Prime Knots Research Articles

On the normal form of knots

This paper is a report on the results of computer experiments with an algorithm that takes classical knots to what we call their “normal form” (and so can be used to identify the knot). The algorithm is implemented in a computer animation that shows the isotopy joining the given knot diagram to its normal form. We describe the algorithm, which is a kind of gradient descent along a functional that we define, present a table of normal forms of prime knots with 7 crossings or less, compare it to the knot table of normal forms of wire knots (obtained in [1] by mechanical experiments with real wire models) and (regretfully) present simple examples showing that normal forms obtained by our algorithm are not unique for a given knot type (sometimes isotopic knots can have different normal forms).

Generalized crossing changes in satellite knots

We show that if K is a satellite knot in the 3-sphere S3 which admits a generalized cosmetic crossing change of order q with |q| ≥ 6, then K admits a pattern knot with a generalized cosmetic crossing change of the same order. As a consequence of this, we find that any prime satellite knot in S3 which admits a torus knot as a pattern cannot admit a generalized cosmetic crossing change of order q with |q| ≥ 6. We also show that if there is any knot in S3 admitting a generalized cosmetic crossing change of order q with |q| ≥ 6, then there must be such a knot which is hyperbolic.

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).

Computational study on the progressive factorization of composite polymer knots into separated prime components.

Using Monte Carlo simulations and advanced knot localization methods, we analyze the length and distribution of prime components in composite knots tied on freely jointed rings. For increasing contour length, we observe the progressive factorization of composite knots into separated prime components. However, we observe that a complete factorization, equivalent to the "decorated ring" picture, is not obtained even for rings of contour lengths N ≃ 3 N(0), about tens of times the most probable length of the prime knots tied on the rings. The decorated ring hypothesis has been used in the literature to justify the factorization of composite knot probabilities into the knotting probabilities of their prime components. Following our results, we suggest that such a hypothesis may not be necessary to explain the factorization of the knotting probabilities, at least when polymers excluding volume is not relevant. We rationalize the behavior of the system through a simple one-dimensional model in which prime knots are replaced by slip links randomly placed on a circle, with the only constraint being that the length of the loops has the same distribution as that of the length of the corresponding prime knots.

Classification of genus 1 virtual knots having at most five classical crossings

The goal of this paper is to tabulate all genus one prime virtual knots having diagrams with ≤ 5 classical crossings. First, we construct all nonlocal prime knots in the thickened torus T × I which have diagrams with ≤ 5 crossings and admit no destabilizations. Then we use a generalized version of the Kauffman polynomial to prove that all those knots are different. Finally, we convert the knot diagrams in T thus obtained into virtual knot diagrams in the plane.

On the groundstate energy spectrum of magnetic knots and links

By using analytical results for the constrained minimum energy of magnetic knots we determine the influence of internal twist on the minimum magnetic energy levels of knots and links, and by using ropelength data from the RIDGERUNNER tightening algorithm (Ashton et al 2011 Exp. Math. 20 57–90) we obtain the groundstate energy spectra of the first 250 prime knots and 130 prime links. The two spectra are found to follow an almost identical logarithmic law. By assuming that the number of knot types grows exponentially with the topological crossing number, we show that this generic behavior can be justified by a general relationship between ropelength and crossing number, which is in good agreement with former analytical estimates (Buck and Simon 1999 Topol. Appl. 91 245–57, Diao 2003 J. Knot Theory Ramifications 12 1–16). Moreover, by considering the ropelength averaged over a given knot family, we establish a new connection between the averaged ropelength and the topological crossing number of magnetic knots.

New evidence on the asymptotics of knotted lattice polygons via local strand-passage models

Two simple-cubic-lattice models of local strand passage in self-avoiding polygons are used to study the asymptotic properties of knot probabilities. Using Monte Carlo simulations of fixed knot-type, variable-length polygons for knot-types , we investigate the limiting (as polygon length goes to ∞) knot-transition probability to go from knot-type K to knot-type K′ after a single strand passage. For knot-types K and Kb, we present numerical evidence that the model-dependent limiting knot-transition probability to go from knot-type K to K′ = K#Kb is independent of the original knot-type K, and that the limiting knot-transition probability is zero for any transition from K to a knot K′ which does not have K in its prime-knot decomposition. These results provide new numerical evidence in support of the conjectured asymptotic form, , for the number of n-edge polygons with fixed knot-type K, where α0 and μ0 are respectively the entropic critical exponent and limiting entropy per edge for unknotted lattice polygons and fK is the number of prime knots in K’s prime-knot decomposition.

Free-Energy Calculations for Semi-Flexible Macromolecules: Applications to DNA Knotting and Looping

We present a method to obtain numerically accurate values of configurational free energies of semiflexible macromolecular systems, based on the technique of thermodynamic integration combined with normal-mode analysis of a reference system subject to harmonic constraints. Compared with previous free-energy calculations that depend on a reference state, our approach introduces two innovations, namely the use of internal coordinates to constrain the reference states and the ability to freely select these reference states. As a consequence, it is possible to explore systems that undergo substantially larger fluctuations than those considered in previous calculations, including semiflexible biopolymers having arbitrary ratios of contour length L to persistence length P. To validate the method, high accuracy is demonstrated for free energies of prime DNA knots with L/P=20 and L/P=40, corresponding to DNA lengths of 3000 and 6000 base pairs, respectively. We then apply the method to study the free-energy landscape for a model of a synaptic nucleoprotein complex containing a pair of looped domains, revealing a bifurcation in the location of optimal synapse (crossover) sites. This transition is relevant to target-site selection by DNA-binding proteins that occupy multiple DNA sites separated by large linear distances along the genome, a problem that arises naturally in gene regulation, DNA recombination, and the action of type-II topoisomerases.

THE CONCORDANCE GENUS OF 11-CROSSING KNOTS

The concordance genus of a knot is the least genus of any knot in its concordance class. It is bounded above by the genus of the knot, and bounded below by the slice genus, two well-studied invariants. In this paper we consider the concordance genus of 11-crossing prime knots. This analysis resolves the concordance genus of 533 of the 552 prime 11-crossing knots. The appendix to the paper gives concordance diagrams for 59 knots found to be concordant to knots of lower genus, including null-concordances for the 30 11-crossing knots known to be slice.

ON NON-ISOTOPIC SPANNING SURFACES FOR A CLASS OF ARBORESCENT KNOTS

We use the methods of Hedden, Juhász and Sarkar to exhibit a set of arborescent knots that bound large numbers of non-isotopic minimal genus spanning surfaces. In particular, we describe a sequence of prime knots Kn which will bound at least 22n-1 non-isotopic minimal spanning surfaces of genus n.

Upper bound on lattice stick number of knots

AbstractThe lattice stick number sL(K) of a knot K is defined to be the minimal number of straight line segments required to construct a stick presentation of K in the cubic lattice. In this paper, we find an upper bound on the lattice stick number of a nontrivial knot K, except the trefoil knot, in terms of the minimal crossing number c(K) which is sL(K) ≤ 3c(K) + 2. Moreover if K is a non-alternating prime knot, then sL(K) ≤ 3c(K) − 4.

Tunnel number degeneration under the connected sum of prime knots

We study 2-string free tangle decompositions of knots with tunnel number two. As an application, we construct infinitely many counter-examples to a conjecture in the literature stating that the tunnel number of the connected sum of prime knots doesnʼt degenerate by more than one: t(K1#K2)⩾t(K1)+t(K2)−1, for K1 and K2 prime knots.

Spontaneous Knotting and Unknotting of Flexible Linear Polymers: Equilibrium and Kinetic Aspects

We report on a computational study of the statics and dynamics of long flexible linear polymers that spontaneously knot and unknot. Specifically, the equilibrium self-entanglement properties, such as the knotting probability, knot length and position, are investigated with extensive Monte Carlo sampling of chains of up to 15,000 beads. Tens of such equilibrated chains of up to 4, 096 beads are next used as starting points for Langevin dynamics simulations. The complex interplay of chain dynamics and self-knotting is addressed by monitoring the time evolution of various metric and entanglement properties. In particular, the extensive duration of the simulations allows for observing the spontaneous formation and disappearance of prime and composite physical knots in linear chains. Notably, a sizeable fraction of self-knotting and unknotting events is found to involve regions that are far away from the chain termini. To the best of our knowledge this represents the first instance where spontaneous changes in knotting for linear homopolymers are systematically characterized using unbiased dynamics simulations.

Upper bounds on the minimum length of cubic lattice knots

Knots have been considered to be useful models for simulating molecular chains such as DNA and proteins. One quantity that we are interested in is the molecular knot and the minimum number of monomers necessary to realize a knot. In this paper, we consider every knot in the cubic lattice. In particular, the minimum length of a knot indicates the minimum length necessary to construct the knot in the cubic lattice. Diao introduced this term (he used ‘minimum edge number’ instead) and proved that the minimum length of the trefoil knot 31 is 24. Also the minimum lengths of the knots 41 and 51 are known to be 30 and 34, respectively. In this paper we find a general upper bound of the minimum length of a nontrivial knot K, except the trefoil knot, in terms of the minimum crossing number c(K). The upper bound is . Moreover, if K is a non-alternating prime knot, then the upper bound is . Our work can be considered a direct consequence of the results obtained by the authors in Hong et al (2012 arXiv:1209.0048). Furthermore, if K is (n + 1, n)-torus knot, then the upper bound is .

Evolution of heliospheric magnetized configurations via topological invariants

The analogy between magnetohydrodynamics (MHD) and knot theory is utilized in presenting a new method for an analysis of stability and evolution of complex magnetic heliospheric flux tubes. Planar projection of a three-dimensional magnetic configuration depicts the structure as a two-dimensional diagram with crossings, to which one may assign mathematical operations leading to robust topological invariants. These invariants enrich the topological information of magnetic configurations beyond helicity. It is conjectured that the field which emerges from the solar photosphere is structured as one of the simplest knots—unknot or prime knot—and these flux ropes are then stretched while carried by the solar wind into the interplanetary medium. Preservation of invariants for small diffusivity and large cross section of the emerging magnetic flux makes them impervious to large scale reconnection, allowing us to predict the observed structures at 1AU as elongated prime knots. Similar structures may be observed in magnetic clouds which got disconnected from their footpoints and in ion drop-out configurations from a compact flare source in solar impulsive solar events. Observation of small scale magnetic features consistent with prime knots may indicate spatial intermittency and non-Gaussian statistics in the turbulent cascade process. For flux tubes with higher resistivity, magnetic energy decay rate should decrease with increased knot complexity as the invariants are then harder to be violated. These observations could be confirmed if adjacent satellites happen to measure distinctly oriented magnetic fields with directionally varying suprathermal particle fluxes.

KNOTS IN THE SOLID TORUS UP TO 6 CROSSINGS

We classify non-affine, prime knots in the solid torus up to 6 crossings. We establish which of these are amphicheiral: almost all knots with symmetric Jones polynomial are amphicheiral, but in a few cases we use stronger invariants, such as HOMFLYPT and Kauffman skein modules, to show that some such knots are not amphicheiral. Examples of knots with the same Jones polynomial that are different in the HOMFLYPT skein module are presented. It follows from our computations, that the wrapping conjecture is true for all knots up to 6 crossings.

Knotty invariants: structure and evolution of magnetized fluids

AbstractIt is conjectured that magnetic configurations may be characterized by the embedded invariants which are extracted from their topological textures. Hence, experimentally observed magnetic structures can be classified by the invariants of the elementary building blocks of the 3-D structures - prime knots. These unique invariants may describe the magnetic configurations in ion drop-outs in solar impulsive events, in photospheric flux ropes carried into the interplanetary medium by the solar wind, and can affect the decay rates for complex magnetic configurations.

PRIME KNOTS WHOSE ARC INDEX IS SMALLER THAN THE CROSSING NUMBER

It is known that the arc index of alternating knots is the minimal crossing number plus two and the arc index of prime nonalternating knots is less than or equal to the minimal crossing number. We study some cases when the arc index is strictly less than the minimal crossing number. We also give minimal grid diagrams of some prime nonalternating knots with 13 crossings and 14 crossings whose arc index is the minimal crossing number minus one.

ERRATA: "A PARTIAL ORDER ON THE SET OF PRIME KNOTS WITH UP TO 11 CROSSINGS"

Let K be a prime knot in S3 and G(K) = π1(S3 - K) the knot group. We write K1 ≥ K2 if there exists a surjective homomorphism from G(K1) onto G(K2). In this paper, we determine this partial order on the set of prime knots with up to 11 crossings. There exist such 801 prime knots and then 640, 800 should be considered. The existence of a surjective homomorphism can be proved by constructing it explicitly. On the other hand, the non-existence of a surjective homomorphism can be proved by the Alexander polynomial and the twisted Alexander polynomial.

Some applications of planar graph in knot theory

The relationship between a link diagram and its corresponding planar graph is briefly reviewed. A necessary and sufficient condition is given to detect when a planar graph corresponds to a knot. The relationship between planar graph and almost planar Seifert surface is discussed. Using planar graph, we construct an alternating amphicheiral prime knot with crossing number n for any even number n ≥ 4. This gives an affirmative answer to problem 1.66(B) on Kirby's problem list.