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Journal articles on the topic 'Tangles'

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Author: Grafiati

Published: 4 June 2021

Last updated: 6 September 2023

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1

O'Keeffe, Michael, and Michael M. J. Treacy. "Tangled piecewise-linear embeddings of trivalent graphs." Acta Crystallographica Section A Foundations and Advances 78, no. 2 (February 18, 2022): 128–38. http://dx.doi.org/10.1107/s2053273322000560.

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A method is described for generating and exploring tangled piecewise-linear embeddings of trivalent graphs under the constraints of point-group symmetry. It is shown that the possible vertex-transitive tangles are either graphs of vertex-transitive polyhedra or bipartite vertex-transitive nonplanar graphs. One tangle is found for 6 vertices, three for 8 vertices (tangled cubes), seven for 10 vertices, and 21 for 12 vertices. Also described are four isogonal embeddings of pairs of cubes and 12 triplets of tangled cubes (16 and 24 vertices, respectively). Vertex 2-transitive embeddings are obtained for tangled trivalent graphs with 6 vertices (two found) and 8 vertices (45 found). Symmetrical tangles of the 10-vertex Petersen graph and the 20-vertex Desargues graph are also described. Extensions to periodic tangles are indicated. These are all interesting and viable targets for molecular synthesis.

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2

Kim, Soo Hwan. "Stability of the Cauchy Additive Functional Equation on Tangle Space and Applications." Advances in Mathematical Physics 2016 (2016): 1–10. http://dx.doi.org/10.1155/2016/4030658.

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We introduce real tangle and its operations, as a generalization of rational tangle and its operations, to enumerating tangles by using the calculus of continued fraction and moreover we study the analytical structure of tangles, knots, and links by using new operations between real tangles which need not have the topological structure. As applications of the analytical structure, we prove the generalized Hyers-Ulam stability of the Cauchy additive functional equation fx⊕y=fx⊕fy in tangle space which is a set of real tangles with analytic structure and describe the DNA recombination as the action of some enzymes on tangle space.

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3

Cochran, Tim D., and Daniel Ruberman. "Invariants of tangles." Mathematical Proceedings of the Cambridge Philosophical Society 105, no. 2 (March 1989): 299–306. http://dx.doi.org/10.1017/s0305004100067785.

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A tangle is a pair of strings (t0, t1) properly embedded in a 3-ball. Tangles have been used in several approaches to the classification of knots (see [1, 4, 15]). In these investigations, one keeps track of the endpoints of the arcs, so that the sum of two tangles along their boundaries is well defined. In particular, the sum of a given tangle with a trivial tangle, and any invariants of the resulting link, are invariants of the tangle under the restricted relation of isotopy keeping the endpoints fixed.

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4

BOGDANOV, ANDREY, VADIM MESHKOV, ALEXANDER OMELCHENKO, and MIKHAIL PETROV. "ENUMERATING THE k-TANGLE PROJECTIONS." Journal of Knot Theory and Its Ramifications 21, no. 07 (April 7, 2012): 1250069. http://dx.doi.org/10.1142/s0218216512500691.

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The paper addresses the enumeration problem for k-tangles. We introduce the notion of a cascade diagram of a k-tangle projection and suggest an effective enumeration algorithm for projections based on the cascade representation. Tangle projections and alternating tangles with up to 12 crossings are tabulated. We also provide pictures of alternating k-tangles with at most five crossings.

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5

Kwon, Bo-Hyun. "Uniqueness of reduced alternating rational 3-tangle diagrams." Journal of Knot Theory and Its Ramifications 24, no. 05 (April 2015): 1550030. http://dx.doi.org/10.1142/s0218216515500303.

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Tangles were introduced by J. Conway. In 1970, he proved that every rational 2-tangle defines a rational number and two rational 2-tangles are isotopic if and only if they have the same rational number. So, from Conway's result we have a perfect classification for rational 2-tangles. However, there is no similar theorem to classify rational 3-tangles. In this paper, we introduce an invariant of rational n-tangles which is obtained from the Kauffman bracket. It forms a vector with Laurent polynomial entries. We prove that the invariant classifies the rational 2-tangles and the reduced alternating rational 3-tangles. We conjecture that it classifies the rational 3-tangles as well.

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6

EMERT, J., and C. ERNST. "N-STRING TANGLES." Journal of Knot Theory and Its Ramifications 09, no. 08 (December 2000): 987–1004. http://dx.doi.org/10.1142/s021821650000058x.

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An n-string tangle (B, T) is a 3-ball B containing n properly embedded arcs T={ti}. An n-string tangle (B, T) is called rational if it is a homeo-morphism of pairs from (B, T) to (D, P)×I where D is the unit disk, P is any set of n points in the interior of D and I is the unit interval on the real line. In this article we begin to generalize the well known classification of 2-string rational tangles to rational tangles with three or more strings. A symbol describing an n-string rational tangle and an algebraic topological invariant is developed.

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7

BRITTENHAM, MARK. "PERSISTENTLY LAMINAR TANGLES." Journal of Knot Theory and Its Ramifications 08, no. 04 (June 1999): 415–28. http://dx.doi.org/10.1142/s0218216599000286.

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We show how to build tangles T in a 3-ball with the property that any knot obtained by tangle sum with T has a persistent lamination in its exterior, and therefore has property P. The construction is based on an example of a persistent lamination in the exterior of the twist knot 61, due to Ulrich Oertel. We also show how the construction can be generalized to n-string tangles.

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8

Kwon, Bo-Hyun. "On the classification of a large set of rational 3-tangle diagrams." Journal of Knot Theory and Its Ramifications 26, no. 13 (November 2017): 1750087. http://dx.doi.org/10.1142/s0218216517500870.

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We note that a rational [Formula: see text]-tangle diagram is obtained from a combination of four generators. There is an algorithm to distinguish two rational [Formula: see text]-tangle diagrams up to isotopy: [B. Kwon, An algorithm to classify rational [Formula: see text]-tangles, J. Knot Theory Ramifications 24(1) (2015) 1550004]. However, there is no perfect classification about rational [Formula: see text]-tangle diagrams such as the classification of rational [Formula: see text]-tangle diagrams corresponding to rational numbers. In this paper, we classify a large set of rational [Formula: see text]-tangles which are generated by only three generators.

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9

Patil, Vishal P., Harry Tuazon, Emily Kaufman, Tuhin Chakrabortty, David Qin, Jörn Dunkel, and M. Saad Bhamla. "Ultrafast reversible self-assembly of living tangled matter." Science 380, no. 6643 (April 28, 2023): 392–98. http://dx.doi.org/10.1126/science.ade7759.

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Tangled active filaments are ubiquitous in nature, from chromosomal DNA and cilia carpets to root networks and worm collectives. How activity and elasticity facilitate collective topological transformations in living tangled matter is not well understood. We studied California blackworms ( Lumbriculus variegatus ), which slowly form tangles in minutes but can untangle in milliseconds. Combining ultrasound imaging, theoretical analysis, and simulations, we developed and validated a mechanistic model that explains how the kinematics of individual active filaments determines their emergent collective topological dynamics. The model reveals that resonantly alternating helical waves enable both tangle formation and ultrafast untangling. By identifying generic dynamical principles of topological self-transformations, our results can provide guidance for designing classes of topologically tunable active materials.

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10

Milani, Vida, Seyed M. H. Mansourbeigi, and Hossein Finizadeh. "Algebraic and topological structures on rational tangles." Applied General Topology 18, no. 1 (April 3, 2017): 1. http://dx.doi.org/10.4995/agt.2017.2250.

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<p>In this paper we present the construction of a group Hopf algebra on the class of rational tangles. A locally finite partial order on this class is introduced and a topology is generated. An interval coalgebra structure associated with the locally finite partial order is specified. Irrational and real tangles are introduced and their relation with rational tangles are studied. The existence of the maximal real tangle is described in detail.</p>

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11

Nelson, Sam, and Veronica Rivera. "Quantum enhancements of involutory birack counting invariants." Journal of Knot Theory and Its Ramifications 23, no. 07 (June 2014): 1460006. http://dx.doi.org/10.1142/s0218216514600062.

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The involutory birack counting invariant is an integer-valued invariant of unoriented tangles defined by counting homomorphisms from the fundamental involutory birack of the tangle to a finite involutory birack over a set of framings modulo the birack rank of the labeling birack. In this first of an anticipated series of several papers, we enhance the involutory birack counting invariant with quantum weights, which may be understood as tangle functors of involutory birack-labeled unoriented tangles.

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12

Alishahi, Akram, and Eaman Eftekhary. "Tangle Floer homology and cobordisms between tangles." Journal of Topology 13, no. 4 (September 24, 2020): 1582–657. http://dx.doi.org/10.1112/topo.12168.

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13

CHUNG, JAE-WOOK. "THE INVARIANT OF n-PUNCTURED BALL TANGLES." Journal of Knot Theory and Its Ramifications 19, no. 10 (October 2010): 1247–90. http://dx.doi.org/10.1142/s021821651000842x.

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Based on the Kauffman bracket at A = eiπ/4, we define an invariant for a special type of n-punctured ball tangles. The invariant Fn takes values in the set PM2 × 2n(ℤ) of 2 × 2n matrices over ℤ modulo the scalar multiplication of ±1. We provide the formula to compute the invariant of the k1 + ⋯ + kn-punctured ball tangle composed of given n, k1, …, kn-punctured ball tangles. Also, we define the horizontal and the vertical connect sums of punctured ball tangles and provide the formulae for their invariants from those of given punctured ball tangles. In addition, we introduce the elementary operations on the class ST of 1-punctured ball tangles, called spherical tangles. The elementary operations on ST induce the operations on PM2 × 2(ℤ), also called the elementary operations. We show that the group generated by the elementary operations on PM2 × 2(ℤ) is isomorphic to a Coxeter group.

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14

Nogueira, João Miguel, and António Salgueiro. "The minimum crossing number of essential tangles." Journal of Knot Theory and Its Ramifications 23, no. 10 (September 2014): 1450054. http://dx.doi.org/10.1142/s0218216514500540.

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In this paper we compute the sharp lower bounds for the crossing number of n-string k-loop essential tangles. For essential tangles with only string components, we characterize the ones with the minimum crossing number for a given number of components, both when the tangle has knotted strings or only unknotted strings.

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15

DARCY, ISABEL K. "SOLVING UNORIENTED TANGLE EQUATIONS INVOLVING 4-PLATS." Journal of Knot Theory and Its Ramifications 14, no. 08 (December 2005): 993–1005. http://dx.doi.org/10.1142/s0218216505004202.

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The system of unoriented tangle equations [Formula: see text] and [Formula: see text] is completely solved for the tangles U and [Formula: see text] as a function of [Formula: see text] where K1 and K2 are 4-plats, and [Formula: see text] and [Formula: see text] rational tangles such that |f1g2 - g1f2| > 1.

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16

Mold, Matthew John, Adam O’Farrell, Benjamin Morris, and Christopher Exley. "Aluminum and Neurofibrillary Tangle Co-Localization in Familial Alzheimer’s Disease and Related Neurological Disorders." Journal of Alzheimer's Disease 78, no. 1 (October 27, 2020): 139–49. http://dx.doi.org/10.3233/jad-200838.

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Background: Protein misfolding disorders are frequently implicated in neurodegenerative conditions. Familial Alzheimer’s disease (fAD) is an early-onset and aggressive form of Alzheimer’s disease (AD), driven through autosomal dominant mutations in genes encoding the amyloid precursor protein and presenilins 1 and 2. The incidence of epilepsy is higher in AD patients with shared neuropathological hallmarks in both disease states, including the formation of neurofibrillary tangles. Similarly, in Parkinson’s disease, dementia onset is known to follow neurofibrillary tangle deposition. Objective: Human exposure to aluminum has been linked to the etiology of neurodegenerative conditions and recent studies have demonstrated a high level of co-localization between amyloid-β and aluminum in fAD. In contrast, in a donor exposed to high levels of aluminum later developing late-onset epilepsy, aluminum and neurofibrillary tangles were found to deposit independently. Herein, we sought to identify aluminum and neurofibrillary tangles in fAD, Parkinson’s disease, and epilepsy donors. Methods: Aluminum-specific fluorescence microscopy was used to identify aluminum in neurofibrillary tangles in human brain tissue. Results: We observed aluminum and neurofibrillary-like tangles in identical cells in all respective disease states. Co-deposition varied across brain regions, with aluminum and neurofibrillary tangles depositing in different cellular locations of the same cell. Conclusion: Neurofibrillary tangle deposition closely follows cognitive-decline, and in epilepsy, tau phosphorylation associates with increased mossy fiber sprouting and seizure onset. Therefore, the presence of aluminum in these cells may exacerbate the accumulation and misfolding of amyloidogenic proteins including hyperphosphorylated tau in fAD, epilepsy, and Parkinson’s disease.

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17

Petit, Nicolas. "The Wriggle polynomial for virtual tangles." Journal of Knot Theory and Its Ramifications 28, no. 14 (December 2019): 1950087. http://dx.doi.org/10.1142/s0218216519500871.

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We generalize the Wriggle polynomial, first introduced by L. Folwaczny and L. Kauffman, to the case of virtual tangles. This generalization naturally arises when considering the self-crossings of the tangle. We prove that the generalizations (and, by corollary, the original polynomial) are Vassiliev invariants of order one for virtual knots, and study some simple properties related to the connected sum of tangles.

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18

CHUNG, JAE-WOOK, and XIAO-SONG LIN. "ON n-PUNCTURED BALL TANGLES." Journal of Knot Theory and Its Ramifications 15, no. 06 (August 2006): 715–48. http://dx.doi.org/10.1142/s0218216506004695.

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We consider a class of topological objects in the 3-sphere S3which will be called n-punctured ball tangles. Using the Kauffman bracket at A = eiπ/4, an invariant for a special type of n-punctured ball tangles is defined. The invariant F takes values in PM2×2n(ℤ), that is the set of 2 × 2nmatrices over ℤ modulo the scalar multiplication of ±1. This invariant leads to a generalization of a theorem of Krebes which gives a necessary condition for a given collection of tangles to be embedded in a link in S3disjointly. We also address the question of whether the invariant F is surjective onto PM2×2n(ℤ). We will show that the invariant F is surjective when n = 0. When n = 1, n-punctured ball tangles will also be called spherical tangles We show that det F(S) ≡ 0 or 1 mod 4 for every spherical tangle S. Thus, F is not surjective when n = 1.

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19

SUZUKI, SAKIE. "On the universal sl2 invariant of Brunnian bottom tangles." Mathematical Proceedings of the Cambridge Philosophical Society 154, no. 1 (October 1, 2012): 127–43. http://dx.doi.org/10.1017/s0305004112000503.

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AbstractThe universal sl2 invariant is an invariant of bottom tangles from which one can recover the colored Jones polynomial of links. We are interested in the relationship between topological properties of bottom tangles and algebraic properties of the universal sl2 invariant. A bottom tangle T is called Brunnian if every proper subtangle of T is trivial. In this paper, we prove that the universal sl2 invariant of n-component Brunnian bottom tangles takes values in a small subalgebra of the n-fold completed tensor power of the quantized enveloping algebra Uh(sl2). As an application, we give a divisibility property of the colored Jones polynomial of Brunnian links.

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20

CABRERA-IBARRA, HUGO, and DAVID A. LIZÁRRAGA-NAVARRO. "BRAID SOLUTIONS TO THE ACTION OF THE GIN ENZYME." Journal of Knot Theory and Its Ramifications 19, no. 08 (August 2010): 1051–74. http://dx.doi.org/10.1142/s0218216510008327.

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The topological analysis of enzymes, an active research topic, has allowed the application of the tangle model of Ernst and Sumners to deduce the action mechanism of several enzymes, modeled as 2-string tangles. By first deriving some results in the theory of 3-braids, in this paper we analyze knotted and linked products of site-specific recombination mediated by the Gin DNA invertase, an enzyme that involves 3-string tangles. Provided that the 3-tangles involved are 3-braids, we determine four families of solutions to its action, two families for each of the directly and inversely repeated site cases. For each case, one of the given solutions had not previously been reported in the related literature. These solutions were found using a computer algorithm, based on our theoretical results, which allows one to solve tangle equations under the assumption that the product of two or more rounds of recombinations is known.

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21

Elbracht, Christian, Jakob Kneip, and Maximilian Teegen. "Obtaining trees of tangles from tangle-tree duality." Journal of Combinatorics 13, no. 2 (2022): 251–87. http://dx.doi.org/10.4310/joc.2022.v13.n2.a3.

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22

Ochiai, Mitsuyuki, and Noriko Morimura. "Base-Tangle Decompositions ofn-String Tangles with 1." Experimental Mathematics 17, no. 1 (January 2008): 1–8. http://dx.doi.org/10.1080/10586458.2008.10129024.

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23

Saito, Toshio. "Examples of nested essential free tangles." Journal of Knot Theory and Its Ramifications 25, no. 04 (April 2016): 1650020. http://dx.doi.org/10.1142/s0218216516500206.

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It is proved by Ozawa that every knot in the 3-sphere has a unique essential tangle decomposition if it admits an essential free 2-tangle sphere. In this paper, we show that for every integer [Formula: see text] with [Formula: see text], there is a knot in the 3-sphere which admits an essential free [Formula: see text]-tangle decomposition and also admits at least [Formula: see text] distinct essential tangle spheres. To this end, we give examples of nested essential free [Formula: see text]-tangles for every integer [Formula: see text] with [Formula: see text].

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24

SHIMOKAWA, KOYA. "PARALLELISM OF TWO STRINGS IN ALTERNATING TANGLES." Journal of Knot Theory and Its Ramifications 07, no. 04 (June 1998): 489–502. http://dx.doi.org/10.1142/s0218216598000255.

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We consider the parallelism of two strings in alternating tangles. We show that if there is a pair of parallel strings in an alternating tangle then its alternating diagrams satify certain conditions. As a corollary, for a knot admitting a decomposition into two alternating tangles with two or three strings, we prove that its non-trivial Dehn surgery yields a 3-manifold with an essential lamination. Hence such a knot has property P and satisfy the cabling conjecture.

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25

DARCY, ISABEL K. "SOLVING ORIENTED TANGLE EQUATIONS INVOLVING 4-PLATS." Journal of Knot Theory and Its Ramifications 14, no. 08 (December 2005): 1007–27. http://dx.doi.org/10.1142/s0218216505004214.

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The system of unoriented tangle equations [Formula: see text] and [Formula: see text] is completely solved for the tangles U and [Formula: see text] as a function of [Formula: see text] where K1 and K2 are 4-plats, and [Formula: see text] and [Formula: see text] rational tangles such that |f1g2 - g1f2| > 1. As an application, it is completely determined when one 4-plat can be obtained from another 4-plat via a signed crossing change.

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26

Jung, E., M. R. Hwang, D. Park, and S. Tamaryan. "Three-party entanglement in tripartite teleportation scheme through noisy channels." Quantum Information and Computation 10, no. 5&6 (May 2010): 377–97. http://dx.doi.org/10.26421/qic10.5-6-2.

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In this paper we have tried to interpret the physical role of three-tangle and $\pi$-tangle in real physical information processes. For the model calculation we adopt the tripartite teleportation scheme through various noisy channels. The three parties consist of sender, accomplice and receiver. It is shown that the $\pi$-tangles for the X- and Z-noisy channels vanish at the limit $\kappa t \rightarrow \infty$, where $\kappa t$ is a decoherence parameter introduced in the master equation in the Lindblad form. At this limit the maximum fidelity of the receiver's state reduces to the classical limit $2/3$. However, this nice feature is not maintained for the Y- and isotropy-noise channels. For the Y-noise channel the $\pi$-tangle vanishes when $0.61 \leq \kappa t$. At $\kappa t = 0.61$ the maximum fidelity becomes $0.57$, which is much less than the classical limit. Similar phenomenon occurs for the isotropic noise channel. We also compute analytically the three-tangles for the X- and Z-noise channels. The remarkable fact is that the three-tangle for the Z-noise channel coincides exactly with the corresponding $\pi$-tangle. In the X-noise channel the three-tangle vanishes when $0.10 \leq \kappa t$. At $\kappa t = 0.10$ the fidelity of the receiver's state can reduce to the classical limit provided that the accomplice performs the measurement appropriately. However, the maximum fidelity becomes $8/9$, which is much larger than the classical limit. Since the Y- and isotropy-noise channels are rank-$8$ mixed states, their three-tangles are not computed explicitly in this paper. Instead, their upper bounds are derived by making use of the analytic formulas of the three-tangle for other noisy channels. Our analysis strongly suggests that different tripartite entanglement measure is needed whose value is between three-tangle and $\pi$-tangle.

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27

Druivenga, Nathan, Charles Frohman, and Sanjay Kumar. "Tangle functors from semicyclic representations." Journal of Knot Theory and Its Ramifications 26, no. 11 (October 2017): 1750065. http://dx.doi.org/10.1142/s0218216517500651.

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Let [Formula: see text] be a [Formula: see text]th root of unity where [Formula: see text] is odd. Let [Formula: see text] denote the quantum group with large center corresponding to the Lie algebra [Formula: see text] with generators [Formula: see text], and [Formula: see text]. A semicyclic representation of [Formula: see text] is an [Formula: see text]-dimensional irreducible representation [Formula: see text], so that [Formula: see text] with [Formula: see text], [Formula: see text] and [Formula: see text]. We construct a tangle functor for framed homogeneous tangles colored with semicyclic representations, and prove that for [Formula: see text]-tangles coming from knots, the invariant defined by the tangle functor coincides with Kashaev’s invariant.

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28

Pemba, Kadidia Adula. "Tangles." Journal of LGBT Youth 6, no. 1 (January 2, 2009): 91. http://dx.doi.org/10.1080/19361650802396924.

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29

LEE, H. C. "ON SEIFERT CIRCLES AND FUNCTORS FOR TANGLES." International Journal of Modern Physics A 07, supp01b (April 1992): 581–610. http://dx.doi.org/10.1142/s0217751x9200394x.

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The properties of the Seifert circles in an oriented tangle diagram are exploited to prove a theorem that asserts that every (n,n)-tangle diagram is isotopic to a partially closed braid, and a second one that facilitates the assignment of wrong-way edges, one on each Seifert circle, in a tangle diagram. These result are used to identify the structure of an abstract algebra on which a functor for the isotopy of general tangles may be constructed. Any finite dimensional irreducible representation of a quasitriangular Hopf algebra is a realization of this algebra.

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30

ERNST, C. "TANGLE EQUATIONS." Journal of Knot Theory and Its Ramifications 05, no. 02 (April 1996): 145–59. http://dx.doi.org/10.1142/s0218216596000114.

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We are given fixed rational tangles P and R and knots or links K1 and K2. Let O be a tangle such that N(O+P)=K1and N(O+R)=K2, where N is the numerator construction on the tangles O+P and O+R, respectively. N(O+P)=K1and N(O+R)=K1 form two equations, in which O is treated as a variable and P, R, K1and K2 are kept fixed. The number and types of solutions for O in the cases where K1and K2 are 4-plats or Montesinos knots are discussed.

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31

Brion, J. P., D. P. Hanger, M. T. Bruce, A. M. Couck, J. Flament-Durand, and B. H. Anderton. "Tau in Alzheimer neurofibrillary tangles. N- and C-terminal regions are differentially associated with paired helical filaments and the location of a putative abnormal phosphorylation site." Biochemical Journal 273, no. 1 (January 1, 1991): 127–33. http://dx.doi.org/10.1042/bj2730127.

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To investigate the extent to which whole tau proteins, structurally abnormal tau and fragments of tau are incorporated into neurofibrillary tangles in Alzheimer's disease, an immunocytochemical mapping study using a panel of antibodies to several synthetic human tau peptides has been performed. Neurofibrillary tangles were immunolabelled in situ, and paired helical filaments (PHF), the principal structural component of tangles, were immunolabelled after isolation and Pronase treatment. N-Terminal and C-terminal domains of tau were found to be present in tangles in situ. SDS-treated PHF were found to contain most of the C-terminal half of tau and were also labelled by antibodies to ubiquitin. Only some of these PHF were labelled by antisera to tau sequences towards the N-terminus, and this enabled the identification of a region of tau in which proteolytic cleavage may occur. The ultrastructural appearance of the immunolabelling suggested that both the N- and C-terminal domains of tau extend outwards from the axis of PHF. After Pronase treatment. PHF were strongly labelled only by an antiserum to PHF and by the antiserum to the most C-terminal tau synthetic peptide. The latter antiserum also strongly labelled extracellular tangles in situ, whereas these extracellular tangles were poorly labelled by the antisera to the other synthetic peptides. One anti-(tau peptide) serum labelled a population of neurofibrillary tangles in situ only after alkaline phosphatase pretreatment of tissue sections. Our results show that, although peptides along the length of the tau molecule are associated with neurofibrillary tangles in situ, only the C-terminal one-third of the molecule is tightly associated with PHF, since this region of tau is resistant to SDS treatment of PHF. We also report the existence in PHF in situ of a masked tau epitope which is partially unmasked by dephosphorylation. These results are indicative of post-translational changes in tangle-associated tau in degenerating neurons in Alzheimer's disease.

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32

Kwon, B. "An algorithm to classify rational 3-tangles." Journal of Knot Theory and Its Ramifications 24, no. 01 (January 2015): 1550004. http://dx.doi.org/10.1142/s0218216515500042.

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A 3-tangle T is the disjoint union of three properly embedded arcs in the unit 3-ball; it is called rational if there is a homeomorphism of pairs from (B3, T) to (D2 × I, {x1, x2, x3} × I). Two rational 3-tangles T and T′ are isotopic if there is an orientation-preserving self-homeomorphism h : (B3, T) → (B3, T′) that is the identity map on the boundary. In this paper, we give an algorithm to check whether or not two rational 3-tangles are isotopic by using a modified version of Dehn's method for classifying simple closed curves on surfaces.

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33

KREBES, DAVID A. "AN OBSTRUCTION TO EMBEDDING 4-TANGLES IN LINKS." Journal of Knot Theory and Its Ramifications 08, no. 03 (May 1999): 321–52. http://dx.doi.org/10.1142/s0218216599000213.

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We consider the ways in which a 4-tangle T inside a unit cube can be extended outside the cube into a knot or link L. We present two links n(T) and d(T) such that the greatest common divisor of the determinants of these two links always divides the determinant of the link L. In order to prove this result we give a two-integer invariant of 4-tangles. Calculations are facilitated by viewing the determinant as the Kauffman bracket at a fourth root of -1, which sets the loop factor to zero. For rational tangles, our invariant coincides with the value of the associated continued fraction.

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34

CABRERA-IBARRA, HUGO. "ON THE CLASSIFICATION OF RATIONAL 3-TANGLES." Journal of Knot Theory and Its Ramifications 12, no. 07 (November 2003): 921–46. http://dx.doi.org/10.1142/s021821650300286x.

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An n-string tangle (B3,T) is a 3-ball B3 containing n properly embedded arcs T={ti} and it is called rational if there is a homeomorphism of pairs from (B3,T) to (D,P)×I where D is the unit disk, P is any set of n points in the interior of D and I is the unit interval. In this article we partially generalize the well known classification of 2-string rational tangles to 3-string rational tangles. Using the results obtained we analyze knotted and linked products of site-specific recombination mediated by the Gin DNA invertase and give a possible solution to its action. Gin is an enzyme that carries out processive recombination.

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35

Saito, Toshio. "Essential tangle decompositions of knots with tunnel number one tangles." Topology and its Applications 196 (December 2015): 815–20. http://dx.doi.org/10.1016/j.topol.2015.05.053.

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36

Jellinger, Kurt A., and Christian Bancher. "Senile Dementia with Tangles (Tangle Predominant Form of Senile Dementia)." Brain Pathology 8, no. 2 (April 5, 2006): 367–76. http://dx.doi.org/10.1111/j.1750-3639.1998.tb00160.x.

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37

Brazil, Melanie. "Preventing tangles." Nature Reviews Drug Discovery 1, no. 2 (February 2002): 100. http://dx.doi.org/10.1038/nrd741.

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Brockett, Tracey Physioc. "Daily Tangles." Massachusetts Review 58, no. 4 (2017): 729–36. http://dx.doi.org/10.1353/mar.2017.0112.

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39

Melzer, Arthur. "Esoteric Tangles." Perspectives on Political Science 44, no. 4 (October 2, 2015): 226–36. http://dx.doi.org/10.1080/10457097.2015.1088769.

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40

Goldman, Jay R., and Louis H. Kauffman. "Rational Tangles." Advances in Applied Mathematics 18, no. 3 (April 1997): 300–332. http://dx.doi.org/10.1006/aama.1996.0511.

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41

LAUDA, AARON D., and HENDRYK PFEIFFER. "OPEN-CLOSED TQFTS EXTEND KHOVANOV HOMOLOGY FROM LINKS TO TANGLES." Journal of Knot Theory and Its Ramifications 18, no. 01 (January 2009): 87–150. http://dx.doi.org/10.1142/s0218216509006793.

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We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to represent a refinement of Bar-Natan's universal geometric complex algebraically, and thereby extend Khovanov homology from links to arbitrary tangles. For every plane diagram of an oriented tangle, we construct a chain complex whose terms are modules of a suitable algebra A such that there is one action of A or A op for every boundary point of the tangle. We give examples of such algebras A for which our tangle homology theory reduces to the link homology theories of Khovanov, Lee and Bar-Natan if it is evaluated for links. As a consequence of the Cardy condition, Khovanov's graded theory can only be extended to tangles if the underlying field has finite characteristic. Whenever the open-closed TQFT arises from a state-sum construction, we obtain honest planar algebra morphisms, and all composition properties of the universal geometric complex carry over to the algebraic complex. We give examples of state-sum open-closed TQFTs for which one can still determine both characteristic p Khovanov homology of links and Rasmussen's s-invariant.

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42

Khesin, Andrey Boris. "A tabulation of ribbon knots in tangle form." Journal of Knot Theory and Its Ramifications 27, no. 02 (February 2018): 1850018. http://dx.doi.org/10.1142/s0218216518500189.

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It is known that there are 21 ribbon knots with 10 crossings or fewer. We show that for every ribbon knot, there exists a tangle that satisfies two properties associated with the knot. First, under a specific closure, the closed tangle is equivalent to its corresponding knot. Second, under a different closure, the closed tangle is equivalent to the unlink. For each of these 21 ribbon knots, we present a 4-strand tangle that satisfies these properties. We provide diagrams of these tangles and also express them in planar diagram notation.

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FUKUMOTO, YOSHIHIRO, PAUL KIRK, and JUANITA PINZÓN-CAICEDO. "Traceless SU(2) representations of 2-stranded tangles." Mathematical Proceedings of the Cambridge Philosophical Society 162, no. 1 (June 3, 2016): 101–29. http://dx.doi.org/10.1017/s0305004116000360.

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AbstractGiven a 2-stranded tangle T contained in a ℤ-homology ball Y, we investigate the character variety R(Y, T) of conjugacy classes of traceless SU(2) representations of π1(Y \ T). In particular we completely determine the subspace of binary dihedral representations, and identify all of R(Y, T) for many tangles naturally associated to knots in S3. Moreover, we determine the image of the restriction map from R(T, Y) to the traceless SU(2) character variety of the 4-punctured 2-sphere (the pillowcase). We give examples to show this image can be non-linear in general, and show it is linear for tangles associated to pretzel knots.

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Ferman, Tanis J., Naoya Aoki, Bradley F. Boeve, Jeremiah A. Aakre, Kejal Kantarci, Jonathan Graff-Radford, Joseph E. Parisi, et al. "Subtypes of dementia with Lewy bodies are associated with α-synuclein and tau distribution." Neurology 95, no. 2 (June 19, 2020): e155-e165. http://dx.doi.org/10.1212/wnl.0000000000009763.

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ObjectiveTo determine whether Lewy body disease subgroups have different clinical profiles.MethodsParticipants had dementia, autopsy-confirmed transitional or diffuse Lewy body disease (TLBD or DLBD) (n = 244), or Alzheimer disease (AD) (n = 210), and were seen at least twice (mean follow-up 6.2 ± 3.8 years). TLBD and DLBD groups were partitioned based on the presence or absence of neocortical neurofibrillary tangles using Braak staging. Four Lewy body disease subgroups and AD were compared on clinical features, dementia trajectory, and onset latency of probable dementia with Lewy bodies (DLB) or a DLB syndrome defined as probable DLB or dementia with one core feature of parkinsonism or probable REM sleep behavior disorder.ResultsIn TLBD and DLBD without neocortical tangles, diagnostic sensitivity was strong for probable DLB (87% TLBD, 96% DLBD) and the DLB syndrome (97% TLBD, 98% DLBD) with median latencies <1 year from cognitive onset, and worse baseline attention-visual processing but better memory-naming scores than AD. In DLBD with neocortical tangles, diagnostic sensitivity was 70% for probable DLB and 77% for the DLB syndrome with respective median latencies of 3.7 years and 2.7 years from cognitive onset, each associated with tangle distribution. This group had worse baseline attention-visual processing than AD, but comparable memory-naming impairment. TLBD with neocortical tangles had 48% diagnostic sensitivity for probable DLB and 52% for the DLB syndrome, with median latencies >6 years from cognitive onset, and were cognitively similar to AD. Dementia trajectory was slowest for TLBD without neocortical tangles, and fastest for DLBD with neocortical tangles.ConclusionsThe phenotypic expression of DLB was associated with the distribution of α-synuclein and tau pathology.

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ZINN-JUSTIN, PAUL, and JEAN-BERNARD ZUBER. "ON THE COUNTING OF COLORED TANGLES." Journal of Knot Theory and Its Ramifications 09, no. 08 (December 2000): 1127–41. http://dx.doi.org/10.1142/s0218216500000669.

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The connection between matrix integrals and links is used to define matrix models which count alternating tangles n which each closed loop is weighted with a factor n, i.e. may be regarded as decorated with n possible colors. For n=2, the corresponding matrix integral is that recently solved in the study of the random lattice six-vertex model. The generating function of alternating 2-color tangle is provided in terms of elliptic functions, expanded to 16-th order (16 crossings) and its asymptotic behaviors is given.

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Reutter, David J., and Jamie Vicary. "Shaded tangles for the design and verification of quantum circuits." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2224 (April 2019): 20180338. http://dx.doi.org/10.1098/rspa.2018.0338.

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We give a scheme for interpreting shaded tangles as quantum circuits, with the property that if two shaded tangles are ambient isotopic, their corresponding computational effects are identical. We analyse 11 known quantum procedures in this way—including entanglement manipulation, error correction and teleportation—and in each case present a fully topological formal verification, yielding generalized procedures in some cases. We also use our methods to identify two new procedures, for topological state transfer and quantum error correction. Our formalism yields in some cases significant new insight into how the procedures work, including a description of quantum entanglement arising from topological entanglement of strands, and a description of quantum error correction where errors are ‘trapped by bubbles’ and removed from the shaded tangle.

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Hashimoto, Naoki, Tohru Takeuchi, Ryoko Ishihara, Katsuyuki Ukai, Hiroshi Kobayashi, Hiromu Iwata, Kiyoshi Iwai, Yutaka Mizuno, Haruyasu Yamaguchi, and Hiroto Shibayama. "Glial fibrillary tangles in diffuse neurofibrillary tangles with calcification." Acta Neuropathologica 106, no. 2 (August 1, 2003): 150–56. http://dx.doi.org/10.1007/s00401-003-0715-0.

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Roberts, Lawrence P. "Planar algebras and the decategorification of bordered Khovanov homology." Journal of Knot Theory and Its Ramifications 26, no. 04 (April 2017): 1750023. http://dx.doi.org/10.1142/s0218216517500237.

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We give a simple, combinatorial construction of a unital, spherical, non-degenerate *-planar algebra over the ring [Formula: see text]. This planar algebra is similar in spirit to the Temperley–Lieb planar algebra, but computations show that they are different. The construction comes from the combinatorics of the decategorifications of the type A and type D structures in the author’s previous work on bordered Khovanov homology. In particular, the construction illustrates how gluing of tangles occurs in the bordered Khovanov homology and its difference from that in Khovanov’s tangle homology without being encumbered by any extra homological algebra. It also provides a simple framework for showing that these theories are not related through a simple process, thereby confirming recent work of Manion. Furthermore, using Khovanov’s conventions and a state sum approach to the Jones polynomial, we obtain new invariant for tangles in [Formula: see text] where [Formula: see text] is a compact, planar surface with boundary, and the tangle intersects each boundary cylinder in an even number of points. This construction naturally generalizes Khovanov’s approach to the Jones polynomial.

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49

Orem, Sarah. "Tangles of Resentment." Signs: Journal of Women in Culture and Society 46, no. 4 (June 1, 2021): 963–85. http://dx.doi.org/10.1086/713296.

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Grohe, Martin, and Pascal Schweitzer. "Computing with Tangles." SIAM Journal on Discrete Mathematics 30, no. 2 (January 2016): 1213–47. http://dx.doi.org/10.1137/15m1027565.

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