Alternating Augmentations of Links

Extending Quasi-Alternating Links

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…

Twisting quasi-alternating links

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.

Generating links that are both quasi-alternating and almost alternating

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 …

A criterion for almost alternating links to be non-splittable

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.

A meridian lemma for fully alternating links in thickened surfaces

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.

Quasi-alternating links and odd homology: computations and conjectures

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…

The Jones polynomial of an almost alternating link

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…

Strongly quasipositive quasi-alternating links and Montesinos links

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…

Classical results for alternating virtual links

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…

Jones Polynomial versus Determinant of Quasi-Alternating Links

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.

Generalized augmented alternating links and hyperbolic volume

Augmented alternating links are links obtained by adding trivial components that bound twice-punctured disks to non-split reduced non-2-braid prime alternating projections. These links are known to be hyperbolic. Here, we extend to show that generalized augmented alternating links, which allow for new trivial components that bound n-punctured disks, are also hyperbolic. As an application we consider generalized belted sums of links and compute their volumes.

Involutions of alternating links

Let $L$ be an alternating prime non-split link in $S^3$. We use the category of flypes between reduced alternating diagrams for $L$ to classify involutions on $L$. As consequences, we show that the quotient of an alternating periodic link is alternating, and that all freely 2-periodic alternating links have an even number of components.

Homologically thin, non-quasi-alternating links

We exhibit the first examples of links which are homologically thin but not quasi-alternating. To show that they are not quasi-alternating, we argue that none of their branched double-covers bounds a negative definite 4-manifold with non-torsion H_1. Using this method, we also complete the determination of the quasi-alternating pretzel links.

Quasi-alternating links and $Q$-polynomials

Qazaqzeh and Chbili showed that for any quasi-alternating link, the degree of $Q$-polynomial is less than its determinant. We give a refinement of their evaluation.

A note on quasi-alternating Montesinos links

Quasi-alternating links are a generalization of alternating links. They are homologically thin for both Khovanov homology and knot Floer homology. Recent work of Greene and joint work of the first author with Kofman resulted in the classification of quasi-alternating pretzel links in terms of their integer tassel parameters. Replacing tassels by rational tangles generalizes pretzel links to Montesinos links. In this paper we establish conditions on the rational parameters of a Montesinos link to be quasi-alternating. Using recent results on left-orderable groups and Heegaard Floer L-spaces, we…

Alternating links and definite surfaces

We establish a characterization of alternating links in terms of definite spanning surfaces. We apply it to obtain a new proof of Tait's conjecture that reduced alternating diagrams of the same link have the same crossing number and writhe. We also deduce a result of Banks and Hirasawa-Sakuma about Seifert surfaces for special alternating links. The appendix, written by Juhász and Lackenby, applies the characterization to derive an exponential time algorithm for alternating knot recognition.

A characterization of alternating link exteriors in terms of cubed complexes

We give a characterization of alternating link exteriors in terms of cubed complexes. To this end, we introduce the concept of a "signed BW cubed-complex", and give a characterization for a signed BW cubed-complex to have the underlying space which is homeomorphic to an alternating link exterior.

A new obstruction of quasi-alternating links

We prove that the degree of the Brandt-Lickorish-Millet polynomial of any quasi-alternating link is less than its determinant. Therefore, we obtain a new and a simple obstruction criterion for quasi-alternateness. As an application, we identify some knots of 12 crossings or less and some links of 9 crossings or less that are not quasi-alternating. Also, we show that there are only finitely many Kanenobu knots which are quasi-alternating. This last result supports Conjecture 3.1 of Greene in [10] which states that there are only finitely many quasi-alternating links with a given determinant. Mo…

扩展拟交错链环 II

Extending Quasi-alternating Links II

本文探讨了通过替换交叉点构造拟交错链环的方法，给出了该方法对相反类型交错缠结有效的条件，并建立了有效的非交错缠结族。

Teichmüller polynomials of fibered alternating links

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.

关于拟交错链环的Jones多项式，II

On the Jones polynomial of quasi-alternating links, II

将Thistlethwaite关于交错链环Jones多项式结构的结果推广至拟交错链环，证明了非$(2,n)$-环面链环的素拟交错链环的Jones多项式及Khovanov同调差分分级无间隙，行列式为多项式宽度上界，并刻画了非素情形有多于一个间隙的条件。

Geometry of alternating links on surfaces

We consider links that are alternating on surfaces embedded in a compact 3-manifold. We show that under mild restrictions, the complement of the link decomposes into simpler pieces, generalising the polyhedral decomposition of alternating links of Menasco. We use this to prove various facts about the hyperbolic geometry of generalisations of alternating links, including weakly generalised alternating links described by the first author. We give diagrammatical properties that determine when such links are hyperbolic, find the geometry of their checkerboard surfaces, bound volume, and exclude ex…

A new proof that alternating links are non-trivial

We use a simple geometric argument and small cancellation properties of link groups to prove that alternating links are non-trivial. This proof uses only classic results in topology and combinatorial group theory.

交错链环补空间的Tangle分解

Tangle decompositions of alternating link complements

本研究聚焦素交错链环能否分解为两个素交错tangle的问题，改进已有结果证明该分解要么在交错链环图中可见，要么对应伪Montesinos链环，为交错链环结构研究提供新结论。

A surgical perspective on quasi-alternating links

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…

Right-angled polyhedra and alternating links

To any prime alternating link, we associate a collection of hyperbolic right-angled ideal polyhedra by relating geometric, topological and combinatorial methods to decompose the link complement. The sum of the hyperbolic volumes of these polyhedra is a new geometric link invariant, which we call the right-angled volume of the alternating link. We give an explicit procedure to compute the right-angled volume from any alternating link diagram, and prove that it is a new lower bound for the hyperbolic volume of the link.

The classification of quasi-alternating Montesinos links

In this note, we complete the classification of quasi-alternating Montesinos links. We show that the quasi-alternating Montesinos links are precisely those identified independently by Qazaqzeh-Chbili-Qublan and Champanerkar-Ording. A consequence of our proof is that a Montesinos link $L$ is quasi-alternating if and only if its double branched cover is an L-space, and bounds both a positive definite and a negative definite 4-manifold with vanishing first homology.

Alternating links with totally geodesic checkerboard surfaces

We prove that alternating links with two totally geodesic checkerboard surfaces are three links with projection the 1-skeleton of the octahedron, the cuboctahedron and the icosidodecahedron. We also characterize these links as right-angled completely realisable links and show that all hyperbolic weaving knots with two exceptions have both checkerboard surfaces not totally geodesic.

A note on amphicheiral alternating links

After reconsidering the Dasbach-Hougardy counterexample to the Kauffman Conjecture on alternating knots, we reformulate the conjecture and consider Dasbach-Hougardy counterexample and similar counterexamples in the light of the reformulated conjecture.

On Coxeter mapping classes and fibered alternating links

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…

关于交错拟正纽结

On alternating quasipositive links

本文证明满足特定条件（无一对赛弗特圆仅由单个交叉连接）的交错拟正纽结图必为正图，且纽结是强拟正的，明确了交错拟正纽结的结构性质。

A Note on Alternating Links and Root Polytopes

In this paper, a relationship between the determinant of an alternating link and a certain polytope obtained from the link diagram is analyzed. We also show that when the underlying graph of the link diagram is properly oriented, the number of its spanning arborescence is equal to the determinant, i.e., the value at -1 of the Jones polynomial.

Quasi-alternating links with small determinant

Quasi-alternating links of determinant 1, 2, 3, and 5 were previously classified by Greene and Teragaito, who showed that the only such links are two-bridge. In this paper, we extend this result by showing that all quasi-alternating links of determinant at most 7 are connected sums of two-bridge links, which is optimal since there are quasi-alternating links not of this form for all larger determinants. We achieve this by studying their branched double covers and characterizing distance-one surgeries between lens spaces of small order, leading to a classification of formal L-spaces with order …

Geometry of biperiodic alternating links

A biperiodic alternating link has an alternating quotient link in the thickened torus. In this paper, we focus on semi-regular links, a class of biperiodic alternating links whose hyperbolic structure can be immediately determined from a corresponding Euclidean tiling. Consequently, we determine the exact volumes of semi-regular links. We relate their commensurability and arithmeticity to the corresponding tiling, and assuming a conjecture of Milnor, we show there exist infinitely many pairwise incommensurable semi-regular links with the same invariant trace field. We show that only two semi-r…

Alternating links have representativity 2

We prove that if $L$ is a non-trivial alternating link embedded (without crossings) in a closed surface $F\subset S^3$, then $F$ has a compressing disk whose boundary intersects $L$ in no more than two points. Moreover, whenever the surface is incompressible and $\partial$-incompressible in the link exterior, it can be isotoped to have a standard tube at some crossing of any reduced alternating diagram.

A New property of quasi-alternating links

We show that the crossing number of any link that is known to be quasi-alternating is less than or equal to its determinant. Based on this, we conjecture that the crossing number of any quasi-alternating link is less than or equal to its determinant. Thus, if this conjecture is proved then it would give an easier obstruction for quasi-alternateness than the ones already known.

双组分交替链接的生成

Genera of two-component alternating links

本文研究双组分交替链接的生成数，提出一个统一量描述定向和非定向生成数，并通过splice序列和修正项确定其性质。

Homological Polynomial Coefficients and the Twist Number of Alternating Surface Links

For $D$ a reduced alternating surface link diagram, we bound the twist number of $D$ in terms of the coefficients of a polynomial invariant. To this end, we introduce a generalization of the homological Kauffman bracket defined by Krushkal. Combined with work of Futer, Kalfagianni, and Purcell, this yields a bound for the hyperbolic volume of a class of alternating surface links in terms of these coefficients.

Determinant density and biperiodic alternating links

Let L be any infinite biperiodic alternating link. We show that for any sequence of finite links that Folner converges almost everywhere to L, their determinant densities converge to the Mahler measure of the 2-variable characteristic polynomial of the toroidal dimer model on an associated biperiodic graph.

加厚曲面中的广义增广胞腔交错链环是双曲的

Generalized Augmented Cellular Alternating Links in Thickened Surfaces are Hyperbolic

本文将广义增广交错链环的双曲性推广到闭曲面 I-丛中的胞腔交错链环，并证明相关链环族在几何拓扑中构成有限体积双曲3-流形的闭子集。

The Jones polynomial and the planar algebra of alternating links

It is a well known result from Thistlethwaite that the Jones polynomial of a non-split alternating link is alternating. We find the right generalization of this result to the case of non-split alternating tangles. More specifically: the Jones polynomial of tangles is valued in a certain skein module, we describe an alternating condition on elements of this skein module, show that it is satisfied by the Jones invariant of the single crossing tangles, and prove that it is preserved by appropriately "alternating" planar algebra compositions. Hence, this condition is satisfied by the Jones polynom…

Essential twisted surfaces in alternating link complements

Checkerboard surfaces in alternating link complements are used frequently to determine information about the link. However, when many crossings are added to a single twist region of a link diagram, the geometry of the link complement stabilizes (approaches a geometric limit), but a corresponding checkerboard surface increases in complexity with crossing number. In this paper, we generalize checkerboard surfaces to certain immersed surfaces, called twisted checkerboard surfaces, whose geometry better reflects that of the alternating link in many cases. We describe the surfaces, show that they a…

An application of non-positively curved cubings of alternating links

By using non-positively curved cubings of prime alternating link exteriors, we prove that certain ideal triangulations of their complements, derived from reduced alternating diagrams, are non-degenerate, in the sense that none of the edges is homotopic relative its endpoints to a peripheral arc. This guarantees that the hyperbolicity equations for those triangulations for hyperbolic alternating links have solutions corresponding to the complete hyperbolic structures. Since the ideal triangulations considered in this paper are often used in the study of the volume conjecture, this result has a …

非分裂交替链接唯一地界定Seifert表面

Non-split, alternating links bound unique Seifert surfaces in the 4-ball

该研究证明非分裂交替链接的Seifert表面在推入4球体后成为光滑同胚，且粘合后的表面总是光滑无结。

A refined upper bound for the hyperbolic volume of alternating links and the colored Jones polynomial

We give a refined upper bound for the hyperbolic volume of an alternating link in terms of the first three and the last three coefficients of its colored Jones polynomial.

高扭转交替链环的 Vol-Det 猜想

The Vol-Det Conjecture for highly twisted alternating links

本文研究高扭转交替链环的 Vol-Det 猜想，改进了 Burton 关于猜想成立交叉数范围的界，并强化了 Stoimenow 关于双曲体积与行列式的不等式。

The Determinant and Volume of 2-Bridge Links and Alternating 3-Braids

We examine the conjecture, due to Champanerkar, Kofman, and Purcell that $\text{vol}(K) < 2 π\log \det (K)$ for alternating hyperbolic links, where $\text{vol}(K) = \text{vol}(S^3\backslash K)$ is the hyperbolic volume and $\det(K)$ is the determinant of $K$. We prove that the conjecture holds for 2-bridge links, alternating 3-braids, and various other infinite families. We show the conjecture holds for highly twisted links and quantify this by showing the conjecture holds when the crossing number of $K$ exceeds some function of the twist number of $K$.

交替扭结链环的枚举

The Enumeration of Alternating Pretzel Links

本文研究交替扭结链环的计数问题，给出按交叉数 $c$ 计算其总数 $\mathcal{P}(c)$ 的闭式公式，并通过数值计算揭示其指数增长规律。

Constructing subdivision rules from alternating links

The study of geometric group theory has suggested several theorems related to subdivision tilings that have a natural hyperbolic structure. However, few examples exist. We construct subdivision tilings for the complement of every nonsingular, prime alternating link. These tilings define a combinatorial space at infinity, similar to the space at infinity for word hyperbolic groups.

交替链的带长新上界

New Upper Bounds on the Ribbonlength of Alternating Links with Bipartite Dual Graphs

本文证明了具有二分图双图的交替链的带长上界，并确定了霍普夫链的精确带长。

Span of the Jones polynomial of an alternating virtual link

For an oriented virtual link, L.H. Kauffman defined the f-polynomial (Jones polynomial). The supporting genus of a virtual link diagram is the minimal genus of a surface in which the diagram can be embedded. In this paper we show that the span of the f-polynomial of an alternating virtual link L is determined by the number of crossings of any alternating diagram of L and the supporting genus of the diagram. It is a generalization of Kauffman-Murasugi-Thistlethwaite's theorem. We also prove a similar result for a virtual link diagram that is obtained from an alternating virtual link diagram by …

On Khovanov Homology of Quasi-Alternating Links

We prove that the length of any gap in the differential grading of the Khovanov homology of any quasi-alternating link is one. As a consequence, we obtain that the length of any gap in the Jones polynomial of any such link is one. This establishes a weaker version of Conjecture 2.3 in [5]. Moreover, we obtain a lower bound for the determinant of any such link in terms of the breadth of its Jones polynomial. This establishes a weaker version of Conjecture 3.8 in [17]. The main tool in obtaining this result is establishing the Knight Move Conjecture [2,Conjecture 1] for the class of quasi-altern…

The Braid Index Of Reduced Alternating Links

It is well known that the minimum crossing number of an alternating link equals the number of crossings in any reduced alternating link diagram of the link. This remarkable result is an application of the Jones polynomial. In the case of the braid index of an alternating link, Murasugi had conjectured that the number of Seifert circles in a reduced alternating diagram of the link equals the braid index of the link. This conjecture turned out to be false. In this paper we prove the next best thing that one could hope for: we characterize exactly those alternating links for which their braid ind…

The number of surfaces of fixed genus in an alternating link complement

Let $L$ be a prime alternating link with $n$ crossings. We show that for each fixed $g$, the number of genus $g$ incompressible surfaces in the complement of $L$ is bounded by a polynomial in $n$. Previous bounds were exponential in $n$.