lens spaces which admit surgeries to the 3-sphere and discuss a potential application to the Berge conjecture.
A generalised Thurston-Bennequin invariant for a Q-singularity of a real algebraic variety is defined as a linking form on the homologies of the real link of the singularity. The main goal of this paper is to present a method to calculate the linking form in terms of the very good resolution graph of a real normal unibranch surface singularity. For such singularities, the value of the linking form is the Thurston-Bennequin number of the real link of the singularity. As a special case of unibranch surface singularities, the behaviour of the linking form is investigated on the Brieskorn double p…
The classification of high-dimensional mu-component boundary links motivates decomposition theorems for the algebraic K-groups of the group ring A[F_mu] and the noncommutative Cohn localization Sigma^{-1}A[F_mu], for any mu>0 and an arbitrary ring A, with F_mu the free group on mu generators and Sigma the set of matrices over A[F_mu] which become invertible over A under the augmentation A[F_mu] to A. Blanchfield A[F_mu]-modules and Seifert A-modules are abstract algebraic analogues of the exteriors and Seifert surfaces of boundary links. Algebraic transversality for A[F_mu]-module chain comple…
We start a systematic analysis of links up to 5-move equivalence. Our motivation is to develop tools which later can be used to study skein modules based on the skein relation being deformation of a 5-move (in an analogous way as the Kauffman skein module is a deformation of a 2-move, i.e. a crossing change). Our main tools are Jones and Kauffman polynomials and the fundamental group of the 2-fold branch cover of S^3 along a link. We use also the fact that a 5-move is a composition of two rational \pm (2,2)-moves (i.e. \pm 5/2-moves) and rational moves can be analyzed using the group of Fox co…
We determine when there exists a ribbon rational homology cobordism between two connected sums of lens spaces, i.e. one without $3$-handles. In particular, we show that if a lens space $L$ admits a ribbon rational homology cobordism to a different lens space, then $L$ must be homeomorphic to $L(n,1)$, up to orientation-reversal. As an application, we classify ribbon $χ$-concordances between connected sums of $2$-bridge links. Our work builds on Lisca's work on embeddings of linear lattices.
We study the geometry of fully augmented link complements in $S^3$ by looking at their link diagrams. We extend the method introduced by Thistlethwaite and Tsvietkova to fully augmented links and define a system of algebraic equations in terms of parameters coming from edges and crossings of the link diagrams. Combining it with the work of Purcell, we show that the solutions to these algebraic equations are related to the cusp shapes of fully augmented link complements. As an application we use the cusp shapes to study the commensurability classes of fully augmented links.
New presentations of a link and a virtual link are introduced and algebraic systems on links and virtual links are constructed respectively. Based on the algebraic systems, Reduction Crossing Algorithms for them are proposed which are used to reduce the number of crossings in a link and virtual link. For known unknots, one can transform them into a trivial knot in a polynomial time by applying corresponding algorithm. As special consequences, Goeritz's unknot and Thistlethwaite's unknot are unknotted. Moreover, an infinite family of knots $K_{G_{2k,2l}}$ are unknotted in $O(n^2)$ time where $n…
Boileau and Rudolph called a link $L$ in the $3$-sphere a $\bf C$-boundary if it can be realized as the intersection of an algebraic curve $A$ in $\bf C^2$ with the boundary of a smooth embedded $4$-ball $B$. They showed that some links are not $\bf C$-boundaries. We say that $L$ is a strong $\bf C$-boundary if $A\setminus B$ is connected. In particular, all quasipositive links are strong $\bf C$-boundaries. In this paper we give examples of non-quasipositive strong $\bf C$-boundaries and non-strong $\bf C$-boundaries. We give a complete classification of (strong) $\bf C$-boundaries with at mo…
Let $B_n$ denote the classical braid group on $n$ strands and let the {\em mixed braid group} $B_{m,n}$ be the subgroup of $B_{m+n}$ comprising braids for which the first $m$ strands form the identity braid. Let $B_{m,\infty}=\cup_nB_{m,n}$. We will describe explicit algebraic moves on $B_{m,\infty}$ such that equivalence classes under these moves classify oriented links up to isotopy in a link complement or in a closed, connected, oriented 3--manifold. The moves depend on a fixed link representing the manifold in $S^3$. More precisely, for link complements the moves are: the two familiar move…
Given an $n$-component link $L$ in any 3-manifold $M$, the space $\mathcal{L} \subset (\mathbb{Q}\cup \mkern-1.5mu\{\infty\})^n$ of rational surgery slopes yielding L-spaces is already fully characterized (in joint work by the author) when $n\!=\!1$ and $\mathcal{L}$ is nontrivial. For $n\mkern-2mu>\mkern-3mu1$, however, there are no previous results for $\mathcal{L}$ as a rational subspace, and only limited results for integer surgeries $\mathcal{L}\cap\mathbb{Z}^n$ on $S^3\mkern-2mu$. Herein, we provide the first nontrivial explicit descriptions of $\mathcal{L}$ for rational surgeries on mul…
Closed 3-string braids admit many bandings to two-bridge links. By way of the Montesinos Trick, this allows us to construct infinite families of knots in the connected sum of lens spaces L(r,1) # L(s,1) that admit a surgery to a lens space for all pairs of integers (r,s) except (0,0). These knots are typically hyperbolic. We also demonstrate that the previously known two families of examples of hyperbolic knots in non-prime manifolds with lens space surgeries of Eudave-Munoz--Wu and Kang all fit this construction. As such, we propose a generalization of the Cabling Conjecture of Gonzales-Acuna…
This paper studies the configuration space of all possible positions of a linkage in R^n. For example, it shows that for every compact algebraic set, there is a linkage whose configuration space is analytically isomorphic to a finite number of copies of the algebraic set. If flexible edges are allowed, any compact set given by polynomial equalities and inequalities is the configuration space of a linkage. This paper also characterizes functions which can be computed by linkages.
It it known that the set of L-space surgeries on a nontrivial L-space knot is always bounded from below. However, already for two-component torus links the set of L-space surgeries might be unbounded from below. For algebraic two-component links we provide three complete characterizations for the boundedness from below: one in terms of the $h$-function, one in terms of the Alexander polynomial, and one in terms of the embedded resolution graph. They show that the set of L-space surgeries is bounded from below for most algebraic links. In fact, the used property of the $h$-function is a suffici…
In this paper we define and study flexible links and flexible isotopy in projective space. Flexible links are meant to capture the topological properties of real algebraic links. We classify all flexible links up to flexible isotopy using Ekholms interpretation of Viros encomplexed writhe.
We construct new topological invariants of three-dimensional manifolds which can, in particular, distinguish homotopy equivalent lens spaces L(7,1) and L(7,2). The invariants are built on the base of a classical (not quantum) solution of pentagon equation, i.e.algebraic relation corresponding to a ``2 tetrahedra to 3 tetrahedra'' local re-building of a manifold triangulation. This solution, found earlier by one of the authors, is expressed in terms of metric characteristics of Euclidean tetrahedra.
We show that if a braid $B$ can be parametrised in a certain way, then previous work can be extended to a construction of a polynomial $f:\mathbb{R}^4\to\mathbb{R}^2$ with the closure of $B$ as the link of an isolated singularity of $f$, showing that the closure of $B$ is real algebraic. In particular, we prove that closures of squares of strictly homogeneous braids and certain lemniscate links are real algebraic. We also show that the constructed polynomials satisfy the strong Milnor condition, providing an explicit fibration of the complement of the closure of $B$ over $S^1$.
We prove that, in order to derive the HOMFLYPT skein module of the lens spaces $L(p,1)$ from the HOMFLYPT skein module of the solid torus, $\mathcal{S}({\rm ST})$, it suffices to solve an infinite system of equations obtained by imposing on the Lambropoulou invariant $X$ for knots and links in the solid torus, braid band moves that are performed only on the first moving strand of elements in a set $Λ^{aug}$, augmenting the basis $Λ$ of $\mathcal{S}({\rm ST})$.
