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The Neumann–Zagier matrices of an ideal triangulation are integer matrices with symplectic properties whose entries encode the number of tetrahedra that wind around each edge of the triangulation. They can be used as input data for the construction of a number of quantum invariants that include the loop invariants, the 3D-index and state-integrals. We define a twisted version of Neumann–Zagier matrices, describe their symplectic properties, and show how to compute them from the combinatorics of an ideal triangulation. As a sample application, we use them to define a twisted version of the 1-loop invariant (a topological invariant) which determines the 1-loop invariant of the cyclic covers of a hyperbolic knot complement, and conjecturally is equal to the adjoint twisted Alexander polynomial. [ABSTRACT FROM AUTHOR]

In this work, we find a closed form formula for the braid index of an n -bridge braid, a class of positive braid knots which simultaneously generalizes torus knots, 1-bridge braids, and twisted torus knots. Our proof is elementary, effective, and self-contained, and partially recovers work of Birman–Kofman. Along the way, we show that the disparate definitions of twisted torus knots in the literature agree. [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]

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 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]

We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a folded ribbon knot. The folded ribbon knot is also a framed knot, and the ribbon linking number is the linking number of the knot and one boundary component of the ribbon. We find the minimum folded ribbonlength for 3 -stick unknots with ribbon linking numbers ± 1 and ± 3 , and we prove that the minimum folded ribbonlength for n -gons with obtuse interior angles is achieved when the n -gon is regular. Among other results, we prove that the minimum folded ribbonlength of any folded ribbon unknot which is a topological annulus with ribbon linking number ± n is bounded from above by 2 n. [ABSTRACT FROM AUTHOR]

Symmetric, elegantly entangled structures are a curious mathematical construction that has found their way into the heart of the chemistry lab and the toolbox of constructive geometry. Of particular interest are those structures—knots, links and weavings—which are composed locally of simple twisted strands and are globally symmetric. This paper considers the symmetric tangling of multiple 2-periodic honeycomb networks. We do this using a constructive methodology borrowing elements of graph theory, low-dimensional topology and geometry. The result is a wide-ranging enumeration of symmetric tangled honeycomb networks, providing a foundation for their exploration in both the chemistry lab and the geometers toolbox. [ABSTRACT FROM AUTHOR]

Previous work [C. Shonkwiler, New computations of the superbridge index, J. Knot Theory Ramifications 29(14) (2020) 2050096] used polygonal realizations of knots to reduce the problem of computing the superbridge number of a realization to a linear programming problem, leading to new sharp upper bounds on the superbridge index of a number of knots. This work extends this technique to polygonal realizations with an odd number of edges and determines the exact superbridge index of many new knots, including the majority of the 9-crossing knots for which it was previously unknown and, for the first time, several 12-crossing knots. Interestingly, at least half of these superbridge-minimizing polygonal realizations do not minimize the stick number of the knot; these seem to be the first such examples. Appendix A gives a complete summary of what is currently known about superbridge indices of prime knots through 10 crossings and Appendix B gives all knots through 16 crossings for which the superbridge index is known. [ABSTRACT FROM AUTHOR]

In this paper, we look at which Alexander and Markov theorems can be defined for generalized knot theories. [ABSTRACT FROM AUTHOR]

A group of matrices with entries in a number field K is integral if it has a finite index subgroup of matrices whose entries are algebraic integers. In this paper, we show that the fundamental groups, G (p / q , (m , n)) , of the orbifolds (p / q , (m , n) are integral as subgroups of both S L (2 , ℂ) and of S L (4 , ℝ) , for all the rational knots and links p / q and all the isotropies with m > 2 , n > 2. We obtain the same result for the fundamental group G (m , m , m) of the orbifold B (m , m , m) , m > 2 , where B are the Borromean rings. The only groups G (m , n , p) with m , n , p ∈ 3 , 4 , 5 , 6 , ∞ , which are integral subgroups of both S L (2 , ℂ) and of S L (4 , ℝ) , are for the following (m , n , p) : { (3 , 3 , 3) , (3 , 3 , ∞) , (3 , 4 , 4) , (3 , 4 , ∞) , (3 , 6 , 6) , (3 , ∞ , ∞) , (4 , 4 , 4) , (4 , 4 , ∞) , (4 , ∞ , ∞) , (6 , 6 , 6) , (6 , 6 , ∞) , (∞ , ∞ , ∞) , (3 , 3 , 5) , (3 , 5 , 5) , (3 , 5 , ∞) , (4 , 4 , 5) , (4 , 5 , ∞) , (5 , 5 , 5) , (5 , 5 , ∞) , (5 , 6 , 6) , (5 , ∞ , ∞) }. [ABSTRACT FROM AUTHOR]

Let M be a connected, closed, oriented three-manifold and K, L two rationally null-homologous oriented simple closed curves in M. We give an explicit algorithm for computing the linking number between K and L in terms of a presentation of M as an irregular dihedral three-fold cover of S 3 branched along a knot α ⊂ S 3 . Since every closed, oriented three-manifold admits such a presentation, our results apply to all (well-defined) linking numbers in all three-manifolds. Furthermore, ribbon obstructions for a knot α can be derived from dihedral covers of α . The linking numbers we compute are necessary for evaluating one such obstruction. This work is a step toward testing potential counter-examples to the Slice-Ribbon Conjecture, among other applications. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Lehmer’s question, an open question about the Mahler measure of monic integral polynomials, is shown to be equivalent to a question about the complexity growth rate of signed 1-periodic graphs. If G is a d-periodic graph (i.e. G has a co-finite free ℤd-action by automorphisms), then a d-variable polynomial ∆G can be defined with Mahler measure equal to the logarithmic growth rate γG of a complexity defined for the finite quotients of G. A plane 1-periodic graph determines a link via projection and the medial graph construction. The polynomial ∆G can be determined from the Alexander polynomial of the link. The complexity growth rate γG of any d-periodic graph is at least log 2. An investigation of plane 1- and 2-periodic graphs yields more connections with knot theory including work of A. Champanerkar and I. Kofman. [ABSTRACT FROM AUTHOR]

Mathematical knots and links are described as piecewise linear – straight, non‐intersecting sticks meeting at corners. Isogonal structures have all corners related by symmetry ('vertex'‐transitive). Corner‐ and stick‐transitive structures are termed regular. No regular knots are found. Regular links are cubic or icosahedral and a complete account of these (36 in number) is given, including optimal (thickest‐stick) embeddings. Stick 2‐transitive isogonal structures are again cubic and icosahedral and also encompass the infinite family of torus knots and links. The major types of these structures are identified and reported with optimal embeddings. The relevance of this work to materials chemistry and biochemistry is noted. [ABSTRACT FROM AUTHOR]

For any classical knot k 1 , we can construct a ribbon 2 -knot spun (k 1) by spinning an arc removed a small segment from k 1 about R 2 in R 4 . A ribbon 2 -knot is an embedded 2 -sphere in R 4 . If k 1 has an n -crossing presentation, by spinning this, we can naturally construct a ribbon presentation with n ribbon crossings for spun (k 1). Thus, we can define naturally a notion on ribbon 2 -knots corresponding to the crossing number on classical knots. It is called the ribbon crossing number. On classical knots, it was a long-standing conjecture that any odd crossing classical knot is not amphicheiral. In this paper, we show that for any odd integer n there exists an amphicheiral ribbon 2 -knot with the ribbon crossing number n. [ABSTRACT FROM AUTHOR]

We prove that the knots 1 3 n 5 9 2 and 1 5 n 4 1 , 1 2 7 both have stick number 10. These are the first non-torus prime knots with more than 9 crossings for which the exact stick number is known. [ABSTRACT FROM AUTHOR]

This paper applies the intuitive techniques of [9] to certain 3-colored knot diagrams, generating corresponding 3-fold branched covering spaces with predictable linking numbers between branch curves. [ABSTRACT FROM AUTHOR]

For a knot or link 𝒦 , let L (𝒦) be the ropelength of 𝒦 and Cr (𝒦) be the crossing number of 𝒦. In this paper, we show that there exists a constant a > 0 such that L (𝒦) ≤ a Cr (𝒦) ln 5 (Cr (𝒦)) for any 𝒦 , i.e. the upper bound of the ropelength of any knot is almost linear in terms of its minimum crossing number. This result is a significant improvement over the best known upper bound established previously, which is of the form L (𝒦) ≤ O (Cr (𝒦) 3 2 ). The proof is based on a divide-and-conquer approach on 4-regular plane graphs: a 4-regular plane graph of n is first repeatedly subdivided into many small subgraphs and then reconstructed from these small subgraphs on the cubic lattice with its topology preserved with a total length of the order O (n ln 5 (n)). The result then follows since a knot can be recovered from a graph that is topologically equivalent to a regular projection of it (which is a 4-regular plane graph). [ABSTRACT FROM AUTHOR]

A ribbon is a smooth mapping (possibly self-intersecting) of an annulus S 1 × I in 3-space having constant width R. Given a regular parametrization x (s) , and a smooth unit vector field u (s) based along x , for a knot K , we may define a ribbon of width R associated to x and u as the set of all points x (s) + r u (s) , r ∈ [ 0 , R ]. For large R , ribbons, and their outer edge curves, may have self-intersections. In this paper, we analyze how the knot type of the outer ribbon edge x (s) + R u (s) relates to that of the original knot K. Generically, as R → ∞ , there is an eventual constant knot type. This eventual knot type is one of only finitely many possibilities which depend just on the vector field u. The particular knot type within the finite set depends on the parametrized curves x (s) , u (s) , and their interactions. We demonstrate a way to control the curves and their parametrizations so that given two knot types K 1 and K 2 , we can find a smooth ribbon of constant width connecting curves of these two knot types. [ABSTRACT FROM AUTHOR]

We review some our results concerning the topology of knotted and linked defects in nematic liquid crystals. We discuss the global topological classification of nematic textures with defects, showing how knotted and linked defect lines have a finite number of 'internal states', counted by the Alexander polynomial of the knot or link. We then give interpretations of these states in terms of umbilic lines, which we also introduce, as well as planar textures. We show how Milnor polynomials can be used to give explicit constructions of these textures. Finally, we discuss some open problems raised by this work. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of a space-efficient knot mosaic. That is, we seek to determine how to create knot mosaics using the least number of non-blank tiles necessary to depict the knot. This least number is called the tile number of the knot. We determine strict bounds for the tile number of a knot in terms of the mosaic number of the knot. We also determine the tile number of several knots and provide space-efficient knot mosaics for each of them. [ABSTRACT FROM AUTHOR]

In this paper, the properties of the Kauffman bracket skein module (KBSM) of are investigated. Links in lens spaces are represented both through band and disk diagrams. The possibility to transform between the diagrams enables us to compute the KBSM on an interesting class of examples consisting of inequivalent links with equivalent lifts in the -sphere. The computations show that the KBSM is an essential invariant, that is, it may take different values on links with equivalent lifts. We also show how the invariant is related to the Kauffman bracket of the lift in the -sphere. [ABSTRACT FROM AUTHOR]

In this review, we provide an overview of entangled proteins. Around 6% of protein structures deposited in the PBD are entangled, forming knots, slipknots, lassos and links. We present theoretical methods and tools that enabled discovering and classifying such structures. We discuss the advantages and disadvantages of the non-trivial topology in proteins, based on available data about folding, stability, biological properties and evolutionary conservation. We also formulate intriguing and challenging questions on the border of biophysics, bioinformatics, biology and mathematics, which arise from the discovery of an entanglement in proteins. Finally, we discuss possible applications of entangled proteins in medicine and nanotechnology, such as the chance to design super stable proteins, whose stability could be controlled by chemical potential. [ABSTRACT FROM AUTHOR]

We prove for a large class of knots that the meridional rank coincides with the bridge number. This class contains all knots whose exterior is a graph manifold. This gives a partial answer to a question of S. Cappell and J. Shaneson [10, pb 1.11] . [ABSTRACT FROM AUTHOR]

Checkerboard framings are an extension of checkerboard colorings for virtual links. From checkerboard framings, we obtain an independent invariant of virtual links: the cut point number. Checkerboard framings and cut points can be used as a tool to extend other classical invariants to virtual links. [ABSTRACT FROM AUTHOR]

We revisit the issue of the existence of infinitely many distinct prime knots with the same Alexander invariant. We present infinitely many distinct families, each family made up of infinitely many distinct knots. Within each family, the Alexander invariant is the same. Unlike, other examples in the literature, ours are elementary and based on a sub-collection of pretzel knots with three tassels. [ABSTRACT FROM AUTHOR]

We introduce an invariant of tangles in Khovanov homology by considering a natural inverse system of Khovanov homology groups. As application, we derive an invariant of strongly invertible knots; this invariant takes the form of a graded vector space that vanishes if and only if the strongly invertible knot is trivial. While closely tied to Khovanov homology — and hence the Jones polynomial — we observe that this new invariant detects non-amphicheirality in subtle cases where Khovanov homology fails to do so. In fact, we exhibit examples of knots that are not distinguished by Khovanov homology but, owing to the presence of a strong inversion, may be distinguished using our invariant. This work suggests a strengthened relationship between Khovanov homology and Heegaard Floer homology by way of two-fold branched covers that we formulate in a series of conjectures. [ABSTRACT FROM AUTHOR]

We extend the concepts of trivializing and knotting numbers for knots to spatial graphs and 2-bouquet graphs, in particular. Furthermore, we calculate the trivializing and knotting numbers for projections and pseudodiagrams of 2-bouquet spatial graphs based on the number of precrossings and the placement of the precrossings in the pseudodiagram of the spatial graph. [ABSTRACT FROM AUTHOR]

In this paper, we classify, up to rigid isotopy, real rational knots of degrees less than or equal to in a real quadric homeomorphic to the 3-sphere. We also study their connections with rigid isotopy classes of real rational knots of low degree in and classify real rational curves of degree 6 in the 3-sphere with exactly one ordinary double point. [ABSTRACT FROM AUTHOR]

For any knot in the 3-sphere S 3 , there exists a diagram such that we have the unknot if we change all crossings on the boundary of some region of the diagram. The minimal number of the crossing changes over all such diagrams is called the region index of a knot. Clearly, the unknotting number is less than or equal to the region index. In this paper, we show that there exists a knot which has a gap between the unknotting number m and the region index for any positive integer m ( m ≥ 2 ) by using the Goeritz invariant. We also show that there exists a knot which has the unknotting number and the region index that are equal to n for any positive integer n ( n ≥ 2 ) by using the Rasmussen invariant. [ABSTRACT FROM AUTHOR]

Smale flows on 3-manifolds can have invariant saddle sets that are suspensions of shifts of finite type. We look at Smale flows with chain recurrent sets consisting of an attracting closed orbit a , a repelling closed orbit r and a saddle set that is a suspension of a full n -shift and draw some conclusions about the knotting and linking of a ∪ r . For example, we show for all values of n it is possible for a and r to be unknots. For any even value of n it is possible for a ∪ r to be the Hopf link, a trefoil and meridian, or a figure-8 knot and meridian. [ABSTRACT FROM AUTHOR]

In this article we show that if a knot diagram admits a non-trivial coloring modulo 13 then there is an equivalent diagram which can be colored with 5 colors. Leaning on known results, this implies that the minimum number of colors modulo 13 is 5. [ABSTRACT FROM AUTHOR]

We show that an alternating knot with unknotting number one has an unknotting crossing in any alternating diagram. We also prove that an alternating knot has unknotting number one if and only if its branched double cover arises as half-integer surgery on a knot in S 3 , thus establishing a converse to the Montesinos trick. Along the way, we reprove a characterisation of almost-alternating diagrams of the unknot originally due to Tsukamoto. These results are established using the obstruction to unknotting number one developed by Greene. [ABSTRACT FROM AUTHOR]

We study the extrinsic geometry of Legendrian knots in the standard tight contact structure on In particular, we show that the total curvature of a Legendrian knot in is bounded below by times, the total number of cusps in the front projection of . We also show that a Legendrian -torus knot has the total curvature bounded below by while that of the Legendrian knots is bounded below by . Furthermore, we find an explicit relation between the Thurston-Bennequin number of a Legendrian knot and the geometric self-linking number, the curvature and the torsion of the knot . [ABSTRACT FROM AUTHOR]

Knots appear frequently in semiflexible (bio)polymers, including double-stranded DNA, and their presence can affect the polymer’s physical and functional properties. In particular, it is possible and indeed often the case that multiple knots appear on a single chain, with effects which have only come under scrutiny in the last few years. In this manuscript, we study the interaction of two knots on a stretched semiflexible polymer, expanding some recent results on the topic. Specifically, we consider an idealization of a typical optical tweezers experiment and show how the bending rigidity of the chain—And consequently its persistence length—Influences the distribution of the entanglements; possibly more importantly, we observe and report how the relative chirality of the otherwise identical knots substantially modifies their interaction. We analyze the free energy of the chain and extract the effective interactions between embedded knots, rationalizing some of their pertinent features by means of simple effective models. We believe the salient aspect of the knot–knot interactions emerging from our study will be present in a large number of semiflexible polymers under tension, with important consequences for the characterization and manipulation of these systems—Be they artificial or biologica in origin—And for their technological application. [ABSTRACT FROM AUTHOR]

We call a delta diagram any diagram of a knot or link whose regions (including the unbounded one) have 3, 4, or 5 sides. We prove that any knot or link admits a delta diagram. We define and estimate combinatorial link invariants stemming from this definition. [ABSTRACT FROM AUTHOR]

There are strong indications that the history of design may have begun with the concept of a meander. This paper explores the application of meanders to new classes of meander and semi-meander knots, meander friezes, labyrinths and mazes. A combinatorial system is introduced to classify meander knots and labyrinths. Mazes are analyzed with the use of graphs. Meanders are also created with the use of simple proto-tiles upon which a series of lines are etched. [ABSTRACT FROM AUTHOR]

Plaited polyhedra, discovered by P. Gerdes in African dance rattle capsules are analyzed from the knot-theory point of view. Every plaited polyhedron can be derived from a knot or link diagram as its dual. In the attempt to classify obtained plaited polyhedra, we propose different methods based on families of knots and links in Conway notation or their corresponding braid families, both leading to the notion of families of plaited polyhedra. [ABSTRACT FROM AUTHOR]

In this paper we propose and discuss implications of a general conjecture that there is a natural action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K\subset S^{3}$. We prove this in a number of nontrivial cases, including all $(2,2p+1)$ torus knots, the figure eight knot, and all 2-bridge knots (when $q=\pm 1$). As the main application of the conjecture, we construct three-variable polynomial knot invariants that specialize to the classical colored Jones polynomials introduced by Reshetikhin and Turaev. We also deduce some new properties of the classical Jones polynomials and prove that these hold for all knots (independently of the conjecture). We furthermore conjecture that the skein module of the unknot is a submodule of the skein module of an arbitrary knot. We confirm this for the same example knots, and we show that this implies that the colored Jones polynomials of $K$ satisfy an inhomogeneous recursion relation. [ABSTRACT FROM PUBLISHER]

We prove that, given any knot γ in a compact 3-manifold M , there exists a Riemannian metric on M such that there is a complex-valued eigenfunction u of the Laplacian, corresponding to the first nontrivial eigenvalue, whose nodal set u − 1 ( 0 ) has a connected component given by γ . Higher dimensional analogs of this result will also be considered. [ABSTRACT FROM AUTHOR]

We illustrate that there are knots for which Heegaard knot Floer homology and Khovanov homology are identical but the Alexander module and torsion invariants differ. The examples are certain symmetric unions. We also give examples of similar flavor, concerning the Kauffman and Q-polynomials in place of the classical homological invariants. This shows there are nonmutant knots with the same knot Floer and Khovanov homology. [ABSTRACT FROM AUTHOR]

We prove that any knot or link in any -manifold can be nicely decomposed ( split) by a filling Dehn sphere. This has interesting consequences in the study of branched coverings over knots and links. We give an algorithm for computing Johansson diagrams of filling Dehn surfaces out from coverings of -manifolds branched over knots or links. [ABSTRACT FROM AUTHOR]

This paper illustrates some classical non-cyclic milestones in mathematical knot theory. [ABSTRACT FROM AUTHOR]

The SH ( 3 ) -move is an unknotting operation on oriented knots, and the SH ( 3 ) -Gordian distance of two knots is the minimum number of SH ( 3 ) -moves needed to transform one into the other, which is half of the coherent band-Gordian distance. We give a table of SH ( 3 ) -Gordian distances between knots with up to seven crossings. [ABSTRACT FROM AUTHOR]

We investigate the question of when distinct branched surfaces in the complement of a 2-bridge knot support essential surfaces with identical boundary slopes. We determine all instances in which this occurs and identify an infinite family of knots for which no boundary slopes are repeated. [ABSTRACT FROM AUTHOR]

We compute the unknotting number of two infinite families of pretzel knots, P ( 3 , 1 , … , 1 , b ) (with b odd and at least 3 and an odd number of 1s) and P ( 3 , 3 , 3 c ) (with c positive and odd). To do this, we extend a technique of Owens using Donaldson's diagonalization theorem, and one of Traczyk using the Jones polynomial, building on work of Lickorish and Millett. [ABSTRACT FROM AUTHOR]

Witten's approach to Khovanov homology of knots is based on the five-dimensional system of partial differential equations, which we call Haydys-Witten equations. We argue for a one-to-one correspondence between its solutions and solutions of the seven-dimensional system of equations. The latter can be formulated on any G2 holonomy manifold and is a close cousin of the monopole equation of Bogomolny. Octonions play the central role in our view, in which both the seven-dimensional equations and the Haydys-Witten equations appear as reductions of the eight-dimensional Spin(7) instanton equation. [ABSTRACT FROM AUTHOR]

We address here the topological equivalence of knots through the so-called Reidemeister moves. These topology-conserving manipulations are recast into dynamical rules on the crossings of knot diagrams. This is presented in terms of a simple graphical representation related to the Gauss code of knots. Drawing on techniques for reaction-diffusion systems, we then develop didactically an operator formalism wherein these rules for crossing dynamics are encoded. The aim is to develop new tools for studying dynamical behaviour and regimes in the presence of topology conservation. This necessitates the introduction of composite paulionic operators. The formalism is applied to calculate some differential equations for the time evolution of densities and correlators of crossings, subject to topology-conserving stochastic dynamics. We consider here the simplified situation of two-dimensional knot projections. However, we hope that this is a first valuable step towards addressing a number of important questions regarding the role of topological constraints and specifically of topology conservation in dynamics through a variety of solution and approximation schemes. Further applicability arises in the context of the simulated annealing of knots. The methods presented here depart significantly from the invariant-based path integral descriptions often applied in polymer systems, and, in our view, offer a fresh perspective on the conservation of topological states and topological equivalence in knots. [ABSTRACT FROM AUTHOR]