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De novo protein design methods can create proteins with folds not yet seen in nature. These methods largely focus on optimizing the compatibility between the designed sequence and the intended conformation, without explicit consideration of protein folding pathways. Deeply knotted proteins, whose topologies may introduce substantial barriers to folding, thus represent an interesting test case for protein design. Here we report our attempts to design proteins with trefoil (31) and pentafoil (51) knotted topologies. We extended previously described algorithms for tandem repeat protein design in order to construct deeply knotted backbones and matching designed repeat sequences (N = 3 repeats for the trefoil and N = 5 for the pentafoil). We confirmed the intended conformation for the trefoil design by X ray crystallography, and we report here on this protein's structure, stability, and folding behaviour. The pentafoil design misfolded into an asymmetric structure (despite a 5-fold symmetric sequence); two of the four repeat-repeat units matched the designed backbone while the other two diverged to form local contacts, leading to a trefoil rather than pentafoil knotted topology. Our results also provide insights into the folding of knotted proteins. This study reports the successful de novo design of a trefoil knotted protein fold for which the crystal structure agrees closely with the intended trefoil knot topology. [ABSTRACT FROM AUTHOR]

This essay is a remembrance of Vaughan Jones and a diagrammatic exposition of the remarkable breakthroughs in knot theory and low-dimensional topology that were catalyzed by his work. [ABSTRACT FROM AUTHOR]

In [B. Kwon and S. Kang, Rectangle conditions and families of 3-bridge prime knots, Topol. Appl. 291 (2021) 107453], using the set E A T k of all essential alternating rational 3-tangles for positive integer k , the authors showed that all knot diagrams in the numerator closure set C N (E A T 2 l + 1) and the denominator closure set C D (E A T 2 l + 2) with l > 0 are 3-bridge prime knot diagrams. In this paper, for n > 4 we construct a set A A T 4 n of additions of alternating rational tangles in E A T 4 . The set A A T 4 n generalizes E A T k and contains it as a subset for some k. We show that any closure set C (A A T 4 n) on A A T 4 n so that the resulting diagrams are reduced and alternating knot diagrams represent alternating 3-bridge prime knot diagrams. Since a tangle diagram in A A T 4 n + 1 is constructed inductively from a tangle diagram in A A T 4 n by adding only one crossing positively, the result of this paper supports the conjecture that 3-bridge property is preserved under one-crossing alternating addition positively to alternating 3-bridge knots in 3-bridge representations. [ABSTRACT FROM AUTHOR]

Here M ̲ denotes a pair (M, A) of a manifold and a subset (e.g. A = ∂ M or A = ∅ ). We construct for each M ̲ its motion groupoid Mot M ̲ , whose object set is the power set P M of M, and whose morphisms are certain equivalence classes of continuous flows of the 'ambient space' M, that fix A, acting on P M . These groupoids generalise the classical definition of a motion group associated to a manifold M and a submanifold N, which can be recovered by considering the automorphisms in Mot M ̲ of N ∈ P M . We also construct the mapping class groupoid MCG M ̲ associated to a pair M ̲ with the same object class, whose morphisms are now equivalence classes of homeomorphisms of M, that fix A. We recover the classical definition of the mapping class group of a pair by taking automorphisms at the appropriate object. For each pair M ̲ we explicitly construct a functor F : Mot M ̲ → MCG M ̲ , which is the identity on objects, and prove that this is full and faithful, and hence an isomorphism, if π 0 and π 1 of the appropriate space of self-homeomorphisms of M are trivial. In particular, we have an isomorphism in the physically important case M ̲ = ([ 0 , 1 ] n , ∂ [ 0 , 1 ] n) , for any n ∈ N . We show that the congruence relation used in the construction Mot M ̲ can be formulated entirely in terms of a level preserving isotopy relation on the trajectories of objects under flows—worldlines (e.g. monotonic 'tangles'). We examine several explicit examples of Mot M ̲ and MCG M ̲ demonstrating the utility of the constructions. [ABSTRACT FROM AUTHOR]

Let X be a closed indefinite 4-manifold with b + (X) = 3 (mod 4) and with non-vanishing mod 2 Seiberg–Witten invariants. We prove a new lower bound on the genus of a properly embedded surface in X \ B 4 representing a given homology class and with boundary a quasipositive knot K ⊂ S 3 . In the null-homologous case our inequality implies that the minimal genus of such a surface is equal to the slice genus of K. If X is symplectic then our lower bound differs from the minimal genus by at most 1 for any homology class that can be represented by a symplectic surface. Along the way, we also prove an extension of the adjunction inequality for closed 4-manifolds to classes of negative self-intersection without requiring X to be of simple type. [ABSTRACT FROM AUTHOR]

For pattern knots admitting genus-one doubly pointed bordered Heegaard diagrams, we give an immersed-curve approach to compute the knot Floer chain complexes of the corresponding satellite knots. In particular, this approach provides a convenient way to compute the knot concordance invariant τ . For patterns P obtained from two-bridge links b(p, q), we derive a formula for the τ -invariant of P (T 2 , 3) and P (- T 2 , 3) in terms of (p, q). We use this formula to study whether such patterns induce homomorphisms on the knot concordance group, providing a glimpse at a conjecture due to Hedden. [ABSTRACT FROM AUTHOR]

This paper discusses some geometric ideas associated with knots in real projective 3-space ℝ P 3 . These ideas are borrowed from classical knot theory. Since knots in ℝ P 3 are classified into three disjoint classes: affine, class- 0 non-affine and class- 1 knots, it is natural to wonder in which class a given knot belongs to. In this paper we attempt to answer this question. We provide a structure theorem for these knots which helps in describing their behavior near the projective plane at infinity. We propose a procedure called space bending surgery, on affine knots to produce several examples of knots. We later show that this operation can be extended on an arbitrary knot in ℝ P 3 . We then study the notion of companionship of knots in ℝ P 3 and using it we provide geometric criteria for a knot to be affine. We also define a notion of "genus" for knots in ℝ P 3 and study some of its properties. We prove that this genus detects knottedness in ℝ P 3 and gives some criteria for a knot to be affine and of class- 1. We also prove a "non-cancellation" theorem for space bending surgery using the properties of genus. Then we show that a knot can have genus 1 if and only if it is a cable knot with a class-1 companion. We produce examples of class- 0 non-affine knots with genus 1. Thus we highlight that, ℝ P 3 admits a knot theory with a truly different flavor than that of S 3 or ℝ 3 . [ABSTRACT FROM AUTHOR]

Although the connected sum of trivial knots is trivial in classical knot theory, there is a nontrivial connected sum of trivial knots in virtual knot theory. In this paper, we give connected sum formulae of the intersection polynomials of virtual knots introduced in the preceding paper [6]. As an application, we show that there are infinitely many connected sums of any pair of virtual knots. [ABSTRACT FROM AUTHOR]

Every oriented closed geodesic on the modular surface has a canonically associated knot in its unit tangent bundle coming from the periodic orbit of the geodesic flow. We study the volume of the associated knot complement with respect to its unique complete hyperbolic metric. We show that there exist sequences of closed geodesics for which this volume is bounded linearly in terms of the period of the geodesic's continued fraction expansion. Consequently, we give a volume's upper bound for some sequences of Lorenz knots complements, linearly in terms of the corresponding braid index. Also, for any punctured hyperbolic surface we give volume's bounds for the canonical lift complement relative to some sequences of sets of closed geodesics in terms of the geodesics length. [ABSTRACT FROM AUTHOR]

The bitopologies have been associated with some knots in the literature with the help of a method called the knot digraph notation. The knot graphs and quasi pseudo metric spaces were used to obtain these bitopologies. With the help of quasi pseudo metrics, two topologies were obtained on a set. In this way, an association between some knots and bitopologies was established. The authors sought an answer to the question "Given the bitopologies associated with knots, can the knot itself be obtained ?" and they gave a method. This mentioned method consists of 6 steps. In this work, it is shown in detail that according to the Alexander-Briggs notation, the reverse of the knot digraph notation is provided for the knots 31, 51, 52, 61, 62, 71, 72, 73, 81, 82, 83, 91, 92, 93, 101, 102, 103. [ABSTRACT FROM AUTHOR]

Knot theory is a distinct branch of topology. In recent years, exciting new application of knot theory to biology and chemistry, especially to DNA structures are being developed. Alexander polynomial is very closely connected with the topological properties of knots. In this paper we have determined the solution of Ak and & knots using the condition of Alexander polynomial and Jones polynomial. [ABSTRACT FROM AUTHOR]

In this short article, I discuss the æsthetics of knot projections and introduce software that creates two-dimensional knot diagrams optimized for visual appearance. The software is in the form of a documented and self-contained suite of functionality written in the R programming language (that is, a 'package'): knotR. The package leverages the graphical capabilities of the popular vectorized graphics software inkscape. Different aspects of knot appearance are discussed, and a framework for objectively optimizing the visual appeal of a knot projection is given. I use the software to create a wide range of knot diagrams. [ABSTRACT FROM AUTHOR]

In this paper we discuss the applications of knotoids to modelling knots in open curves and produce new knotoid invariants. We show how invariants of knotoids generally give rise to well-behaved measures of how much an open curve is knotted. We define biframed planar knotoids, and construct new invariants of these objects that can be computed in polynomial time. As an application of these invariants we improve the classification of planar knotoids with up to five crossings by distinguishing two pairs of prime knotoids that were conjectured to be distinct by Goundaroulis et al. [ABSTRACT FROM AUTHOR]

Experimental data from Dunfield et al. using random grid diagrams suggest that the genus of a knot grows linearly with respect to the crossing number. Using billiard table diagrams of Chebyshev knots developed by Koseleff and Pecker and a random model of 2-bridge knots via these diagrams developed by the author with Krishnan and then with Even-Zohar and Krishnan, we introduce a further-truncated model of all 2-bridge knots of a given crossing number, almost all counted twice. We present a convenient way to count Seifert circles in this model and use this to compute a lower bound for the average Seifert genus of a 2-bridge knot of a given crossing number. [ABSTRACT FROM AUTHOR]

In the theory of topology of words, the move to swap two adjacent letters in words is a simple operator that makes an S -homotopy class of a nanoword into the S -homotopy class of the empty word. In this paper, we study the relation between the swap move of letters and the forbidden moves of virtual diagrams, that is, local moves of virtual diagrams which unknot any virtual knot. [ABSTRACT FROM AUTHOR]

We consider the Lorenz equations, a system of three-dimensional ordinary differential equations modeling atmospheric convection. These equations are chaotic and hard to study even numerically, and so a simpler "geometric model" has been introduced in the seventies. One of the classical problems in dynamical systems is to relate the original equations to the geometric model. This has been achieved numerically by Tucker for the classical parameter values and remains open for general values. In this paper, we establish analytically a relation to the geometric model for a different set of parameter values that we prove must exist. This is facilitated by finding a way to apply topological tools developed for the study of surface dynamics to the more intricate case of three-dimensional flows. [ABSTRACT FROM AUTHOR]

Machine knitting is a well-established fabrication technique for complex soft objects, and both companies and researchers have developed tools for generating machine knitting patterns. However, existing representations for machine knitted objects are incomplete (do not cover the complete domain of machine knittable objects) or overly specific (do not account for symmetries and equivalences among knitting instruction sequences). This makes it difficult to define correctness in machine knitting, let alone verify the correctness of a given program or program transformation. The major contribution of this work is a formal semantics for knitout, a low-level Domain Specific Language for knitting machines. We accomplish this by using what we call the fenced tangle, which extends concepts from knot theory to allow for a mathematical definition of knitting program equivalence that matches the intuition behind knit objects. Finally, using this formal representation, we prove the correctness of a sequence of rewrite rules; and demonstrate how these rewrite rules can form the foundation for higher-level tasks such as compiling a program for a specific machine and optimizing for time/reliability, all while provably generating the same knit object under our proposed semantics. By establishing formal definitions of correctness, this work provides a strong foundation for compiling and optimizing knit programs. [ABSTRACT FROM AUTHOR]

We present an algorithmic approach to discover, study, and design multistable elastic knots. Elastic knots are physical realizations of closed curves embedded in 3-space. When endowed with the material thickness and bending resistance of a physical wire, these knots settle into equilibrium states that balance the forces induced by elastic deformation and self-contacts of the wire. In general, elastic knots can have many distinct equilibrium states, i.e. they are multistable mechanical systems. We propose a computational pipeline that combines randomized spatial sampling and physics simulation to efficiently find stable equilibrium states of elastic knots. Leveraging results from knot theory, we run our pipeline on thousands of different topological knot types to create an extensive data set of multistable knots. By applying a series of filters to this data, we discover new transformable knots with interesting geometric and physical properties. A further analysis across knot types reveals geometric and topological patterns, yielding constructive principles that generalize beyond the currently tabulated knot types. We show how multistable elastic knots can be used to design novel deployable structures and engaging recreational puzzles. Several physical prototypes at different scales highlight these applications and validate our simulation. [ABSTRACT FROM AUTHOR]

Dielectric function representation by B-splines became quite popular and widely used in the context of spectroscopic ellipsometry data interpretation. B-splines are defined by a polynomial degree and a sequence of knots (i.e., the number and positions of knots). Defining the knot sequence is non-trivial, and this task has a significant effect on the actual effectiveness of the B-spline parameterization in spectroscopic ellipsometry data analysis. In this paper, we propose a simple, practical, and systematic knot placement scheme that improves ordinary trial-and-error technique in establishing the knot spacing. The approach suggested here is based on an integral span, a measure introduced in this work. The proposed procedure provides a possibility to determine the knot locations automatically (or, at least, semi-automatically) and excludes widespread modeling ambiguities associated with uncertain knot vector. Moreover, our approach guarantees the absence of non-monotonic behavior of the mean-squared-error function and thereby improves the accuracy of our analysis. The performance of the proposed method has been tested for several real cases of the study. [ABSTRACT FROM AUTHOR]

Vaughan Jones was my mentor and friend. I provide here a few personal recollections including the “knotted antennas” project which brought me to Berkeley. [ABSTRACT FROM AUTHOR]

Solomon rings, upholding the symbol of wisdom with profound historical roots, were widely used as decorations in ancient architecture and clothing. However, it was only recently discovered that such topological structures can be formed by self-organization in biological/chemical molecules, liquid crystals, etc. Here, we report the observation of polar Solomon rings in a ferroelectric nanocrystal, which consist of two intertwined vortices and are mathematically equivalent to a 4 1 2 link in topology. By combining piezoresponse force microscopy observations and phase-field simulations, we demonstrate the reversible switching between polar Solomon rings and vertex textures by an electric field. The two types of topological polar textures exhibit distinct absorption of terahertz infrared waves, which can be exploited in infrared displays with a nanoscale resolution. Our study establishes, both experimentally and computationally, the existence and electrical manipulation of polar Solomon rings, a new form of topological polar structures that may provide a simple way for fast, robust, and high-resolution optoelectronic devices. Polar Solomon rings in a ferroelectric nanocrystal have been observed and electrically manipulated, which consist of two intertwined vortices and are mathematically equivalent to a 4 1 2 link in topology based on knots and links theory. [ABSTRACT FROM AUTHOR]

A link at the origin of an isolated singularity of a two-dimensional semialgebraic surface in R 4 is a topological knot (or link) in S 3 . We study the connection between the ambient Lipschitz geometry of semialgebraic surface germs in R 4 and knot theory. Namely, for any knot K, we construct a surface X K in R 4 such that: the link at the origin of X K is a trivial knot; the germs X K are outer bi-Lipschitz equivalent for all K; two germs X K and X K ′ are ambient semialgebraic bi-Lipschitz equivalent only if the knots K and K ′ are isotopic. We show that the Jones polynomial can be used to recognize ambient bi-Lipschitz non-equivalent surface germs in R 4 , even when they are topologically trivial and outer bi-Lipschitz equivalent. [ABSTRACT FROM AUTHOR]

We prove that if K is a nontrivial null-homotopic knot in a closed oriented 3–manfiold Y such that $Y-K$ does not have an $S^1\times S^2$ summand, then the zero surgery on K does not have an $S^1\times S^2$ summand. This generalises a result of Hom and Lidman, who proved the case when Y is an irreducible rational homology sphere. [ABSTRACT FROM AUTHOR]

A new approach of splines to examine climatological time series is introduced and different applications are shown. Instead of the classical spline‐procedure to find polynomials which fit given values at certain points (knots), a variational formulation is used here, where a cost function is minimized to obtain smooth interpolated values at arbitrary points. The innovation of the method consists in the selection of the constraint, namely the conservation of mean values within certain smoothing intervals. The methodology allows a sound derivation of the past and present climate conditions in times of climate change. It allows a refined computation of extreme values and of the probability of certain temperature thresholds. The methodology furthermore allows to detect break points in climatological time series. Finally the method leads to a consistent interpolation of time series, which are only given by mean values. Several applications of the method, using climatological time series of Vienna are shown and discussed. [ABSTRACT FROM AUTHOR]

It is known that Morse–Smale diffeomorphisms with two hyperbolic periodic orbits exist only on the sphere and they are all topologically conjugate to each other. However, if we allow three orbits to exist then the range of manifolds admitting them widens considerably. In particular, the surfaces of arbitrary genus admit such orientation-preserving diffeomorphisms. In this article we find a complete invariant for the topological conjugacy of Morse–Smale diffeomorphisms with three periodic orbits. The invariant is completely determined by the homotopy type (a pair of coprime numbers) of the torus knot which is the space of orbits of an unstable saddle separatrix in the space of orbits of the sink basin. We use the result to calculate the exact number of the topological conjugacy classes of diffeomorphisms under consideration on a given surface as well as to relate the genus of the surface to the homotopy type of the knot. [ABSTRACT FROM AUTHOR]

We introduce an alternative stratification of knots: by the size of the lattice on which a knot can be first met. Using this classification, we find the fraction of unknots and knots with more than minimal crossings inside different lattices and answer the question of which knots can be realized inside and lattices. In accordance with previous research, the fraction of unknots decreases exponentially with the growth of the lattice size. Our computational results are consistent with theoretical estimates for the number of knots with a fixed crossing number inside lattices of a given size. [ABSTRACT FROM AUTHOR]

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To ensure the smoothness of spatial Pythagorean-hodograph (PH) curve and minimize its rotation, spatial quintic PH curve construction was proposed. Quartic Bézier curve was constructed by quaternion method based on the given four spatial control pouts and the endpoint attributes. Quintic PH curve of Interpolated endpoint was obtained by integrating the quartic Bézier curve. Control points of PH curve can be obtained by inserting two new knots into the original control points, and both control points can be expressed by quaternion operation The PH curve contains many degrees off freedom, which can be used for optimization and adjustment. On the one hand, free parameters can be used to optimize the fairness and minimum rotation of the curve according to the curve geometry. On the other hand, reasonable are length can also be restricted by assigning a corresponding free parameter. Moreover, the PH curve is Better curve, which has the freedom to realize the smoothness splicing of these piecewise curves. [ABSTRACT FROM AUTHOR]

The article offers information on the importance of the strength of surgical knots in a surgery. Topics discussed include role of establishing physics-based operational and safety guidelines for strength of surgical sliding knots including focusing on commercial polypropylene suturing monofilaments, the article also characterizes sliding knots by quantifying their slippage resistance and discussed dependence of the knot strength on pre-tension and topology.

Building on the theory of circuit topology for intra-chain contacts in entangled proteins, we introduce tiles as a way to rigorously model local entanglements which are held in place by molecular forces. We develop operations that combine tiles so that entangled chains can be represented by algebraic expressions. Then we use our model to show that the only knot types that such entangled chains can have are 3 1 , 4 1 , 5 1 , 5 2 , 6 1 , 6 2 , 6 3 , 7 7 , 8 12 and connected sums of these knots. This includes all proteins knots that have thus far been identified. [ABSTRACT FROM AUTHOR]

This paper is an introduction to combinatorial knot theory via state summation models for the Jones polynomial and its generalizations. It is also a story about the developments that ensued in relation to the discovery of the Jones polynomial and a remembrance of Vaughan Jones and his mathematics. [ABSTRACT FROM AUTHOR]

Let r be an odd integer, r ≥ 3. Then the petal number of the torus knot of type (r , r + 2) is equal to 2 r + 3. [ABSTRACT FROM AUTHOR]

We use the Blanchfield form to obtain a lower bound on the equivariant slice genus of a strongly invertible knot. For our main application, let K$K$ be a strongly invertible genus one slice knot with nontrivial Alexander polynomial. We show that the equivariant slice genus of an equivariant connected sum #nK$\#^n K$ is at least n/4$n/4$. We also formulate an equivariant algebraic concordance group, and show that the kernel of the forgetful map to the classical algebraic concordance group is infinite rank. [ABSTRACT FROM AUTHOR]

We give an infinite family of knots that are not rationally concordant to their reverses. More precisely, if τ$\tau$ denotes the involution of the rational knot concordance group CQ$\mathcal {C}_\mathbb {Q}$ induced by reversal and Fix(τ)$\mbox{Fix}(\tau)$ denotes the subgroup of knots fixed under τ$\tau$ in CQ$\mathcal {C}_\mathbb {Q}$, then CQ/Fix(τ)$\mathcal {C}_\mathbb {Q}/\mbox{Fix}(\tau)$ contains an infinite rank subgroup. As a corollary, we show that there exists a knot K$K$ such that for every pair of coprime integers p$p$ and q$q$, the (p,q)$(p,q)$‐cable of K$K$ is not concordant to the reverse of the (p,q)$(p,q)$‐cable of K$K$. [ABSTRACT FROM AUTHOR]

We deal here with three metastable systems: 1° – the dowser texture, 2° – supertwisted cholesterics and 3° – hypotwisted cholesterics. We outline remarkable properties of the tropisms of the dowser texture stemming from its low symmetry, and we show that, using setups called Dowsons Colliders, nematic monopoles can be nucleated, set into motion and brought into collisions in the dowser texture. Subsequently, we point out that nucleation of dislocation loops occurs in cholesteric layers compressed or dilated between cylindrical mica sheets. Under compression, three modes of nucleation of dislocation loops have been identified: individual, serial and continuous. The serial mode generates dislocation nets made of L circular loops connected by C radial crossing into superloops. Similar superloops can also be generated under dilation. We analyse the topology of superloops in terms of the theory of knots and point out that they can be reduced to the unknot by Reidemeister moves. In other words, superloops are multiply folded loops that can be continuously unfolded. [ABSTRACT FROM AUTHOR]

A point source of light is placed inside an oval. The nth caustic by reflection is the envelope of the light rays emanating from the light source after n reflections off the curve. We show that, for a generic point light source, each of these caustics has at least 4 cusps. This is a billiard variation on Jacobi's Last Geometric Statement concerning the number of cusps of the conjugate locus of a point on a convex surface. We present various proofs, using different ideas, including the curve shortening flow and Legendrian knot theory. [ABSTRACT FROM AUTHOR]

The trip matrix, introduced by Louis Zulli [6], gives a method for computing the Jones polynomial of a knot using basic Linear Algebra. We use this method to provide an alternative proof of the formula for the Jones polynomial of T (n , 2) torus knots. In doing so, we provide a partial solution to an open question related to finding an elementary proof of the formula for general T (m , n) torus knots. [ABSTRACT FROM AUTHOR]

We formulate large N duality of U (N) refined Chern–Simons theory with a torus knot/link in S 3 . By studying refined BPS states in M-theory, we provide the explicit form of low-energy effective actions of Type IIA string theory with D4-branes on the Ω -background. This form enables us to relate refined Chern–Simons invariants of a torus knot/link in S 3 to refined BPS invariants in the resolved conifold. Assuming that the extra U (1) global symmetry acts on BPS states trivially, the duality predicts graded dimensions of cohomology groups of moduli spaces of M2–M5 bound states associated to a torus knot/link in the resolved conifold. Thus, this formulation can be also interpreted as a positivity conjecture of refined Chern–Simons invariants of torus knots/links. We also discuss about an extension to non-torus knots. [ABSTRACT FROM AUTHOR]

We recently proposed that topological quantum computing might be based on SL(2,C) representations of the fundamental group π1(S3\K) for the complement of a link K in the three-sphere. The restriction to links whose associated SL(2,C) character variety V contains a Fricke surface κd=xyz−x2−y2−z2+d is desirable due to the connection of Fricke spaces to elementary topology. Taking K as the Hopf link L2a1, one of the three arithmetic two-bridge links (the Whitehead link 512, the Berge link 622 or the double-eight link 632) or the link 732, the V for those links contains the reducible component κ4, the so-called Cayley cubic. In addition, the V for the latter two links contains the irreducible component κ3, or κ2, respectively. Taking ρ to be a representation with character κd (d<4), with |x|,|y|,|z|≤2, then ρ(π1) fixes a unique point in the hyperbolic space H3 and is a conjugate to a SU(2) representation (a qubit). Even though details on the physical implementation remain open, more generally, we show that topological quantum computing may be developed from the point of view of three-bridge links, the topology of the four-punctured sphere and Painlevé VI equation. The 0-surgery on the three circles of the Borromean rings L6a4 is taken as an example.

Im et al. [Signature, nullity and determinant of checkerboard colorable virtual links, J. Knot Theory Ramifications 19(8) (2010) 1093–1114] introduced how to define Goeritz matrices for checkerboard colorable virtual links. In this paper, we extend this for the Goeritz matrices of virtual knots. And we consider its signature and determinant and show they are invariants for virtual knots. [ABSTRACT FROM AUTHOR]

We investigate the nonorientable 4-genus γ 4 of a special family of 2-bridge knots, the double twist knots C (m , n). Because the nonorientable 4-genus is bounded by the nonorientable 3-genus, it is known that γ 4 (C (m , n)) ≤ 3. By using explicit constructions to obtain upper bounds on γ 4 and known obstructions derived from Donaldson's diagonalization theorem to obtain lower bounds on γ 4 , we produce infinite subfamilies of C (m , n) where γ 4 = 0 , 1 , 2 , and 3 , respectively. However, there remain infinitely many double twist knots where our work only shows that γ 4 lies in one of the sets { 1 , 2 } , { 2 , 3 } , or { 1 , 2 , 3 }. We tabulate our results for all C (m , n) with | m | and | n | up to 50. We also provide an infinite number of examples which answer a conjecture of Murakami and Yasuhara. [ABSTRACT FROM AUTHOR]

The Gordian complex of knots is a simplicial complex whose vertices consist of all knot types in 3 . 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 K 0 and for any given natural number n , there exists a family of knots { K 0 , K 1 , ... , K n } such that for any pair (K i , K j) of distinct elements of the family, the Gordian distance of knots by 4-move is d f (K i , K j) = 1. We also show the existence of an arbitrarily high dimensional simplex in the Gordian complexes. [ABSTRACT FROM AUTHOR]

We prove that parities on virtual knots come from invariant 1-cycles on the arcs of knot diagrams. In turn, the invariant cycles are determined by quasi-indices on the crossings of the diagrams. The found connection between the parities and the (quasi)-indices allows one to define a new series of parities on virtual knots. [ABSTRACT FROM AUTHOR]

When an overhand knot tied in an elastic rod is tightened, it can undergo a sudden change in shape through snap buckling. In this article, we use a combination of discrete differential geometry (DDG)-based simulations and tabletop experiments to explore the onset of buckling as a function of knot topology, rod geometry, and friction. In our setup, two open ends of an overhand knot are slowly pulled apart, which leads to snap buckling in the knot loop. We call this phenomenon "inversion" since the loop appears to move dramatically from one side of the knot to the other. This inversion occurs due to the coupling of elastic energy between the braid (the portion of the knot in self-contact) and the loop (the portion of the knot with two ends connected to the braid). A numerical framework is implemented that combines discrete elastic rods with a constraint-based method for frictional contact to explore inversion in overhand knots. The numerical simulation robustly captures inversion in the knot and is found to be in good agreement with experimental results. In order to gain physical insight into the inversion process, we also develop a simplified model of the knot that does not require simulation of self-contact, which allows us to visualize the bifurcation that results in snap buckling. [ABSTRACT FROM AUTHOR]

Consider two parallel lines ℓ 1 and ℓ 2 in ℝ 3 . A rail arc is an embedding of an arc in ℝ 3 such that one endpoint is on ℓ 1 , the other is on ℓ 2 , and its interior is disjoint from ℓ 1 ∪ ℓ 2 . Rail arcs are considered up to rail isotopies, ambient isotopies of ℝ 3 with each self-homeomorphism mapping ℓ 1 and ℓ 2 onto themselves. When the manifolds and maps are taken in the piecewise linear category, these rail arcs are called stick rail arcs. The stick number of a rail arc class is the minimum number of sticks, line segments in a p.l. arc, needed to create a representative. This paper calculates the stick number of rail arcs classes with a crossing number at most 2 and uses a winding number invariant to calculate the stick numbers of infinitely many rail arc classes. Each rail arc class has two canonically associated knot classes, its under and over companions. This paper also introduces the rail stick number of knot classes, the minimum number of sticks needed to create a rail arcs whose under or over companion is the knot class. The rail stick number is calculated for 29 knot classes with crossing number at most 9. The stick number of multi-component rail arcs classes is considered as well as the lattice stick number of rail arcs. [ABSTRACT FROM AUTHOR]

In 2013, Adams introduced the n -crossing number of a knot K , denoted by c n (K). Inequalities between the 2 -, 3 -, 4 - and 5 -crossing numbers have been previously established. We prove c 9 (K) ≤ c 3 (K) − 2 for all knots K that are not the trivial, trefoil, or figure-eight knot. We show this inequality is optimal and obtain previously unknown values for c 9 (K). [ABSTRACT FROM AUTHOR]