Nontrivial Knot Research Articles

Knotting probability of self-avoiding polygons under a topological constraint.

We define the knotting probability of a knot K by the probability for a random polygon or self-avoiding polygon (SAP) of N segments having the knot type K. We show fundamental and generic properties of the knotting probability particularly its dependence on the excluded volume. We investigate them for the SAP consisting of hard cylindrical segments of unit length and radius rex. For various prime and composite knots, we numerically show that a compact formula describes the knotting probabilities for the cylindrical SAP as a function of segment number N and radius rex. It connects the small-N to the large-N behavior and even to lattice knots in the case of large values of radius. As the excluded volume increases, the maximum of the knotting probability decreases for prime knots except for the trefoil knot. If it is large, the trefoil knot and its descendants are dominant among the nontrivial knots in the SAP. From the factorization property of the knotting probability, we derive a sum rule among the estimates of a fitting parameter for all prime knots, which suggests the local knot picture and the dominance of the trefoil knot in the case of large excluded volumes. Here we remark that the cylindrical SAP gives a model of circular DNA which is negatively charged and semiflexible, where radius rex corresponds to the screening length.

Eigenvalue varieties of Brunnian links

In this article, it is proved that the eigenvalue variety of the exterior of a nontrivial, non-Hopf, Brunnian link in $\mathbb{S}^{3}$ contains a nontrivial component of maximal dimension. This generalises, for Brunnian links, the nontriviality of the $A$-polynomial of a nontrivial knot in $\mathbb{S}^{3}$.

On chirality of toroidal embeddings of polyhedral graphs

We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [2, 3]. Building on this and using the chirality of torus knots and links [9, 10], we prove that the nontrivial embeddings of simple 3-connected planar graphs in the standard torus are chiral. For the case that the spatial graph contains a nontrivial knot, the statement was shown by Castle et al. [5]. We give an alternative proof using minors instead of the Euler characteristic. To prove the case in which the graph embedding contains a nonsplit link, we show the chirality of Hopf ladders with at least three rungs, thus generalizing a theorem of Simon [12].

Coverings of torus knots in S2 × S1 and universals

Let [Formula: see text] be an ordinary fiber of a Seifert fibering of [Formula: see text] with two exceptional fibers of order [Formula: see text]. We show that any Seifert manifold with Euler number zero is a branched covering of [Formula: see text] with branching [Formula: see text] if [Formula: see text]. We compute the Seifert invariants of the Abelian covers of [Formula: see text] branched along a [Formula: see text]. We also show that [Formula: see text], a non-trivial torus knot in [Formula: see text], is not universal.

The Elastic Trefoil is the Doubly Covered Circle

We investigate the elastic behavior of knotted loops of springy wire. To this end we minimize the classic bending energy $E_{\text{bend}}=\int\kappa^2$ together with a small multiple of ropelength $\mathcal R=\text{length}/\text{thickness}$ in order to penalize selfintersection. Our main objective is to characterize elastic knots, i.e., all limit configurations of energy minimizers of the total energy $E_\vartheta:=E_{\text{bend}}+\vartheta\mathcal R$ as $\vartheta$ tends to zero. The elastic unknot turns out to be the round circle with bending energy $(2\pi)^2$. For all (non-trivial) knot classes for which the natural lower bound $(4\pi)^2$ for the bending energy is sharp, the respective elastic knot is the twice covered circle. The knot classes for which $(4\pi)^2$ is sharp are precisely the $(2,b)$-torus knots for odd $b$ with $|b|\ge 3$ (containing the trefoil). In particular, the elastic trefoil is the twice covered circle.

2-Adjacency between pretzel links and the trivial link

We study 2-adjacency between a 3-strand pretzel link with two components and the trivial link, and whether the trivial knot is 2-adjacent to a pretzel knot. By investigating mainly the coefficients of their Conway polynomials, Jones polynomials and etc., we show that any nontrivial pretzel link with two-components is not 2-adjacent to the trivial link, and vice versa. Moreover, we also show that the trivial knot is not 2-adjacent to any nontrivial pretzel knot.

PN sequences obtained from signs of crossings of knot diagrams

For an oriented knot diagram, we define a cyclic word in letters [Formula: see text] and [Formula: see text] corresponding to positive and negative crossings along the diagram, respectively. We give a necessary and sufficient condition for a cyclic word in [Formula: see text] and [Formula: see text] to be obtained from some knot diagram. Also, we prove that any PN sequence of a diagram of a nontrivial knot contains at least four subsequences [Formula: see text] and [Formula: see text].

The SL(2, C) Casson Invariant for Knots and the Â-polynomial

AbstractIn this paper, we extend the definition of the SL(2,ℂ) Casson invariant to arbitrary knots K in integral homology 3-spheres and relate it to the m-degree of the Â-polynomial of K. We prove a product formula for the Â-polynomial of the connected sum K1#K2 of two knots in S3 and deduce additivity of the SL(2,ℂ) Casson knot invariant under connected sums for a large class of knots in S3. We also present an example of a nontrivial knot K in S3 with trivial Â-polynomial and trivial SL(2,ℂ) Casson knot invariant, showing that neither of these invariants detect the unknot.

There exist no minimally knotted planar spatial graphs on the torus

We show that all nontrivial embeddings of planar graphs on the torus contain a nontrivial knot or a nonsplit link. This is equivalent to showing that no minimally knotted planar spatial graphs on the torus exist that contain neither a nontrivial knot nor a nonsplit link all of whose components are unknots.

Toroidal embeddings of abstractly planar graphs are knotted or linked

We give explicit deformations of embeddings of abstractly planar graphs that lie on the standard torus \(T^2 \subset \mathbb {R}^3\) and that contain neither a nontrivial knot nor a nonsplit link into the plane. It follows that ravels do not embed on the torus. Our results provide general insight into properties of molecules that are synthesized on a torus.

Two-stage directed self-assembly of a cyclic [3]catenane.

Interlocked molecules possess properties and functions that depend upon their intricate connectivity. In addition to the topologically trivial rotaxanes, whose structures may be captured by a planar graph, the topologically non-trivial knots and catenanes represent some of chemistry's most challenging synthetic targets because of the three-dimensional assembly necessary for their construction. Here we report the synthesis of a cyclic [3]catenane, which consists of three mutually interpenetrating rings, via an unusual synthetic route. Five distinct building blocks self-assemble into a heteroleptic triangular framework composed of two joined Fe(II)3L3 circular helicates. Subcomponent exchange then enables specific points in the framework to be linked together to generate the cyclic [3]catenane product. Our method represents an advance both in the intricacy of the metal-templated self-assembly procedure and in the use of selective imine exchange to generate a topologically complex product.

The A-polynomial and holonomy perturbations

Dunfield-Garoufalidis and Boyer-Zhang proved that the A-polynomial of a nontrivial knot in $S^{3}$ is nontrivial. In this paper, we use holonomy perturbations to prove the non-triviality of the A-polynomial for a nontrivial, null-homotopic knot in an irreducible 3-manifold. Also, we give a strong constraint on the A-polynomial of a knot in the 3-sphere.

Transverse string topology and the cord algebra

We define a coalgebra structure for open strings transverse to any framed codimension 2 submanifold. When the submanifold is a knot in R^3, we show this structure recovers a specialization of the Ng cord algebra, a non-trivial knot invariant which is not determined by a number of other knot invariants.

Berge–Gabai knots and L–space satellite operations

Let $P(K)$ be a satellite knot where the pattern, $P$, is a Berge-Gabai knot (i.e., a knot in the solid torus with a non-trivial solid torus Dehn surgery), and the companion, $K$, is a non-trivial knot in $S^3$. We prove that $P(K)$ is an L-space knot if and only if $K$ is an L-space knot and $P$ is sufficiently positively twisted relative to the genus of $K$. This generalizes the result for cables due to Hedden and the first author.

Knot Invariants from Topological Recursion on Augmentation Varieties

Using the duality between Wilson loop expectation values of SU(N) Chern-Simons theory on $S^3$ and topological open-string amplitudes on the local mirror of the resolved conifold, we study knots on $S^3$ and their invariants encoded in colored HOMFLY polynomials by means of topological recursion. In the context of the local mirror Calabi-Yau threefold of the resolved conifold, we generalize the topological recursion of the remodelled B-model in order to study branes beyond the class of toric Harvey-Lawson special Lagrangians -- as required for analyzing non-trivial knots on $S^3$. The basic ingredients for the proposed recursion are the spectral curve, given by the augmentation variety of the knot, and the calibrated annulus kernel, encoding the topological annulus amplitudes associated to the knot. We present an explicit construction of the calibrated annulus kernel for torus knots and demonstrate the validity of the topological recursion. We further argue that -- if an explicit form of the calibrated annulus kernel is provided for any other knot -- the proposed topological recursion should still be applicable. We study the implications of our proposal for knot theory, which exhibit interesting consequences for colored HOMFLY polynomials of mutant knots.

Splicing knot complements and bordered Floer homology

Abstract We show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere L-spaces has Heegaard Floer homology of rank strictly greater than one. In particular, splicing the complements of nontrivial knots in the 3-sphere never produces an L-space. The proof uses bordered Floer homology.

Mosaic number of knots

Lomonaco and Kauffman developed knot mosaics to give a definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot n-mosaic is an n × n matrix of 11 kinds of specific mosaic tiles representing a knot or a link. The mosaic number m(K) of a knot K is the smallest integer n for which K is representable as a knot n-mosaic. In this paper, we establish an upper bound on the mosaic number of a knot or a link K in terms of the crossing number c(K). Let K be a nontrivial knot or a non-split link except the Hopf link. Then m(K) ≤ c(K) + 1. Moreover if K is prime and non-alternating except [Formula: see text] link, then m(K) ≤ c(K) - 1.

Minimum lattice length and ropelength of 2-bridge knots and links

Knots are commonly found in molecular chains such as DNA and proteins, and they have been considered to be useful models for structural analysis of these molecules. One interested quantity is the minimum number of monomers necessary to realize a molecular knot. The minimum lattice length Len(K) of a knot K indicates the minimum length necessary to construct K in the cubic lattice. Another important quantity in physical knot theory is the ropelength which is one of the knot energies measuring the complexity of knot conformation. The minimum ropelength Rop(K) is the minimum length of an ideally flexible rope necessary to tie a given knot K. Much effort has been invested in the research project for finding upper bounds on both quantities in terms of the minimum crossing number c(K) of the knot. It is known that Len(K) and Rop(K) lie between \documentclass[12pt]{minimal}\begin{document}$\mbox{O}(c(K)^{\frac{3}{4}})$\end{document}O(c(K)34) and O(c(K)[ln (c(K))]5), but unknown yet whether any family of knots has superlinear growth. In this paper, we focus on 2-bridge knots and links. Linear growth upper bounds on the minimum lattice length and minimum ropelength for nontrivial 2-bridge knots or links are presented as Len(K) ⩽ 8c(K) + 2 and Rop(K) ⩽ 11.39c(K) + 12.37.

Taut foliations in knot complements

We show that for any nontrivial knot in $S^3$, there is an open interval containing zero such that a Dehn surgery on any slope in this interval yields a 3-manifold with taut foliations. This generalizes a theorem of Gabai on zero frame surgery.

Equilateral stick number of knots

An equilateral stick number s=(K) of a knot K is defined to be the minimal number of sticks required to construct a polygonal knot of K which consists of equal length sticks. Rawdon and Scharein [Upper bounds for equilateral stick numbers, in Physical Knots: Knotting, Linking, and Folding Geometric Objects in ℝ3, Contemporary Mathematics, Vol. 304 (American Mathematical Society, Providence, RI, 2002), pp. 55–76] found upper bounds for the equilateral stick numbers of all prime knots through 10 crossings by using algorithms in the software KnotPlot. In this paper, we find an upper bound on the equilateral stick number of a non-trivial knot K in terms of the minimal crossing number c(K) which is s=(K) ≤ 2c(K) + 2. Moreover if K is a non-alternating prime knot, then s=(K) ≤ 2c(K) - 2. Furthermore we find another upper bound on the equilateral stick number for composite knots which is s=(K1♯K2) ≤ 2c(K1) + 2c(K2).