Composite Knot Research Articles

Shortest paths of Rubik’s snake composite knots with 9 crossings

A Rubik’s Snake is an interesting toy that was invented over 40 years ago. It can be twisted to many interesting shapes, including non-trivial knots. In this paper we proposed a systematic way to study how many blocks are needed to form a composite knot with 9 crossings. We also improved some results from our previous papers.

Shortest paths of Rubik’s snake composite knots up to 8 crossings

A Rubik’s Snake is a toy that has been around for 40 years. It can be twisted to many interesting shapes. In particular a Rubik’s snake can be twisted to form a knot. In this paper we study how many blocks are needed to form a composite knot with up to 8 crossings. Also, we improved a couple of previous results for Rubik’s snake prime knots.

Social Self-Sorting Synthesis of Molecular Knots.

We report the synthesis of molecular prime and composite knots by social self-sorting of 2,6-pyridinedicarboxamide (pdc) ligands of differing topicity and stereochemistry. Upon mixing achiral monotopic and ditopic pdc-ligand strands in a 1:1:1 ratio with Lu(III), a well-defined heteromeric complex featuring one of each ligand strand and the metal ion is selectively formed. Introducing point-chiral centers into the ligands leads to single-sense helical stereochemistry of the resulting coordination complex. Covalent capture of the entangled structure by ring-closing olefin metathesis then gives a socially self-sorted trefoil knot of single topological handedness. In a related manner, a heteromeric molecular granny knot (a six-crossing composite knot featuring two trefoil tangles of the same handedness) was assembled from social self-sorting of ditopic and tetratopic multi-pdc strands. A molecular square knot (a six-crossing composite knot of two trefoil tangles of opposite handedness) was assembled by social self-sorting of a ditopic pdc strand with four (S)-centers and a tetratopic strand with two (S)- and six (R)-centers. Each of the entangled structures was characterized by 1H and 13C NMR spectroscopy, mass spectrometry, and circular dichroism spectroscopy. The precise control of composition and topological chirality through social self-sorting enables the rapid assembly of well-defined sequences of entanglements for molecular knots.

A short proof of a theorem of Tanaka on composite knots with symmetric union presentations

We present a short proof of a theorem of Tanaka that if a composite ribbon knot admits a symmetric union presentation with one twist region, then it has a non-trivial knot and its mirror image as connected summands.

Persistently foliar composite knots

A knot $\kappa$ in $S^3$ is persistently foliar if, for each non-trivial boundary slope, there is a co-oriented taut foliation meeting the boundary of the knot complement transversely in a foliation by curves of that slope. For rational slopes, these foliations may be capped off by disks to obtain a co-oriented taut foliation in every manifold obtained by non-trivial Dehn surgery on that knot. We show that any composite knot with a persistently foliar summand is persistently foliar and that any nontrivial connected sum of fibered knots is persistently foliar. As an application, it follows that any composite knot in which each of two summands is fibered or at least one summand is nontorus alternating or Montesinos is persistently foliar. We note that, in constructing foliations in the complements of fibered summands, we build branched surfaces whose complementary regions agree with those of Gabai's product disk decompositions, except for the one containing the boundary of the knot complement. It is this boundary region which provides for persistence.

Knot probabilities in equilateral random polygons

We consider the probability of knotting in equilateral random polygons in Euclidean three-dimensional space, which model, for instance, random polymers. Results from an extensive Monte Carlo dataset of random polygons indicate a universal scaling formula for the knotting probability with the number of edges. This scaling formula involves an exponential function, independent of knot type, with a power law factor that depends on the number of prime components of the knot. The unknot, appearing as a composite knot with zero components, scales with a small negative power law, contrasting with previous studies that indicated a purely exponential scaling. The methodology incorporates several improvements over previous investigations: our random polygon data set is generated using a fast, unbiased algorithm, and knotting is detected using an optimised set of knot invariants based on the Alexander polynomial.

Concordances to prime hyperbolic virtual knots

Let $$\Sigma _0,\Sigma _1$$ be closed oriented surfaces. Two oriented knots $$K_0 \subset \Sigma _0 \times [0,1]$$ and $$K_1 \subset \Sigma _1 \times [0,1]$$ are said to be (virtually) concordant if there is a compact oriented 3-manifold W and a smoothly and properly embedded annulus A in $$W \times [0,1]$$ such that $$\partial W=\Sigma _1 \sqcup -\Sigma _0$$ and $$\partial A=K_1 \sqcup -K_0$$ . This notion of concordance, due to Turaev, is equivalent to concordance of virtual knots, due to Kauffman. A prime virtual knot, in the sense of Matveev, is one for which no thickened surface representative $$K \subset \Sigma \times [0,1]$$ admits a nontrivial decomposition along a separating vertical annulus that intersects K in two points. Here we prove that every knot $$K \subset \Sigma \times [0,1]$$ is concordant to a prime satellite knot and a prime hyperbolic knot. For homologically trivial knots in $$\Sigma \times [0,1]$$ , we prove this can be done so that the Alexander polynomial is preserved. This generalizes the corresponding results for classical knot concordance, due to Bleiler, Kirby–Lickorish, Livingston, Myers, Nakanishi, and Soma. The new challenge for virtual knots lies in proving primeness. Contrary to the classical case, not every hyperbolic knot in $$\Sigma \times [0,1]$$ is prime and not every composite knot is a satellite. Our results are obtained using a generalization of tangles in 3-balls we call complementary tangles. Properties of complementary tangles are studied in detail.

Distinguishing slice disks using knot Floer homology

We study the classification of slice disks of knots up to isotopy and diffeomorphism using an invariant in knot Floer homology. We compute the invariant of a slice disk obtained by deform-spinning, and show that it can be effectively used to distinguish non-isotopic slice disks with diffeomorphic complements. Given a slice disk of a composite knot, we define a numerical stable diffeomorphism invariant called the rank. This can be used to show that a slice disk is not a boundary connected sum, and to give lower bounds on the complexity of certain hyperplane sections of the slice disk.

On the complexity of torus knot recognition

We show that the problem of recognizing that a knot diagram represents a specific torus knot, or any torus knot at all, is in the complexity class ${\sf NP} \cap {\sf co\text{-}NP}$, assuming the generalized Riemann hypothesis. We also show that satellite knot detection is in ${\sf NP}$ under the same assumption, and that cabled knot detection and composite knot detection are unconditionally in ${\sf NP}$. Our algorithms are based on recent work of Kuperberg and of Lackenby on detecting knottedness.

A Six‐Crossing Doubly Interlocked [2]Catenane with Twisted Rings, and a Molecular Granny Knot

AbstractA molecular link (a six crossing, doubly interlocked, [2]catenane with twisted rings) and a 31#31 granny knot (a composite knot made up of two trefoil tangles of the same handedness) were constructed by ring‐closing olefin metathesis of an iron(II)‐coordinated 2×2 interwoven grid. The connections were directed by pendant phenyl groups to be between proximal ligand ends on the same faces of the grid. The link was separated from the topoisomeric granny knot by recycling size‐exclusion chromatography. The identity of each topoisomer was determined by tandem mass spectrometry and the structure of the link confirmed by X‐ray crystallography, which revealed two 82‐membered macrocycles, each in figure‐of‐eight conformations, linked through both pairs of loops.

A Six-Crossing Doubly Interlocked [2]Catenane with Twisted Rings, and a Molecular Granny Knot.

A molecular 623 link (a six crossing, doubly interlocked, [2]catenane with twisted rings) and a 31#31 granny knot (a composite knot made up of two trefoil tangles of the same handedness) were constructed by ring‐closing olefin metathesis of an iron(II)‐coordinated 2×2 interwoven grid. The connections were directed by pendant phenyl groups to be between proximal ligand ends on the same faces of the grid. The 623 link was separated from the topoisomeric granny knot by recycling size‐exclusion chromatography. The identity of each topoisomer was determined by tandem mass spectrometry and the structure of the 623 link confirmed by X‐ray crystallography, which revealed two 82‐membered macrocycles, each in figure‐of‐eight conformations, linked through both pairs of loops.

Stereoselective synthesis of a composite knot with nine crossings.

The simultaneous synthesis of a molecular nine-crossing composite knot that contains three trefoil tangles of the same handedness and a [Formula: see text] link (a type of cyclic [3]catenane topologically constrained to always have at least three twists within the links) is reported. Both compounds contain high degrees of topological writhe (w = 9), a structural feature of supercoiled DNA. The entwined products are generated from the cyclization of a hexameric Fe(II) circular helicate by ring-closing olefin metathesis, with the mixture of topological isomers formed as a result of different ligand connectivity patterns. The metal-coordinated composite knot was isolated by crystallization, the topology unambiguously proven by tandem mass spectrometry, with X-ray crystallography confirming that the 324-atom loop crosses itself nine times with matching handedness (all Δ or all Λ) at every metal centre within each molecule. Controlling the connectivity of the ligand end groups on circular metal helicate scaffolds provides an effective synthetic strategy for the stereoselective synthesis of composite knots and other complex molecular topologies.

What tangled webs we weave

Knots have been rigorously studied since the 1860s, but only in the past 30 years have they been made in the laboratory in molecular form. Now, the most complex small-molecule examples so far — a composite knot and an isomeric link, each with nine crossings — have been prepared.

Scaling behavior of knotted random polygons and self-avoiding polygons: Topological swelling with enhanced exponent.

We show that the average size of self-avoiding polygons (SAPs) with a fixed knot is much larger than that of no topological constraint if the excluded volume is small and the number of segments is large. We call it topological swelling. We argue an "enhancement" of the scaling exponent for random polygons with a fixed knot. We study them systematically through SAP consisting of hard cylindrical segments with various different values of the radius of segments. Here we mean by the average size the mean-square radius of gyration. Furthermore, we show numerically that the topological balance length of a composite knot is given by the sum of those of all constituent prime knots. Here we define the topological balance length of a knot by such a number of segments that topological entropic repulsions are balanced with the knot complexity in the average size. The additivity suggests the local knot picture.

Knot contact homology detects cabled, composite, and torus knots

Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot. Further, if the knot contact homology of a knot is isomorphic to that of a cable (respectively composite) knot, then the knot is a cable (respectively composite).

Stick numbers in the simple hexagonal lattice

In the simple hexagonal lattice, bridge number is used to establish a lower bound on stick numbers of knots. This result aids in giving a new proof that the minimal stick number is 11. In addition, the authors establish upper bounds for the stick number of a composite knot. Constructions for. p; pC1/-torus knots and some 3-bridge knots are given requiring one more stick than the lower bound guarantees.

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.

Knots in the canonical book representation of complete graphs

We describe which knots can be obtained as cycles in the canonical book representation of K_n, the complete graph on n vertices. We show that the canonical book representation of K_n contains a Hamiltonian cycle that is a composite knot if and only if n>11 and we show that when p and q are relatively prime, the (p,q) torus knot is a Hamiltonian cycle in the canonical book representation of K_{2p+q}. Finally, we list the number and type of all non-trivial knots that occur as cycles in the canonical book representation of K_n for n<12. We conjecture that the canonical book representation of K_n attains the least possible number of knotted cycles for any embedding of K_n.

RECTANGLE CONDITION FOR COMPRESSION BODIES AND 2-FOLD BRANCHED COVERINGS

We give the rectangle condition for strong irreducibility of Heegaard splittings of 3-manifolds with non-empty boundary. We apply this to a generalized Heegaard splitting of 2-fold covering of S3 branched along a link. The condition implies that any thin meridional level surface in the link complement is incompressible. We also show that the additivity of width holds for a composite knot satisfying the condition.

Knotting of random ring polymers in confined spaces

Stochastic simulations are used to characterize the knotting distributions of random ring polymers confined in spheres of various radii. The approach is based on the use of multiple Markov chains and reweighting techniques, combined with effective strategies for simplifying the geometrical complexity of ring conformations without altering their knot type. By these means we extend previous studies and characterize in detail how the probability to form a given prime or composite knot behaves in terms of the number of ring segments N and confining radius R. For 50 < or =N < or =450 we show that the probability of forming a composite knot rises significantly with the confinement, while the occurrence probability of prime knots are, in general, nonmonotonic functions of 1R. The dependence of other geometrical indicators, such as writhe and chirality, in terms of R and N is also characterized. It is found that the writhe distribution broadens as the confining sphere narrows.