Let $L$ be an oriented link with an alternating diagram $D$. It is known that $L$ is a fibered link if and only if the surface $R$ obtained by applying Seifert's algorithm to $D$ is a Hopf plumbing. Here, we call $R$ a Hopf plumbing if $R$ is obtained by successively plumbing finite number of Hopf bands to a disk.
In this paper, we discuss its extension so that we show the following theorem. Let $R$ be a Seifert surface obtained by applying Seifert's algorithm to an almost alternating diagrams. Then $R$ is a fiber surface if and only if $R$ is a Hopf plumbing.
We also show that the above theorem can not be extended to 2-almost alternating diagrams, that is, we give examples of 2-almost alternating diagrams for knots whose Seifert surface obtained by Seifert's algorithm are fiber surfaces that are not Hopf plumbing. This is shown by using a criterion of Melvin-Morton.
We give an algorithm for computing the Teichmüller polynomial for a certain class of fibered alternating links associated to trees. Furthermore, we exhibit a mutant pair of such links distinguished by the Teichmüller polynomial.
Alternating-sign Hopf plumbing along a tree yields fibered alternating links whose homological monodromy is, up to a sign, conjugate to some alternating-sign Coxeter transformation. Exploiting this tie, we obtain results about the location of zeros of the Alexander polynomial of the fibered link complement implying a strong case of Hoste's conjecture, the trapezoidal conjecture, bi-orderability of the link group, and a sharp lower bound for the homological dilatation of the monodromy of the fibration. The results extend to more general hyperbolic fibered 3-manifolds associated to alternating-s…
We give a complete classification of toroidal Seifert fibered surgeries on alternating knots. Precisely, we show that if an alternating knot admits a toroidal Seifert fibered surgery, then the knot is either the trefoil knot and the surgery slope is zero, or the connected sum of a (2,p)-torus knot and a (2,q)-torus knot and the surgery slope is 2(p+q) with |p|, |q| at least three.
There is one generalization of fibered links in 3-manifolds, called homologically fibered links. It is known that the existence of homologically fibered links whose fiber surface has a given homeomorphic type is determined by the first homology group and its torsion linking form of the ambient 3-manifold. In this paper, we interpret the existence of homologically fibered links with that of a solution of some equation, in terms of the first homology group and its torsion linking form or a surgery diagram of the ambient manifold. As an application, we compute the ${\rm hc}(\cdot)$, defined throu…
We show that all two-bridge knot and link complements are virtually fibered. We also show that spherical Montesinos knot and link complements are virtually fibered. This is accomplished by showing that such knot complements are finitely covered by great circle link complements.
Every $L$-space knot is fibered and strongly quasi-positive, but this does not hold for $L$-space links. In this paper, we use the so called H-function, which is a concordance link invariant, to introduce a subfamily of fibered strongly quasi-positive $L$-space links. Furthermore, we present an infinite family of $L$-space links which are not quasi-positive.
We introduce generalized arrow diagrams and generalized Reidemeister moves for diagrams of links in Seifert fibered spaces. We give a presentation of the fundamental group of the link complement. As a corollary we are able to compute the first homology group of the complement and the twisted Alexander polynomials of the 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 $p \leq -1$.
We study compatible contact structures of fibered Seifert multilinks in homology 3-spheres and especially give a necessary and sufficient condition for the contact structure to be tight in the case where the Seifert fibration is positively twisted. As a corollary we determine the strongly quasipositivity of fibered Seifert links in the 3-sphere. We also study the compatible contact structures of cablings along links in any 3-manifolds.
We characterize cutting arcs on fiber surfaces that produce new fiber surfaces, and the changes in monodromy resulting from such cuts. As a corollary, we characterize band surgeries between fibered links and introduce an operation called Generalized Hopf banding. We further characterize generalized crossing changes between fibered links, and the resulting changes in monodromy.
From a fibered link in the 3-sphere may be constructed a field of not everywhere tangent 2-planes; when the fibered link is the link of an isolated critical point of a map from 4-space to the plane, the plane field is essentially the field of kernels of the derivative of the map. Homotopically, such a plane field determines two integers. I show that the sum of these integers is the Milnor number of the fibered link. Taking the mirror image of a link exchanges the integers. Various examples are computed. It is noted (proof given elsewhere) that these integers are not determined by the algebraic…
Let F be R or C, d the dimension of F over R. Denote by P(F) either the affine plane A(F) or the hyperbolic plane H(F) over F. An arrangement L of k lines in P(F) (pairwise non-parallel in the hyperbolic case) has a link at infinity K(L) comprising k unknotted (d-1)-spheres in the (2d-1)-sphere, whose topology reflects the combinatorics of L `at infinity'. The class of links at infinity of affine F-line arrangements is properly included in the class of links at infinity of hyperbolic F-line arrangements. Many links at infinity of (essentially non-affine) connected hyperbolic C-line arrangement…
A fibered concordance of knots, introduced by Harer, is a concordance between fibered knots that is well-behaved with respect to the fibrations. We consider semi-fibered concordance of two component ordered links $L=J \sqcup K$ with $J$ fibered. These are concordances that restrict to fibered concordances on the first component. Motivated by some examples of Gompf-Scharlemann-Thompson, we further limit our attention to those links $L$ where $K$ is "close to" a fiber of $J$. Such $L$ are studied with virtual covers, where a virtual knot $\upsilon$ is associated to $L$. We show that the concorda…
Let $\barγ$ be a link in a Seifert fibered space $M$ over a hyperbolic $2$-orbifolds $\mathcal O$ that projects injectively to a filling multicurve of closed geodesics $γ$ in $\mathcal O.$ We prove that the complement $M_{\barγ}$ of $\barγ$ in $M$ admits a hyperbolic structure of finite volume and give combinatorial bounds of its volume.
We determine the Thurston unit ball of a family of $n$-chained link, denoted by $C(n,p)$, where $n$ is the number of link components and $p$ is the number of twists. When $p$ is strictly positive, we prove that the Thurston unit ball for $C(n,p)$ is an $n$-dimensional cocube, for arbitrary $n$. Moreover, we clarify the condition for which $C(n,p)$ is fibered and find at least one fibered face for any $p$. Finally we provide the Teichmüller polynomial for the face of Thurston unit ball of $C(n, -2)$ with $n\geq 3$.
We give a new upper bound on the maximum degree of the Jones polynomial of a fibered positive link. In particular, we prove that the maximum degree of the Jones polynomial of a fibered positive knot is at most four times the minimum degree. Using this result, we can complete the classification of all knots of crossing number $\leq 12$ as positive or not positive, by showing that the seven remaining knots for which positivity was unknown are not positive. That classification was also done independently at around the same time by Stoimenow.
It is known that every closed oriented 3-manifold is homology cobordant to a hyperbolic 3-manifold. By contrast we show that many homology cobordism classes contain no Seifert fibered 3-manifold. This is accomplished by determining the isomorphism type of the rational cohomology ring of all Seifert fibered 3-manifolds with no 2-torsion in their first homology. Then we exhibit families of examples of 3-manifolds (obtained by surgery on links), with fixed linking form and cohomology ring, that are not homology cobordant to any Seifert fibered space (as shown by their rational cohomology ring). T…
We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the original knot. We classify those generalized augmented links that are Seifert fibered, and give a torus decomposition for those that are toroidal. In particular, we find that each component of the torus decomposition is either "trivial", in some sense, or homeomorphic to the complement of a generalized augmented link. We show this structure persists under high D…
If there are any 2-component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions. The simplest plausible counterexample to the Generalized Property R Conjecture could be a 2-component link containing the square knot. We characterize all two-component links that contain the square knot and which surger to (S^1 x S^2) # (S^1 x S^2). We exhibit a family of such links that are probably counterexamples to General…
The aim of this article is to detect new classes of quasi-alternating links. Quasi-alternating links are a natural generalization of alternating links. Their knot Floer and Khovanov homology are particularly easy to compute. Since knot Floer homology detects the genus of a knot as well as whether a knot is fibered, as provided bounds on unknotting number and slice genus, characterization of quasi-alternating links becomes an interesting open problem. We show that there exist classes of non-alternating Montesinos links, which are quasi-alternating.
The open book decompositions of the 3-sphere whose pages are pairs of pants have been fully understood for some time, through the lens of contact geometry. The purpose of this note is to exhibit a purely topological derivation of the classification of such open books, in terms of the links that form their bindings and the corresponding monodromies. We construct all of the links and their pair-of-pants fiber surfaces from the simplest example, a connected sum of two Hopf links, through performing (generalized) Stallings twists. Then, by applying the now-classical theory of genus two Heegaard di…
We use the Birman-Ko-Lee presentation of the braid group to show that all closures of strongly quasipositive braids whose normal form contains a positive power of the dual Garside element $δ$ are fibered. We classify links which admit such a braid representative in geometric terms as boundaries of plumbings of positive Hopf bands to a disk. Rudolph constructed fibered strongly quasipositive links as closures of positive words on certain generating sets of $B_n$ and we prove that Rudolph's condition is equivalent to ours. Finally, we show that the braid index is a strict upper bound for the num…
Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level $r$). In this paper, we relate the AMU conjecture to a question about the growth of the Turaev-Viro invariants $TV_r$ of hyperbolic 3-manifolds. We show that if the $r$-growth of $|TV_r(M)|$ for a hyperbolic 3-manifold $M$ that fibers over the circle is exponential, then the monodromy of the fibration of $M$ satisfies the AMU conjecture. Building on earlier work \cite{DK} we give broad constru…
We use virtual knot theory to detect the non-invertibility of some classical links in $\mathbb{S}^3$. These links appear in the study of virtual covers. Briefly, a virtual cover associates a virtual knot $\upsilon$ to a knot $K$ in a $3$-manifold $N$, under certain hypotheses on $K$ and $N$. Virtual covers of links in $\mathbb{S}^3$ come from taking $K$ to be in the complement $N$ of a fibered link $J$. If $J \sqcup K$ is invertible and $K$ is "close to" a fiber of $J$, then $\upsilon$ satisfies a symmetry condition to which some virtual knot polynomials are sensitive. We also discuss virtual …
We show that quasi-alternating links arise naturally when considering surgery on a strongly invertible L-space knot (that is, a knot that yields an L-space for some Dehn surgery). In particular, we show that for many known classes of L-space knots, every sufficiently large surgery may be realized as the two-fold branched cover of a quasi-alternating link. Consequently, there is considerable overlap between L-spaces obtained by surgery on $S^3$, and L-spaces resulting as two-fold branched covers of quasi-alternating links. By adapting this approach to certain Seifert fibered spaces, it is possi…
Given a fibered link, consider the characteristic polynomial of the monodromy restricted to first homology. This generalizes the notion of the Alexander polynomial of a knot. We define a construction, called iterated plumbing, to create a sequence of fibered links from a given one. The resulting sequence of characteristic polynomials has the same form as those arising in work of Salem and Boyd in their study of distributions of Salem and P-V numbers. From this we deduce information about the asymptotic behavior of the large roots of the generalized Alexander polynomials, and define a new poset…
Rationally null-homologous links in Seifert fibered spaces may be represented combinatorially via labeled diagrams. We introduce an additional condition on a labeled link diagram and prove that it is equivalent to the existence of a rational Seifert surface for the link. In the case when this condition is satisfied, we generalize Seifert's algorithm to explicitly construct a rational Seifert surface for any rationally null-homologous knot. As an application of the techniques developed in the paper, we derive closed formulae for the rational Thurston-Bennequin and rotation numbers of a Legendri…
Associated to every state surface for a knot or link is a state graph, which embeds as a spine of the state surface. A state graph can be decomposed along cut-vertices into graphs with induced planar embeddings. Associated with each such planar graph is a checkerboard surface, and each state surface is a fiber if and only if all of its associated checkerboard surfaces are fibers. We give an algebraic condition that characterizes which checkerboard surfaces are fibers directly from their state graphs. We use this to classify fibering of checkerboard surfaces for several families of planar graph…
Geometric interpretations of some virtual knot invariants are given in terms of invariants of links in $\mathbb{S}^3$. Alexander polynomials of almost classical knots are shown to be specializations of the multi-variable Alexander polynomial of certain two-component boundary links of the form $J \sqcup K$ with $J$ a fibered knot. The index of a crossing, a common ingredient in the construction of virtual knot invariants, is related to the Milnor triple linking number of certain three-component links $J \sqcup K_1 \sqcup K_2$ with $J$ a connected sum of trefoils or figure-eights. Our main techn…
Using an obstruction based on Donaldson's theorem, we derive strong restrictions on when a Seifert fibered space $Y = F(e; \frac{p_1}{q_1}, \ldots, \frac{p_k}{q_k})$ over an orientable base surface $F$ can smoothly embed in $S^4$. This allows us to classify precisely when $Y$ smoothly embeds provided $e > k/2$, where $e$ is the normalized central weight and $k$ is the number of singular fibers. Based on these results and an analysis of the Neumann-Siebenmann invariant $\overlineμ$, we make some conjectures concerning Seifert fibered spaces which embed in $S^4$. Finally, we also provide some ap…
We prove that a nicely fibered link (by which we mean the binding of an open book) in a tight contact manifold $(M,ξ)$ with zero Giroux torsion has a transverse representative realizing the Bennequin bound if and only if the contact structure it supports (since it is also the binding of an open book) is $ξ.$ This gives a geometric reason for the non-sharpness of the Bennequin bound for fibered links. We also note that this allows the classification, up to contactomorphism, of maximal self-linking number links in these knot types. Moreover, in the standard tight contact structure on $S^3$ we cl…
We construct an infinite family of quasi-alternating links from a given quasi-alternating link by replacing a crossing by a product of rational tangles each of which extends that crossing. Consequently, we determine an infinite family of quasi-alternating Montesinos links. This family contains all the classes of quasi-alternating Montesinos links that have been detected by Widmar in \cite{W}. We conjecture that this family contains all quasi-alternating Montesinos links up to mirror image that are not alternating and this will characterize all quasi-alternating Montesinos links.
We prove that twisting any quasi-alternating link $L$ with no gaps in its Jones polynomial $V_L(t)$ at the crossing where it is quasi-alternating produces a link $L^{*}$ with no gaps in its Jones polynomial $V_{L^*}(t)$. This leads us to conjecture that the Jones polynomial of any prime quasi-alternating link, other than $(2,n)$-torus links, has no gaps. This would give a new property of quasi-alternating links and a simple obstruction criterion for a link to be quasi-alternating. We prove that the conjecture holds for quasi-alternating Montesinos links as well as quasi-alternating links with …
An alternating distance is a link invariant that measures how far away a link is from alternating. We study several alternating distances and demonstrate that there exist families of links for which the difference between certain alternating distances is arbitrarily large. We also show that two alternating distances, the alternation number and the alternating genus, are not comparable.
Champanerkar and Kofman introduced an interesting way to construct new examples of quasi-alternating links from existing ones. Actually, they proved that replacing a quasi-alternating crossing c in a quasi-alternating link by a rational tangle of same type yields a new quasi-alternating link. This construction has been extended to alternating algebraic tangles and applied to characterize all quasi-alternating Montesinos links. In this paper, we extend this technique to any alternating tangle of same type as c. As an application, we give new examples of quasi-alternating knots of 13 and 14 cros…
Quasi-alternating links are homologically thin for both Khovanov homology and knot Floer homology. We show that every quasi-alternating link gives rise to an infinite family of quasi-alternating links obtained by replacing a crossing with an alternating rational tangle. Consequently, we show that many pretzel links are quasi-alternating, and we determine the thickness of Khovanov homology for "most" pretzel links with arbitrarily many strands.
We construct an infinite family of links which are both almost alternating and quasi-alternating from a given either almost alternating diagram representing a quasi-alternating link, or connected and reduced alternating tangle diagram. To do that we use what we call a dealternator extension which consists in replacing the dealternator by a rational tangle extending it. We note that all not alternating and quasi-alternating Montesinos links can be obtained in that way. We check that all the obtained quasi-alternating links satisfy Conjecture 3.1 of Qazaqzeh et al. (JKTR 22 (06), 2013), that is …
In "Tunnel one, fibered links", the second author showed that the tunnel of a tunnel number one, fibered link can be isotoped to lie as a properly embedded arc in the fiber surface of the link. In this paper, we analyze how the arc behaves under the monodromy action, and show that the tunnel arc is nearly clean, with the possible exception of twisting around the boundary of the fiber.
We show that one can interweave an unknot into any non-alternating connected projection of a link so that the resulting augmented projection is alternating.
We give a sufficient condition for an almost alternating link diagram to represent a non-splittable link. The main theorem gives us a way to see if a given almost alternating link diagram represents a splittable link without increasing numbers of crossings of diagrams in the process. Moreover, we show that almost alternating links with more than two components are non-trivial.
Menasco showed that a closed surface in the complement of a non-split prime alternating link in $S^3$ contains a circle isotopic in the link complement to a meridian of the links. This result is known as the meridian lemma for alternating links. We give a meridian lemma for the class of fully alternating links in the thickened orientable surfaces of positive genus.
We present computational results about quasi-alternating knots and links and odd homology obtained by looking at link families in the Conway notation. More precisely, we list quasi-alternating links up to 12 crossings and the first examples of quasi-alternating knots and links with at least two different minimal diagrams, where one is quasi-alternating and the other is not. We provide examples of knots and links with $n\le 12$ crossings which are homologically thin and have no minimal quasi-alternating diagrams. These links are candidates for homologically thin links that are not quasi-alterna…
A link is almost alternating if it is non-alternating and has a diagram that can be transformed into an alternating diagram via one crossing change. We give formulas for the first two and last two potential coefficients of the Jones polynomial of an almost alternating link. Using these formulas, we show that the Jones polynomial of an almost alternating link is nontrivial. We also show that either the first two or last two coefficients of the Jones polynomial of an almost alternating link alternate in sign. Finally, we describe conditions that ensure an almost alternating diagram has the fewes…
The aim of this article is to give a characterization of strongly quasipositive quasi-alternating links and detect new classes of strongly quasipositive Montesinos links and non-strongly quasipositive Montesinos links. In this direction, we show that, if $L$ is an oriented quasi-alternating link with a quasi-alternating crossing $c$ such that $L_0$ is alternating (where $L_0$ has the induced orientation), then $L$ is definite if and only if it is strongly quasipositive (up to mirroring). We also show that if $L$ is an oriented quasi-alternating link with a quasi-alternating crossing $c$ such t…
We extend some classical results of Bankwitz, Crowell, and Murasugi to the setting of virtual links. For instance, we show that an alternating virtual link is split if and only if it is visibly split, and that the Alexander polynomial of any almost classical alternating virtual link is alternating. The first result is a consequence of an inequality relating the link determinant and crossing number for any non-split alternating virtual link. The second is a consequence of the matrix-tree theorem of Bott and Mayberry. We extend the first result to semi-alternating virtual links. We discuss the T…
A knot K in the 3-sphere is said to have Property nR if, whenever K is a component of an n-component link L and some integral surgery on L produces the connected sum of n copies of S^1 x S^2, there is a sequence of handle slides on L that converts L into a 0-framed unlink. The Generalized Property R Conjecture is that all knots have Property nR for all n. The simplest plausible counterexample could be the square knot. Exploiting the remarkable symmetry of the square knot, we characterize all two-component links that contain it and which surger to S^1 x S^2 # S^1 x S^2. We argue that at least o…
We prove that there are only finitely many values of the Jones polynomial of quasi-alternating links of a given determinant. Consequently, we prove that there are only finitely many quasi-alternating links of a given Jones polynomial iff there are only finitely many quasi-alternating links of a given determinant.
