Family Of Knots Research Articles

Torsion in the knot concordance group and cabling

We define a nontrivial modulo 2 valued additive concordance invariant defined on the torsion subgroup of the knot concordance group using involutive knot Floer package. For knots not contained in its kernel, we prove that their iterated (\mathrm{odd},1) -cables have infinite order in the concordance group and, among them, infinitely many are linearly independent. Furthermore, by taking (2,1) -cables of the aforementioned knots, we present an infinite family of knots which are strongly rationally slice but not slice.

Non-Orientable Lagrangian Fillings of Legendrian Knots

AbstractWe investigate when a Legendrian knot in the standard contact ${{\mathbb{R}}}^3$ has a non-orientable exact Lagrangian filling. We prove analogs of several results in the orientable setting, develop new combinatorial obstructions to fillability, and determine when several families of knots have such fillings. In particular, we completely determine when an alternating knot (and more generally a plus-adequate knot) is decomposably non-orientably fillable and classify the fillability of most torus and 3-strand pretzel knots. We also describe rigidity phenomena of decomposable non-orientable fillings, including finiteness of the possible normal Euler numbers of fillings and the minimisation of crosscap numbers of fillings, obtaining results which contrast in interesting ways with the smooth setting.

Constructing knots with specified geometric limits

It is known that any tame hyperbolic 3-manifold with infinite volume and a single end is the geometric limit of a sequence of finite volume hyperbolic knot complements. Purcell and Souto showed that if the original manifold embeds in the 3-sphere, then such knots can be taken to lie in the 3-sphere. However, their proof was nonconstructive; no examples were produced. In this paper, we give a constructive proof in the geometrically finite case. That is, given a geometrically finite, tame hyperbolic 3-manifold with one end, we build an explicit family of knots whose complements converge to it geometrically. Our knots lie in the (topological) double of the original manifold. The construction generalises the class of fully augmented links to a Kleinian groups setting.

The Gordian complexes of knots given by 4-move

The Gordian complex of knots is a simplicial complex whose vertices consist of all knot types in [Formula: see text]. Local moves play an important role in defining knot invariants. There are many local moves known as unknotting operations for knots. In this paper, we discuss the 4-move operation. We show that for any knot [Formula: see text] and for any given natural number [Formula: see text], there exists a family of knots [Formula: see text] such that for any pair [Formula: see text] of distinct elements of the family, the Gordian distance of knots by 4-move is [Formula: see text]. We also show the existence of an arbitrarily high dimensional simplex in the Gordian complexes.

The cosmetic crossing conjecture for split links

Given a band sum of a split two-component link along a nontrivial band, we obtain a family of knots indexed by the integers by adding any number of full twists to the band. We show that the knots in this family have the same Heegaard knot Floer homology and the same instanton knot Floer homology. In contrast, a generalization of the cosmetic crossing conjecture predicts that the knots in this family are all distinct. We verify this prediction by showing that any two knots in this family have distinct Khovanov homology. Along the way, we prove that each of the three knot homologies detects the trivial band.

Evolution for Khovanov polynomials for figure-eight-like family of knots

We look at how evolution method deforms, when one considers Khovanov polynomials instead of Jones polynomials. We do this for the figure-eight-like knots (also known as ’double braid’ knots, see arXiv:1306.3197) — a two-parametric family of knots which “grows” from the figure-eight knot and contains both two-strand torus knots and twist knots. We prove that parameter space splits into four chambers, each with its own evolution, and two isolated points. Remarkably, the evolution in the Khovanov case features an extra eigenvalue, which drops out in the Jones [Formula: see text] limit.

Knot reversal and rational concordance

AbstractWe give an infinite family of knots that are not rationally concordant to their reverses. More precisely, if denotes the involution of the rational knot concordance group induced by reversal and denotes the subgroup of knots fixed under in , then contains an infinite rank subgroup. As a corollary, we show that there exists a knot such that for every pair of coprime integers and , the ‐cable of is not concordant to the reverse of the ‐cable of .

Colored Jones polynomials without tails

We exhibit an infinite family of knots with the property that the first coefficient of the n-colored Jones polynomial grows linearly with n. This shows that the concept of stability and tail seen in the colored Jones polynomials of alternating knots does not generalize naively.

The tightest knot is not necessarily the smallest

In this paper, we attempt to find counterexamples to the conjecture that the ideal form of a knot, that which minimizes its contour length while respecting a no-overlap constraint, also minimizes the volume of the knot, as determined by its convex hull. We measure the convex hull volume of knots during the length annealing process, identifying local minima in the hull volume that arise due to buckling and symmetry breaking. We use [Formula: see text] torus knots as an illustrative example of a family of knots whose locally minimal-length embeddings are not necessarily ordered by volume. We identify several knots whose central curve has a convex hull volume that is not minimized in the ideal configuration, and find that [Formula: see text] has a non-ideal global minimum in its convex hull volume even when the thickness of its tube is taken into account.

A lower bound on the stable 4–genus of knots

We present a lower bound on the stable $4$-genus of a knot based on Casson-Gordon $\tau$-signatures. We compute the lower bound for an infinite family of knots, the twist knots, and show that a twist knot is torsion in the knot concordance group if and only if it has vanishing stable $4$-genus.

Colored HOMFLY-PT polynomials of quasi-alternating 3-braid knots

Obtaining a closed form expression for the colored HOMFLY-PT polynomials of knots from 3-strand braids carrying arbitrary SU(N) representation is a challenging problem. In this paper, we confine our interest to twisted generalized hybrid weaving knots which we denote hereafter by Qˆ3(m1,−m2,n,ℓ). This family of knots not only generalizes the well-known class of weaving knots but also contains an infinite family of quasi-alternating knots. Interestingly, we obtain a closed form expression for the HOMFLY-PT polynomial of Qˆ3(m1,−m2,n,ℓ) using a modified version of the Reshitikhin-Turaev method. In addition, we compute the exact coefficients of the Jones polynomials and the Alexander polynomials of quasi-alternating knots Qˆ3(1,−1,n,±1). For these homologically-thin knots, such coefficients are known to be the ranks of their Khovanov and link Floer homologies, respectively. We also show that the asymptotic behavior of the coefficients of the Alexander polynomial is trapezoidal. On the other hand, we compute the [r]-colored HOMFLY-PT polynomials of quasi alternating knots for small values of r. Remarkably, the study of the determinants of certain twisted weaving knots leads to establish a connection with enumerative geometry related to mth Lucas numbers, denoted hereafter as Lm,2n. At the end, we verify that the reformulated invariants satisfy Ooguri-Vafa conjecture and we express certain BPS integers in terms of hyper-geometric functions F12[a,b,c;z].

Notes on constructions of knots with the same trace

The m-trace of a knot is the 4-manifold obtained from $B^4$ by attaching a 2-handle along the knot with m-framing. In 2015, Abe, Jong, Luecke and Osoinach introduced a technique to construct infinitely many knots with the same (diffeomorphic) m-trace, which is called the operation (∗m). In 2018, Miller and Piccirillo gave pairs of knots with diffeomorphic m-traces by utilizing Gompf and Miyazaki’s dualizable pattern. In this paper, we clarify the relation between the two techniques. In particular, we prove that the “twistings” appearing in both techniques are corresponding. In addition, we show that the family of knots admitting the same 4-surgery given by Teragaito can be explained by the operation (∗m).

Knots and signal transmission in topological quantum devices

We discuss the basic problem of signal transmission in quantum mechanics in terms of topological theories. Using the analogy between knot diagrams and quantum amplitudes we show how one can define and calculate the transmission coefficients of concept topological quantum devices that realize the knot topology. We observe that the problem is in different ways similar to that of transmission on quantum graphs. The desired transmission or filtering properties can be attained by the variation of topology of the device, or an external parameter, which in our model controls the topological phase. One interesting property of the transmission coefficients is the existence of ‘self-averaging’ phases, in which the value of the coefficient is independent from all the representatives in a chosen family of knots, and hence can be used for family classification. We briefly discuss physical realizations of the concept devices.

On the secondary Upsilon invariant

In this paper we construct an infinite family of knots with vanishing Upsilon invariant $\Upsilon$, although their secondary Upsilon invariants $\Upsilon^2$ show that they are linearly independent in the smooth knot concordance group. We also prove a conjecture in a paper by Allen.

Piecewise-linear embeddings of knots and links with rotoinversion symmetry.

This article describes the simplest members of an infinite family of knots and links that have achiral piecewise-linear embeddings in which linear segments (sticks) meet at corners. The structures described are all corner- and stick-2-transitive - the smallest possible for achiral knots.

Hyperbolic torsion polynomials of pretzel knots

Abstract In this paper, we explicitly calculate the highest degree term of the hyperbolic torsion polynomial of an infinite family of pretzel knots. This gives supporting evidence for a conjecture of Dunfield, Friedl and Jackson that the hyperbolic torsion polynomial determines the genus and fiberedness of a hyperbolic knot. The verification of the genus part of the conjecture for this family of knots also follows from the work of Agol and Dunfield [1] or Porti [19].

Homotopy ribbon concordance, Blanchfield pairings, and twisted Alexander polynomials

AbstractWe establish homotopy ribbon concordance obstructions coming from the Blanchfield form and Levine–Tristram signatures. Then, as an application of twisted Alexander polynomials, we show that for every knot K with nontrivial Alexander polynomial, there exists an infinite family of knots that are all concordant to K and have the same Blanchfield form as K, such that no pair of knots in that family is homotopy ribbon concordant.

Representations of the (-2,3,7)-pretzel knot and orderability of Dehn surgeries

We construct a 1-parameter family of SL2(R) representations of the pretzel knot P(−2,3,7). As a consequence, we conclude that Dehn surgeries on this knot are left-orderable for all rational surgery slopes less than 6. Furthermore, we discuss a family of knots and exhibit similar orderability results for a few other examples.

Linear independence of cables in the knot concordance group

We produce infinite families of knots { K i } i ≥ 1 \{K^i\}_{i\ge 1} for which the set of cables { K p , 1 i } i , p ≥ 1 \{K^i_{p,1}\}_{i,p\ge 1} is linearly independent in the knot concordance group, C \mathcal {C} . We arrange that these examples lie arbitrarily deep in the solvable and bipolar filtrations of C \mathcal {C} , denoted by { F n } \{\mathcal {F}_n\} and { B n } \{\mathcal {B}_n\} respectively. As a consequence, this result cannot be reached by any combination of algebraic concordance invariants, Casson-Gordon invariants, and Heegaard-Floer invariants such as τ \tau , ε \varepsilon , and Υ \Upsilon . We give two applications of this result. First, for any n ≥ 0 n\ge 0 , there exists an infinite family { K i } i ≥ 1 \{K^i\}_{i\geq 1} such that for each fixed i i , { K 2 j , 1 i } j ≥ 0 \{K^i_{2^j,1}\}_{j\geq 0} is a basis for an infinite rank summand of F n \mathcal {F}_n and { K p , 1 i } i , p ≥ 1 \{K^i_{p,1}\}_{i, p\geq 1} is linearly independent in F n / F n .5 \mathcal {F}_{n}/\mathcal {F}_{n.5} . Second, for any n ≥ 1 n\ge 1 , we give filtered counterexamples to Kauffman’s conjecture on slice knots by constructing smoothly slice knots with genus one Seifert surfaces where one derivative curve has nontrivial Arf invariant and the other is nontrivial in both F n / F n .5 \mathcal {F}_n/\mathcal {F}_{n.5} and B n − 1 / B n + 1 \mathcal {B}_{n-1}/\mathcal {B}_{n+1} . We also give examples of smoothly slice knots with genus one Seifert surfaces such that one derivative has nontrivial Arf invariant and the other is topologically slice but not smoothly slice.

Doubly slice knots and metabelian obstructions

An [Formula: see text]-dimensional knot [Formula: see text] is called doubly slice if it occurs as the cross section of some unknotted [Formula: see text]-dimensional knot. For every [Formula: see text] it is unknown which knots are doubly slice, and this remains one of the biggest unsolved problems in high-dimensional knot theory. For [Formula: see text], we use signatures coming from [Formula: see text]-cohomology to develop new obstructions for [Formula: see text]-dimensional knots with metabelian knot groups to be doubly slice. For each [Formula: see text], we construct an infinite family of knots on which our obstructions are nonzero, but for which double sliceness is not obstructed by any previously known invariant.