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Critical inquiry into sensitive social problems requires risk-taking by researchers and research participants alike. Detailed in this chapter are insights into risks as experienced by the scholar within a study involving Aboriginal participants in remote Australia. Research is both personal and political, requiring researchers to move deftly between each realm as they negotiate their legitimacy as agents of social change through this medium. Rather than avoiding personal vulnerability, the scholar can navigate personal and professional risks positively. Arguably, through actively embracing the knot of discomfort associated with risk-taking, researchers open themselves to deeper engagement with Aboriginal participant voice and expertise, thereby discovering innovative solutions to complex problems.

In this paper is provided, along with its simulation with knot points, a design and development of a mathematical model to create a straight-line motion and trajectory for any mechanical robotic handler using the boundary deviation principle. Here in the 3-dimensional Euclidean space, an algorithm for direct line motion is designed. An algorithm called the Constricted Non-conformity Algorithm (CNA) is introduced that to achieve a straight line, it can be used (the shortest path between the source & the target) through any type of robot (whether stationary robots or mobile robots). When the computer has been processed, this algorithm supplies the software interface for the mechanical robot. To get the robot to drive within the shortest distance situation, the mathematical model is used for straight line motion. C++ also executes a simulation to demonstrate the workings of the algorithm. A computational validation is often done by doing a case study to find a midpoint. The outcome of the simulation demonstrates how effective the technique is.

The art of knitting using precise machinery to bind individual threads into a single entity cannot be learned alone. To enhance my professional works, I sought an opportunity to learn machine knitting. I also learned to knit cords using six-needle cord knitting machine. After returning to my own institution, the lack of knitting equipment posed a challenge for my practicing, progressing, and perfecting of the techniques I learned. Nevertheless, I undertook the challenge and utilized my new understanding in knitting. The purpose of the design was to create a cocktail dress using an unconventional material— specifically yarn. The dress features bias drapes of yellow and blue ottoman stitch pieces with a blue all-knit piece and red cords strap forming a knot with tassels on the back. The yarn colors were selected for the contrast effect. The content of yarns is 100% acrylic with characteristics of durability, easy care, and softness.

We show that any two bridge positions of a handlebody-knot are stably equivalent., 14 pages, 18 figures

MJ0366 from Methanocaldococcus jannaschii is the smallest topologically knotted protein known to date. 92 residues in length, MJ0366 ties a trefoil (3(1)) knot by threading its C-terminal helix through a buttonhole formed by the remainder of the secondary structure elements. By generating a library of point mutations at positions pertinent to the knot formation, we systematically evaluated the contributions of individual residues to the folding stability and kinetics of MJ0366. The experimental Φ-values were used as restraints to computationally generate an ensemble of conformations that correspond to the transition state of MJ0366, which revealed several nonnative contacts. The importance of these nonnative contacts in stabilizing the transition state of MJ0366 was confirmed by a second round of mutagenesis, which also established the pivotal role of F15 in stapling the network of hydrophobic interactions around the threading C-terminal helix. Our converging experimental and computational results show that, despite the small size, the transition state of MJ0366 is formed at a very late stage of the folding reaction coordinate, following a polarized pathway. Eventually, the formation of extensive native contacts, as well as a number of nonnative ones, leads to the threading of the C-terminal helix that defines the topological knot.

We consider 32 homotopy classifications of knot projections (images of generic immersions from a circle into a 2-sphere). These 32 equivalence relations are obtained based on which moves are forbidden among the five type of Reidemeister moves. We show that 32 cases contain 20 non-trivial cases that are mutually different. To complete the proof, we obtain new tools, i.e., new invariants., 11 pages, 13 figures

The Gukov-Manolescu series, denoted by $F_K$, is a conjectural invariant of knot complements that, in a sense, analytically continues the colored Jones polynomials. In this paper we use the large color $R$-matrix to study $F_K$ for some simple links. Specifically, we give a definition of $F_K$ for positive braid knots, and compute $F_K$ for various knots and links. As a corollary, we present a class of `strange identities' for positive braid knots., 27 pages, 13 figures. v2 minor corrections

In 2001, \"Ostlund formulated the question: are Reidemeister moves of types 1 and 3 sufficient to describe a homotopy from any generic immersion of a circle in a two-dimensional plane to an embedding of the circle? The positive answer to this question was treated as a conjecture (\"Ostlund conjecture). In 2014, Hagge and Yazinski disproved the conjecture by showing the first counterexample with a minimal crossing number of 16. This example is naturally extended to counterexamples with given even minimal crossing numbers more than 14. This paper obtains the first counterexample with a minimal crossing number of 15. This example is naturally extended to counterexamples with given odd minimal crossing numbers more than 13., Comment: 8 pages, 14 figures

Background: Cardiac implantable electric device (CIED) implantation is a minor surgical procedure. One post-surgery complication often experienced is a suture knot projecting out of the skin 1-3 months after surgery. A unidirectional barbed suture (V-Loc; Covidien, Mansfield, MA) is a suture with barbs, which enables securing sutures without knots. The conventional sutures often leave a knot protruding at the suture line end; however, V-Loc eliminates the need to tie surgical knots and leaves a clean surface. Objective: We aimed to evaluate the safety and efficacy of V-Loc use in CIED implantation. Methods: A total of 153 patients underwent CIED implantation. Eighty-one patients were sutured using conventional string (PDS; Ethicon, Somerville, NJ) and V-Loc was used in 72 cases. A conventional method is to close a wound in three layers. After approximating the subcutaneous tissue in two layers with 3/0 PDS, the skin was closed with 4/0 PDS. Meanwhile, the new technique involves an over-and-over running stitch on the subcutaneous tissue with 3/0 V-Loc, followed by skin closure with 4/0 V-Loc. Results: The average suturing time was shorter in the V-Loc group than in the conventional group (9.9±1.7 vs. 18.0±4.1 mins. P Conclusion: V-loc suture material offers the possibility of a knotless stitch. We noted positive results in patients who were sutured with V-loc, and suggest its usefulness in CIED implantation.

Link/knot invariants are series with integer coefficients, and it is a long-standing problem to get them positive and possessing cohomological interpretation. Constructing positive "superpolynomials" is not straightforward, especially for colored invariants. A simpler alternative is a multi-parametric generalization of the character expansion, which leads to colored "hyperpolynomials". The third construction involves branes on resolved conifolds, which gives rise to still another family of invariants associated with composite representations. We revisit this triality issue in the simple case of the Hopf link and discover a previously overlooked way to produce positive colored superpolynomials from the DIM-governed four-point functions, thus paving a way to a new relation between super- and hyperpolynomials., Comment: 12 pages

We define the affinization of an arbitrary monoidal category $\mathcal{C}$, corresponding to the category of $\mathcal{C}$-diagrams on the cylinder. We also give an alternative characterization in terms of adjoining dot generators to $\mathcal{C}$. The affinization formalizes and unifies many constructions appearing in the literature. In particular, we describe a large number of examples coming from Hecke-type algebras, braids, tangles, and knot invariants. When $\mathcal{C}$ is rigid, its affinization is isomorphic to its horizontal trace, although the two definitions look quite different. In general, the affinization and the horizontal trace are not isomorphic., 32 pages; v2: minor corrections, published version

Surgical knot tying is one of the most fundamental and important procedures in surgery, and a high-quality knot can significantly benefit the postoperative recovery of the patient. However, a longtime operation may easily cause fatigue to surgeons, especially during the tedious wound closure task. In this paper, we present a vision-based method to automate the suture thread grasping, which is a sub-task in surgical knot tying and an intermediate step between the stitching and looping manipulations. To achieve this goal, the acquisition of a suture's three-dimensional (3D) information is critical. Towards this objective, we adopt a transfer-learning strategy first to fine-tune a pre-trained model by learning the information from large legacy surgical data and images obtained by the on-site equipment. Thus, a robust suture segmentation can be achieved regardless of inherent environment noises. We further leverage a searching strategy with termination policies for a suture's sequence inference based on the analysis of multiple topologies. Exact results of the pixel-level sequence along a suture can be obtained, and they can be further applied for a 3D shape reconstruction using our optimized shortest path approach. The grasping point considering the suturing criterion can be ultimately acquired. Experiments regarding the suture 2D segmentation and ordering sequence inference under environmental noises were extensively evaluated. Results related to the automated grasping operation were demonstrated by simulations in V-REP and by robot experiments using Universal Robot (UR) together with the da Vinci Research Kit (dVRK) adopting our learning-driven framework.

Monte Carlo simulations are used to study the conformational behavior of a semiflexible polymer confined to cylindrical and conical channels. The channels are sufficiently narrow that the conditions for the Odijk regime are marginally satisfied. For cylindrical confinement, we examine polymers with a single knot of topology $3_1$, $4_1$, or $5_1$, as well as unknotted polymers that are capable of forming S-loops. We measure the variation of the free energy $F$ with the end-to-end polymer extension length $X$ and examine the effect of varying the polymer topology, persistence length $P$ and cylinder diameter $D$ on the free energy functions. Similarly, we characterize the behavior of the knot span along the channel. We find that increasing the knot complexity increases the typical size of the knot. In the regime of low $X$, where the knot/S-loop size is large, the conformational behavior is independent of polymer topology. In addition, the scaling properties of the free energy and knot span are in agreement with predictions from a theoretical model constructed using known properties of interacting polymers in the Odijk regime. We also examine the variation of $F$ with position of a knot in conical channels for various values of the cone angle $\alpha$. The free energy decreases as the knot moves in a direction where the cone widens, and it also decreases with increasing $\alpha$ and with increasing knot complexity. The behavior is in agreement with predictions from a theoretical model in which the dominant contribution to the change in $F$ is the change in the size of the hairpins as the knot moves to the wider region of the channel., Comment: 15 pages, 11 figures, supplemental information

Lab Study Script and Protocols for manual tracing study using Tibetan knot pattern (Prime) and asymmetrical pattern (Anti-prime) and Control (non-tracing).

Background: Vascular ligation and its detachment remain a necessary skill for a surgeon. We often use two threads of the same color to ligate a vessel. However, some problems (grasping the mixed-up threads and/or difficulty in releasing the entanglement) may occur while using same colored ligatures. To solve these problems, we devised to use ligatures of different colors, and here, we present this ingenious idea. Methods: When a vessel is encountered that needs to be cut, we pass two different colored threads behind it. Results: We can grasp the two threads without mixing them up. In addition, in case of entanglement, unwinding of the entangled threads is quicker. We then finally tie a knot with these threads and cut the vessel visible between them. Conclusion: Using different colored ligatures enhance surgical convenience in the operating room.

We prove the analyticity of smooth critical points for O’Hara’s knot energies [Formula: see text], with [Formula: see text] and [Formula: see text], subject to a fixed length constraint. This implies, together with the already established regularity results for O’Hara’s knot energies, that bounded energy critical points of [Formula: see text] subject to a fixed length constraint are not only [Formula: see text] but also analytic. Our approach is based on Cauchy’s method of majorants and a decomposition of the gradient that was adapted from the Möbius energy case [Formula: see text].

Background In our institute, all elderly patients with displaced femoral neck fracture were treated with cemented bipolar hemiarthroplasty (BHA) using the modified Dall approach. To our knowledge, there are no reports on the knot position of the greater trochanter reattachment. The aim of this study was to determine influence of two knot positions (anterior or posterior) on the complications of the greater trochanter. Methods This is a prospective non-randomized study conducted on 95 elderly patients (95 hips) from September 2013 to December 2017. The knot position was changed from anterior to posterior alternately. The X-ray images obtained immediately after the operation were compared with those obtained at 3 months postoperatively; thereafter, the status of the greater trochanter was classified into three types: type A, no apparent shifting and fracture; type C, over 1-mm shifting of the fragment; and type F, fracture of the greater trochanter. Results Regarding age at operation, sex, BMI, size of the greater trochanteric fragment, stem type, and surgeon, there was no significant difference between two groups. In the anterior group, 34 hips (72.3%), 5 hips (10.6%), and 8 hips (17.0%) were classified under type A, C, and F, respectively. In the posterior group, 44 hips (91.7%), 1 hip (2.1%), and 3 hips (6.3%) were classified under type A, C, and F, respectively. There were significantly fewer greater trochanteric complications in the posterior group. Conclusions The posterior knot position improved the union of the greater trochanter after BHA compared with the anterior knot position. Trial registration We had approved IRB at our hospital clinical research review committee. Retrospectively registered.

We recently defined an invariant of contact manifolds with convex boundary in Kronheimer and Mrowka's sutured monopole Floer homology theory. Here, we prove that there is an isomorphism between sutured monopole Floer homology and sutured Heegaard Floer homology which identifies our invariant with the contact class defined by Honda, Kazez and Mati\'c in the latter theory. One consequence is that the Legendrian invariants in knot Floer homology behave functorially with respect to Lagrangian concordance. In particular, these invariants provide computable and effective obstructions to the existence of such concordances. Our work also provides the first proof which does not rely on the relative Giroux correspondence that the vanishing or non-vanishing of Honda, Kazez and Mati\'c's contact class is a well-defined invariant of contact manifolds., Comment: 63 pages, 13 figures; v2: corrected Lemma 3.3 and subsequent material, many other small changes; v3: accepted version, substantially revised to correct the proof of the main theorem

We consider a large class of q-series that have the structure of Nahm sums, or equivalently motivic generating series for quivers. First, we initiate a systematic analysis and classification of classical and quantum A-polynomials associated to such q-series. These quantum quiver A-polynomials encode recursion relations satisfied by the above series, while classical A-polynomials encode asymptotic expansion of those series. Second, we postulate that those series, as well as their quantum quiver A-polynomials, can be reconstructed by means of the topological recursion. There is a large class of interesting quiver A-polynomials of genus zero, and for a number of them we confirm the above conjecture by explicit calculations. In view of recently found dualities, for an appropriate choice of quivers, these results have a direct interpretation in topological string theory, knot theory, counting of lattice paths, and related topics. In particular it follows, that various quantities characterizing those systems, such as motivic Donaldson-Thomas invariants, various knot invariants, etc., have the structure compatible with the topological recursion and can be reconstructed by its means.

We argue that the usual Bloch sphere is insufficient in various aspects for the representation of qubits in quantum information theory. For example, spin flip operations with the quaternions I J K = e 2 &pi, i 2 = &minus, 1 and J I K = + 1 cannot be distinguished on the Bloch sphere. We show that a simple knot theoretic extension of the Bloch sphere representation is sufficient to track all unitary operations for single qubits. Next, we extend the Bloch sphere representation to entangled states using knot theory. As applications, we first discuss contextuality in quantum physics&mdash, in particular the Kochen-Specker theorem. Finally, we discuss some arguments against many-worlds theories within our knot theoretic model of entanglement. The key ingredients of our approach are symmetries and geometric properties of the unitary group.

In ordinary logistic regression, the logit of the conditional probability of the response given the covariates is modelled as a linear function of the covariates. In this study, a more general logistic regression model is considered, where linearity is not assumed. Since the linear function of the covariates is replaced by a general function of the covariates, spline approximation is used. A knot selection algorithm is proposed to determine the knot locations in spline approximation. Simulation experiments have been carried out to check the performance of the proposed algorithm. The proposed algorithm performs reasonably well.

A classic knot is an embedding of an one-dimensional sphere S1 in a real three-dimensional environment, usually R3. Under these conditions it is possible to consider the knot diagram, that is, the planar projection of the embedding. This resembles a curve in which the intersections are exchanged for interruptions in its trace, thus indicating that one arc passes over the other. In Knot Theory we study algebraic invariants extracted from the complement of the embedding, and this complement is visible in the case of embedding S1 --> R3. One of the invariants extracted from the knot diagram is the Alexander Polynomial. In this article we show how the process of determining the Alexander Polynomial of a knot can be transported to an algorithm implemented in Python Language and obtained from a drawn diagram with the aid of a computer mouse.

Letting $C$ be a compact $C^\omega$-curve embedded in $\boldsymbol R^3$ ($C^\omega$ means real analyticity), we consider a $C^\omega$-cuspidal edge $f$ along $C$. When $C$ is non-closed, in the authors' previous works, the local existence of three distinct cuspidal edges along $C$ whose first fundamental forms coincide with that of $f$ was shown, under a certain reasonable assumption on $f$. In this paper, if $C$ is closed, that is, $C$ is a knot, we show that there exist infinitely many cuspidal edges along $C$ having the same first fundamental form as that of $f$ such that their images are non-congruent to each other, in general., Comment: 12 pages, 2 figures

We study $q$-series-valued invariants of 3-manifolds that depend on the choice of a root system $G$. This is a natural generalization of the earlier works by Gukov-Pei-Putrov-Vafa [arXiv:1701.06567] and Gukov-Manolescu [arXiv:1904.06057] where they focused on $G={\rm SU}(2)$ case. Although a full mathematical definition for these ''invariants'' is lacking yet, we define $\hat{Z}^G$ for negative definite plumbed 3-manifolds and $F_K^G$ for torus knot complements. As in the $G={\rm SU}(2)$ case by Gukov and Manolescu, there is a surgery formula relating $F_K^G$ to $\hat{Z}^G$ of a Dehn surgery on the knot $K$. Furthermore, specializing to symmetric representations, $F_K^G$ satisfies a recurrence relation given by the quantum $A$-polynomial for symmetric representations, which hints that there might be HOMFLY-PT analogues of these 3-manifold invariants.

Given a knot in the three-sphere, is it possible to unknot it by performing a single twist, and if so, what are the possible linking numbers of such a twist? We develop obstructions to unknotting using a twist of a specified linking number. The obstructions we describe are built using classical knot invariants, Casson-Gordon invariants, and Heegaard Floer theory., 26 pages, 4 figures

The isogeometric boundary element method (IGABEM) has great potential for the simulation of elasticity problems because of its exact geometric representation and good approximation properties. These advantages can be extended to thin structures, including coatings, but the development of an accurate and efficient method to deal with the large number of nearly singular integrals existing in the IGABEM presents a great challenge for very thin sections. In this paper, we propose a new sinh + scheme for weakly, strongly and hyper near-singular integrals arising in 3D IGABEM for thermoelastic problems, based on the sinh transformation method and adaptive integral method. The presented scheme is efficient, since it combines the advantages of both methods: (1) when the thickness δ of coatings/thin structures is moderately small, an accurate and efficient integral result will be obtained by the adaptive integral method; (2) when δ is very small, the nearly singular integrals are computed by the sinh + scheme efficiently. With the introduction of NURBS in IGABEM, truncation errors arising in the Taylor expansion cannot be ignored. Based on the values of these errors, the computed knot spans are further divided into several sub-knot spans and different methods will be used to evaluate the integral over each sub-knot span in the new scheme. In addition, based on the analytical extension of the NURBS surface, an adaptation of the sinh transformation method is proposed which can evaluate the near-singular integrals accurately for cases in which the projection point lies outside of the considered knot span. Several numerical examples are presented to validate the accuracy and efficiency of the 3D IGABEM based on the sinh + scheme in the analysis of thermoelastic problems.

We study random knotting by considering knot and link diagrams as decorated, (rooted) topological maps on spheres and sampling them with the counting measure on from sets of a fixed number of vertices n. We prove that random rooted knot diagrams are highly composite and hence almost surely knotted (this is the analogue of the Frisch-Wasserman-Delbruck conjecture) and extend this to unrooted knot diagrams by showing that almost all knot diagrams are asymmetric. The model is similar to one of Dunfield, et al.

In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 1). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. 3). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary 3)., 13 pages, 25 figures

A quandle is an algebra with two binary operations satisfying three conditions which are related to Reidemeister moves in knot theory. In this paper we introduce the notion of the (canonical) tensor product of a quandle. The tensor product of the knot quandle or the knot symmetric quandle of a surface-link in 4-space can be used to classify or construct invariants of 1-handles attaching to the surface-link. We also compute the tensor products for dihedral quandles and their symmetric doubles.

Topological nodal line semimetals host stable chained, linked, or knotted line degeneracies in momentum space protected by symmetries. In this Letter, we use the Jones polynomial as a general topological invariant to capture the global knot topology of the oriented nodal lines. We show that every possible change in Jones polynomial is attributed to the local evolutions around every point where two nodal lines touch. As an application of our theory, we show that nodal chain semimetals with four touching points can evolve to a Hopf link. We extend our theory to 3D non-Hermitian multiband exceptional line semimetals. Our work provides a recipe to understand the transition of the knot topology for protected nodal lines.

In this manuscript, we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves. We define the bracket polynomial of curves in 3-space and show that it has real coefficients and is a continuous function of the curve coordinates. This is used to define the Jones polynomial in a way that it is applicable to both open and closed curves in 3-space. For open curves, the Jones polynomial has real coefficients and it is a continuous function of the curve coordinates and as the endpoints of the curve tend to coincide, the Jones polynomial of the open curve tends to that of the resulting knot. For closed curves, it is a topological invariant, as the classical Jones polynomial. We show how these measures attain a simpler expression for polygonal curves and provide a finite form for their computation in the case of polygonal curves of 3 and 4 edges.

We employ a recently developed method for constructing rational electromagnetic field configurations in Minkowski space to investigate several properties of these source-free finite-action Maxwell ("knot") solutions. The construction takes place on the Penrose diagram but uses features of de Sitter space, in particular its isometry group. This admits a classification of all knot solutions in terms of $S^3$ harmonics, labelled by a spin $2j\in{\mathbb N}_0$, which in fact provides a complete "knot basis" of finite-action Maxwell fields. We display a $j{=}1$ example, compute the energy for arbitrary spin-$j$ configurations, derive a linear relation between spin and helicity and characterize the subspace of null fields. Finally, we present an expression for the electromagnetic flux at null infinity and demonstrate its equality with the total energy., 1+14 pages LaTeX, 4 figures in 6 parts; v2: additional j=1 knot solution, discussion and illustration of energy densitites and field lines, 3 more refs., published in PLA

We present a new conjectural symmetry of the colored Alexander polynomial, that is the specialization of the quantum $$\mathfrak {sl}_N$$ invariant widely known as the colored HOMFLY-PT polynomial. We provide arguments in support of the existence of the symmetry by studying the loop expansion and the character expansion of the colored HOMFLY-PT polynomial. We study the constraints this symmetry imposes on the group theoretic structure of the loop expansion and provide solutions to those constraints. The symmetry is a powerful tool for research on polynomial knot invariants and in the end we suggest several possible applications of the symmetry.

By studying Rozansky-Witten theory with non-compact target spaces we find new connections with knot invariants whose physical interpretation was not known. This opens up several new avenues, which include a new formulation of $q$-series invariants of 3-manifolds in terms of affine Grassmannians and a generalization of Akutsu-Deguchi-Ohtsuki knot invariants., Comment: 33 pages, 1 figure, 7 tables

Spór Cypryjski jest jednym z najważniejszych zagadnień w tureckiej polityce zagranicznej od lat pięćdziesiątych. Jest on przeważnie postrzegany jako sprawa ponad podziałami politycznymi, i prawie wszystkie tureckie partie polityczne popierają zaangażowanie państwa tureckiego na Cyprze od lat sześćdziesiątych i od Cypryjskiej Operacji Pokojowej w 1974 roku. Jednakże, po tym jak Partia AK (Partia Sprawiedliwości i Rozwoju) doszła do władzy w 2002 roku, z głównym motywem zapobieżenia świeckiego narodowego zamachu stanu, jak również mając na celu osiągnięcie pełnego członkostwa w Unii Europejskiej, partia ta przyjęła pro-aktywną politykę zagraniczną sprzyjającą rozładowaniu sytuacji na tej wyspie. Właśnie dlatego Turcja i tureccy Cypryjczycy aktywnie poparli referendum w ramach planu Annana w 2004 roku. Jednakże, po odrzuceniu planu ugody przez greckich Cypryjczyków, i po decyzji Unii Europejskiej o zaakceptowaniu rządu grecko-cypryjskiego jako członka UE i jedynego reprezentanta Republiki Cypru w tej wspólnocie, Spór Cypryjski stał się problemem, psującym stosunki nie tylko turecko-cypryjskie i grecko-cypryjskie, ale również te pomiędzy Turcją a Unią Europejską. W 2010 roku, wraz z odkryciem złóż gazu ziemnego i porozumieniami zawartymi przez administrację grecko-cypryjską, Spór Cypryjski nabrał nowego wymiaru. Jak dotąd polityka energetyczna nie mogła być użyta jako instrument w promowaniu rozwiązania na wyspie, tymczasem spór przekształcił się w o wiele bardziej skomplikowany problem, z wieloma wymiarami. Cyprus Dispute is one of the fundamental foreign policy issues in Turkish foreign policy since the 1950s. Cyprus Dispute has often been perceived as an issue above petty politics in Turkey, and almost all Turkish political parties supported the Turkish State’s involvement in Cyprus since the 1960s and Cyprus Peace Operation in 1974. However, after AK Party came to power in 2002, with the main motive of preventing a secular nationalist military coup, as well as with the aim of becoming a full member of the European Union (EU), the party adopted a proactive foreign policy favoring the solution in the island. That is why Turkey and Turkish Cypriots actively supported the Annan Plan referendum in 2004. However, upon the rejection of the settlement plan by Greek Cypriots and EU’s decision to accept Greek Cypriot government as an EU member and the only representative of the Cyprus Republic, Cyprus Dispute has transformed into a problem spoiling not only Turkish-Greek and Turkish-Cypriot relations but also Turkish-EU relations. In the 2010s, the Cyprus Dispute gained a new dimension with the gas discoveries and agreements made by the Greek Cypriot administration. So far, energy politics could not be used as a leverage to promote the solution on the island, and the dispute has transformed into a more complex problem with many layers.

For the approximation of B-spline or T-spline surfaces to point clouds the computation of a parametrization is required. Since the construction of both surface types is based upon the concept of B-spline curves, we begin by developing parametrization methods for curves. Parametrization for B-Spline curve approximation includes the computation of a point parametrization t as well as the knot vector u for a given set of ordered points.

We show that a knot whose minimum crossing number $c(K)$ is even and greater than $30$ is not fertile; there exists a knot $K'$ with crossing number less than $c$ such that $K'$ is not obtained from a minimum crossing number diagram of $K$ by suitably changing the over-under information., Comment: 3 pages, no figures