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Journal articles on the topic 'Transverse knots and links'

Author: Grafiati

Published: 10 March 2023

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1

OGASA, EIJI. "THE INTERSECTION OF SPHERES IN A SPHERE AND A NEW GEOMETRIC MEANING OF THE ARF INVARIANT." Journal of Knot Theory and Its Ramifications 11, no. 08 (December 2002): 1211–31. http://dx.doi.org/10.1142/s0218216502002104.

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Let [Formula: see text] be a 3-sphere embedded in the 5-sphere S5 (i = 1,2). Let [Formula: see text] and [Formula: see text] intersect transversely. Then the intersection [Formula: see text] is a disjoint collection of circles. Thus we obtain a pair of 1-links, C in [Formula: see text], and a pair of 3-knots, [Formula: see text] in S5 (i = 1, 2). Conversely let (L1, L2) be a pair of 1-links and (X1, X2) be a pair of 3-knots. It is natural to ask whether the pair of 1-links (L1, L2) is obtained as the intersection of the 3-knots X1 and X2 as above. We give a complete answer to this question. Our answer gives a new geometric meaning of the Arf invariant of 1-links. Let f : S3 → S5 be a smooth transverse immersion such that the self-intersection C consists of double points. Suppose that C is a single circle in S5. Then f-1(C) in S3 is a 1-knot or a 2-component 1-link. There is a similar realization problem. We give a complete answer to this question.

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2

DING, FAN, and HANSJÖRG GEIGES. "LEGENDRIAN KNOTS AND LINKS CLASSIFIED BY CLASSICAL INVARIANTS." Communications in Contemporary Mathematics 09, no. 02 (April 2007): 135–62. http://dx.doi.org/10.1142/s0219199707002381.

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It is shown that Legendrian (respectively transverse) cable links in S3 with its standard tight contact structure, i.e. links consisting of an unknot and a cable of that unknot, are classified by their oriented link type and the classical invariants (Thurston–Bennequin invariant and rotation number in the Legendrian case, self-linking number in the transverse case). The analogous result is proved for torus knots in the 1-jet space J1(S1) with its standard tight contact structure.

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3

Chmutov, S., S. Jablan, K. Karvounis, and S. Lambropoulou. "On the link invariants from the Yokonuma–Hecke algebras." Journal of Knot Theory and Its Ramifications 25, no. 09 (August 2016): 1641004. http://dx.doi.org/10.1142/s0218216516410042.

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In this paper, we study properties of the Markov trace tr[Formula: see text] and the specialized trace [Formula: see text] on the Yokonuma–Hecke algebras, such as behavior under inversion of a word, connected sums and mirror imaging. We then define invariants for framed, classical and singular links through the trace [Formula: see text] and also invariants for transverse links through the trace tr[Formula: see text]. In order to compare the invariants for classical links with the Homflypt polynomial, we develop computer programs and we evaluate them on several Homflypt-equivalent pairs of knots and links. Our computations lead to the result that these invariants are topologically equivalent to the Homflypt polynomial on knots. However, they do not demonstrate the same behavior on links.

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4

Vance, Katherine. "Tau invariants for balanced spatial graphs." Journal of Knot Theory and Its Ramifications 29, no. 09 (August 2020): 2050066. http://dx.doi.org/10.1142/s0218216520500662.

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In 2003, Ozsváth and Szabó defined the concordance invariant [Formula: see text] for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of [Formula: see text] for knots in [Formula: see text] and a combinatorial proof that [Formula: see text] gives a lower bound for the slice genus of a knot. Recently, Harvey and O’Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory for transverse spatial graphs in [Formula: see text], extending HFK for knots. We define a [Formula: see text]-filtered chain complex for balanced spatial graphs whose associated graded chain complex has homology determined by Harvey and O’Donnol’s graph Floer homology. We use this to show that there is a well-defined [Formula: see text] invariant for balanced spatial graphs generalizing the [Formula: see text] knot concordance invariant. In particular, this defines a [Formula: see text] invariant for links in [Formula: see text]. Using techniques similar to those of Sarkar, we show that our [Formula: see text] invariant is an obstruction to a link being slice.

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5

Bode, B., M. R. Dennis, D. Foster, and R. P. King. "Knotted fields and explicit fibrations for lemniscate knots." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2202 (June 2017): 20160829. http://dx.doi.org/10.1098/rspa.2016.0829.

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We give an explicit construction of complex maps whose nodal lines have the form of lemniscate knots. We review the properties of lemniscate knots, defined as closures of braids where all strands follow the same transverse (1, ℓ) Lissajous figure, and are therefore a subfamily of spiral knots generalizing the torus knots. We then prove that such maps exist and are in fact fibrations with appropriate choices of parameters. We describe how this may be useful in physics for creating knotted fields, in quantum mechanics, optics and generalizing to rational maps with application to the Skyrme–Faddeev model. We also prove how this construction extends to maps with weakly isolated singularities.

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6

Ito, Tetsuya. "Braids, chain of Yang–Baxter like operations, and (transverse) knot invariants." Journal of Knot Theory and Its Ramifications 27, no. 11 (October 2018): 1843009. http://dx.doi.org/10.1142/s0218216518430095.

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We introduce a notion of a chain of Yang–Baxter like operations. This is a sequence of solutions of an asymmetric variant of the Yang–Baxter equation and is a multi-operator generalization of (bi)rack/quandles. We discuss knot and link invariants coming from a chain of Yang–Baxter like operations, and give potential applications. Among them, we define a cocycle invariant for transverse links.

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7

Xie, C., S. Y. Haffert, J. de Boer, M. A. Kenworthy, J. Brinchmann, J. Girard, I. A. G. Snellen, and C. U. Keller. "A MUSE view of the asymmetric jet from HD 163296." Astronomy & Astrophysics 650 (June 2021): L6. http://dx.doi.org/10.1051/0004-6361/202140602.

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Context. Jets and outflows are thought to play important roles in regulating star formation and disk evolution. An important question is how the jets are launched. HD 163296 is a well-studied Herbig Ae star that hosts proto-planet candidates, a protoplanetary disk, a protostellar jet, and a molecular outflow, which makes it an excellent laboratory for studying jets. Aims. We aim to characterize the jet at the inner regions and check if there are large differences with the features at large separations. A secondary objective is to demonstrate the performance of Multi Unit Spectroscopic Explorer (MUSE) in high-contrast imaging of extended line emission. Methods. MUSE in the narrow field mode (NFM) can provide observations at optical wavelengths with high spatial (∼75 mas) and medium spectral (R ∼ 2500) resolution. With the high-resolution spectral differential imaging technique, we can characterize the kinematic structures and physical conditions of jets down to 100 mas. Results. We detect multiple atomic lines in two new knots, B3 and A4, at distances of < 4″ from the host star with MUSE. The derived Ṁjet/Ṁacc is about 0.08 and 0.06 for knots B3 and A4, respectively. The observed [Ca II]/[S II] ratios indicate that there is no sign of dust grains at distances of < 4″. Assuming the A4 knot traced the streamline, we can estimate a jet radius at the origin by fitting the half width half maximum of the jet, which sets an upper limit of 2.2 au on the size of the launching region. Although MUSE has the ability to detect the velocity shifts caused by high- and low-velocity components, we found no significant evidence of velocity decrease transverse to the jet direction in our 500 s MUSE observation. Conclusions. Our work demonstrates the capability of using MUSE NFM observations for the detailed study of stellar jets in the optical down to 100 mas. The derived Ṁjet/Ṁacc, no dust grain, and jet radius at the star support the magneto-centrifugal models as a launching mechanism for the jet.

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8

Coe, Tom, Jim Mackey, and Hyde Marine. "Controlling Oil Spills in Fast Currents with the Flow∼Diverter." International Oil Spill Conference Proceedings 2003, no. 1 (April 1, 2003): 833–41. http://dx.doi.org/10.7901/2169-3358-2003-1-833.

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ABSTRACT Sixty nine percent (645 million tons annually) of the oil transported in United States waters is on waterways where currents routinely exceed one knot. Conventional skimmers and booms lose their effectiveness when current speeds exceed 1 knot. The US Coast Guard recognized that this threat could not be easily controlled, and so they initiated a project that led to the successful development of a novel spill control device, the Oil Spill Flow~Diverter. The Flow~Diverter system is effective at diverting and converging oil at speeds up to 5+ knots. In more moderate currents it can also be used in place of an anchor, towboat or outrigger arm to deploy and position the outboard end of a deflection boom. It may also have application to dispersant and in-situ burn operations. The diverter is a unique stable catamaran design that consists of two hulls each comprised of symmetrical foils with integral buoyancy. The foils are pinned to a rigid connecting structure such that they can pivot but always remain parallel to each other. Two or more diverter catamarans can be connected together with cables to increase the total sweep width of the system. Two control lines are anchored to shore or secured to a boat and are used to deploy the system by adjusting the foils’ angle to the oncoming water. With the control lines securely anchored, the system is launched into the current and “flies out” into a stable operating position. It remains in equilibrium, balanced by the hydrodynamic lift forces of the passing water and the tension in the lines. The foils create a strong transverse surface current downstream to achieve the desired diversion and consolidation affect on floating oil. Unlike most skimmers and deflection boom, the diverters are not adversely affected as currents increase. The oil is diverted by the same lateral distance irrespective of the current or speed of advance. This paper presents the development of the Flow~Diverter prototype, its testing and operational evaluations. Several applications of the diverter technology in various response tactics are discussed. Use of the Diverter during a recent US Coast Guard Spill Exercise on the Ohio River is also presented. Production model enhancements are presented that will enhance performance in 7+ knot currents and shallow water applications.

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9

Sebastian, K. L. "Knots and links." Resonance 11, no. 3 (March 2006): 25–35. http://dx.doi.org/10.1007/bf02835965.

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10

Radovic, Ljiljana, and Slavik Jablan. "Meander knots and links." Filomat 29, no. 10 (2015): 2381–92. http://dx.doi.org/10.2298/fil1510381r.

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We introduced concept of meander knots, 2-component meander links and multi-component meander links and derived different families of meander knots and links from open meanders with n ? 16 crossings. We also defined semi-meander knots (or knots with ordered Gauss code) and their product.

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11

Salhua Moreno, César Augusto. "Application of Cubic B-Spline Curves for Hull Meshing." Ciencia y tecnología de buques 14, no. 28 (January 31, 2021): 53–62. http://dx.doi.org/10.25043/19098642.215.

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This paper describes the development of a regular hull meshing code using cubic B-Spline curves. The discretization procedure begins by the definition of B-Spline curves over stations, bow and stern contours of the hull plan lines. Thus, new knots are created applying an equal spaced subdivision procedure on defined B-spline curves. Then, over these equal transversal space knots, longitudinal B-spline curves are defined and subdivided into equally spaced knots, too. Subsequently, new transversal knots are created using the longitudinal equally spaced knots. Finally, the hull mesh is composed by quadrilateral panels formed by these new transversal and longitudinal knots. This procedure is applied in the submerged Wigley hulls Series 60 Cb=0.60. Their mesh volumes are calculated using the divergence theorem, for mesh quality evaluation.

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12

Bataineh, Khaled. "Stuck Knots." Symmetry 12, no. 9 (September 21, 2020): 1558. http://dx.doi.org/10.3390/sym12091558.

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Singular knots and links have projections involving some usual crossings and some four-valent rigid vertices. Such vertices are symmetric in the sense that no strand overpasses the other. In this research we introduce stuck knots and links to represent physical knots and links with projections involving some stuck crossings, where the physical strands get stuck together showing which strand overpasses the other at a stuck crossing. We introduce the basic elements of the theory and we give some isotopy invariants of such knots including invariants which capture the chirality (mirror imaging) of such objects. We also introduce another natural class of stuck knots, which we call relatively stuck knots, where each stuck crossing has a stuckness factor that indicates to the value of stuckness at that crossing. Amazingly, a generalized version of Jones polynomial makes an invariant of such quantized knots and links. We give applications of stuck knots and links and their invariants in modeling and understanding bonded RNA foldings, and we explore the topology of such objects with invariants involving multiplicities at the bonds. Other perspectives are also discussed.

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13

Ozsváth, Peter, Zoltán Szabó, and Dylan P. Thurston. "Legendrian knots, transverse knots and combinatorial Floer homology." Geometry & Topology 12, no. 2 (May 12, 2008): 941–80. http://dx.doi.org/10.2140/gt.2008.12.941.

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14

Kauffman, L. H., and V. O. Manturov. "Virtual knots and links." Proceedings of the Steklov Institute of Mathematics 252, no. 1 (January 2006): 104–21. http://dx.doi.org/10.1134/s0081543806010111.

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15

DOBROWOLSKI, JAN CZ. "DNA knots and links." Polimery 48, no. 01 (January 2003): 3–15. http://dx.doi.org/10.14314/polimery.2003.003.

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16

Battye, Richard A., and Paul M. Sutcliffe. "Solitons, links and knots." Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 455, no. 1992 (December 8, 1999): 4305–31. http://dx.doi.org/10.1098/rspa.1999.0502.

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17

Bystrom, Kerry. "Humanitarianism, Responsibility,Links,Knots." Interventions 16, no. 3 (June 4, 2013): 405–23. http://dx.doi.org/10.1080/1369801x.2013.798141.

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18

Bataineh, Khaled, and Hadeel Ghaith. "Involutory biquandles and singular knots and links." Open Mathematics 16, no. 1 (May 10, 2018): 469–89. http://dx.doi.org/10.1515/math-2018-0031.

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AbstractWe define a new algebraic structure for singular knots and links. It extends the notion of a bikei (or involutory biquandle) from regular knots and links to singular knots and links. We call this structure a singbikei. This structure results from the generalized Reidemeister moves representing singular isotopy. We give several examples on singbikei and we use singbikei to distinguish several singular knots and links.

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19

Etnyre, John, Lenhard Ng, and Vera Vértesi. "Legendrian and transverse twist knots." Journal of the European Mathematical Society 15, no. 3 (2013): 969–95. http://dx.doi.org/10.4171/jems/383.

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20

Plamenevskaya, Olga. "Transverse knots and Khovanov homology." Mathematical Research Letters 13, no. 4 (2006): 571–86. http://dx.doi.org/10.4310/mrl.2006.v13.n4.a7.

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21

Kawamura, Kengo. "Ribbon-clasp T2-knots and semi-welded knots." Journal of Knot Theory and Its Ramifications 27, no. 14 (December 2018): 1850079. http://dx.doi.org/10.1142/s0218216518500797.

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The notion of ribbon-clasp surface-links is a generalization of ribbon surface-links. It is known that any ribbon [Formula: see text]-knot is presented by a welded knot, and then its knot group is isomorphic to the group of the welded knot. In this paper, we define the group of a semi-welded knot. We show that any ribbon-clasp [Formula: see text]-knot is presented by a semi-welded knot and that the knot group of the ribbon-clasp [Formula: see text]-knot is isomorphic to the group of the semi-welded knot.

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22

Jablan, Slavik, Ljiljana Radović, and Radmila Sazdanović. "Knots and links in architecture." Pollack Periodica 7, Supplement 1 (January 2012): 65–76. http://dx.doi.org/10.1556/pollack.7.2012.s.6.

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23

DIAO, YUANAN, CLAUS ERNST, and ANTHONY MONTEMAYOR. "NULLIFICATION OF KNOTS AND LINKS." Journal of Knot Theory and Its Ramifications 21, no. 06 (April 7, 2012): 1250046. http://dx.doi.org/10.1142/s0218216511009984.

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It is known that a knot/link can be nullified, i.e. can be made into the trivial knot/link, by smoothing some crossings in a projection diagram of the knot/link. The nullification of knots/links is believed to be biologically relevant. For example, in DNA topology, the nullification process may be the pathway for a knotted circular DNA to unknot itself (through recombination of its DNA strands). The minimum number of such crossings to be smoothed in order to nullify the knot/link is called the nullification number. It turns out that there are several different ways to define such a number, since different conditions may be applied in the nullification process. We show that these definitions are not equivalent, thus they lead to different nullification numbers for a knot/link in general, not just one single nullification number. Our aim is to explore some mathematical properties of these nullification numbers. First, we give specific examples to show that the nullification numbers we defined are different. We provide a detailed analysis of the nullification numbers for the well known 2-bridge knots and links. We also explore the relationships among the three nullification numbers, as well as their relationships with other knot invariants. Finally, we study a special class of links, namely those links whose general nullification number equals one. We show that such links exist in abundance. In fact, the number of such links with crossing number less than or equal to n grows exponentially with respect to n.

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24

KUPERBERG, GREG. "QUADRISECANTS OF KNOTS AND LINKS." Journal of Knot Theory and Its Ramifications 03, no. 01 (March 1994): 41–50. http://dx.doi.org/10.1142/s021821659400006x.

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We show that every non-trivial tame knot or link in ℝ3 has a quadrisecant, i.e. four collinear points. The quadrisecant must be topologically non-trivial in a precise sense. As an application, we show that a nonsingular, algebraic surface in ℝ3 which is a knotted torus must have degree at least eight.

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25

Adams, Colin C. "Toroidally alternating knots and links." Topology 33, no. 2 (April 1994): 353–69. http://dx.doi.org/10.1016/0040-9383(94)90017-5.

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26

Fintushel, Ronald, and Ronald J. Stern. "Knots, links, and 4-manifolds." Inventiones Mathematicae 134, no. 2 (October 16, 1998): 363–400. http://dx.doi.org/10.1007/s002220050268.

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27

DAMIANO, DAVID B., and ELIZABETH M. SENNOTT. "COLORING ALGEBRAIC KNOTS AND LINKS." Journal of Knot Theory and Its Ramifications 17, no. 05 (May 2008): 553–78. http://dx.doi.org/10.1142/s0218216508006324.

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Let K be a knot or link and let p = det (K). Using integral colorings of rational tangles, Kauffman and Lambropoulou showed that every rational K has a mod p coloring with distinct colors. If p is prime this holds for all mod p colorings. Harary and Kauffman conjectured that this should hold for prime, alternating knot diagrams without nugatory crossings for p prime. Asaeda, Przyticki and Sikora proved the conjecture for Montesinos knots. In this paper, we use an elementary combinatorial argument to prove the conjecture for prime alternating algebraic knots with prime determinant. We also give a procedure for coloring any prime alternating knot or link diagram and demonstrate the conjecture for non-algebraic examples.

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28

Jiang, Boju, Xiao-Song Lin, Shicheng Wang, and Ying-Qing Wu. "Achirality of knots and links." Topology and its Applications 119, no. 2 (April 2002): 185–208. http://dx.doi.org/10.1016/s0166-8641(01)00062-1.

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29

CHENG, ZHIYUN, and HONGZHU GAO. "A POLYNOMIAL INVARIANT OF VIRTUAL LINKS." Journal of Knot Theory and Its Ramifications 22, no. 12 (October 2013): 1341002. http://dx.doi.org/10.1142/s0218216513410022.

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In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms [Parity in knot theory, Sb. Math.201 (2010) 693–733] to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of the odd writhe polynomial defined by the first author in [A polynomial invariant of virtual knots, preprint (2012), arXiv:math.GT/1202.3850v1]. The relation between this new polynomial invariant and the affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramification22 (2013) 1340007; A linking number definition of the affine index polynomial and applications, preprint (2012), arXiv:1211.1747v1] is discussed. In the second part we introduce a polynomial invariant for long flat virtual knots. In the third part we define a polynomial invariant for 2-component virtual links. This polynomial invariant can be regarded as a generalization of the linking number.

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30

Adams, Colin C., Bevin M. Brennan, Deborah L. Greilsheimer, and Alexander K. Woo. "Stick Numbers and Composition of Knots and Links." Journal of Knot Theory and Its Ramifications 06, no. 02 (April 1997): 149–61. http://dx.doi.org/10.1142/s0218216597000121.

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We address the concept of stick number for knots and links under various restrictions concerning the length of the sticks, the angles between sticks, and placements of the vertices. In particular, we focus on the effect of composition on the various stick numbers. Ultimately, we determine the traditional stick number for an infinite class of knots, which are the (n,n-1)-torus knots together with all of the possible compositions of such knots. The exact stick number was previously known for only seven knots.

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31

Jin, Gyo Taek, and Ho Lee. "Mutation invariance of the arc index for some montesinos knots." Journal of Knot Theory and Its Ramifications 26, no. 10 (September 2017): 1750058. http://dx.doi.org/10.1142/s0218216517500584.

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For the alternating knots or links, mutations do not change the arc index. In the case of nonalternating knots, some semi-alternating knots or links have this property. We mainly focus on the problem of mutation invariance of the arc index for nonalternating knots which are not semi-alternating. In this paper, we found families of infinitely many mutant pairs/triples of Montesinos knots with the same arc index.

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32

Bettersworth, Zac, and Claus Ernst. "Incoherent nullification of torus knots and links." Journal of Knot Theory and Its Ramifications 28, no. 05 (April 2019): 1950033. http://dx.doi.org/10.1142/s0218216519500330.

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In the paper, we study the incoherent nullification number [Formula: see text] of knots and links. We establish an upper bound on the incoherent nullification number of torus knots and links and conjecture that this upper bound is the actual incoherent nullification number of this family. Finally, we establish the actual incoherent nullification number of particular subfamilies of torus knots and links.

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33

Shimizu, Ayaka. "Unknotting operations on knots and links." Journal of Knot Theory and Its Ramifications 30, no. 02 (February 2021): 2171001. http://dx.doi.org/10.1142/s0218216521710012.

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By considering unknotting operations, we obtain ways of measuring how knotted a knot is. Unknotting phenomena can be seen not only in knot theory, but also in various settings such as DNA knots, mind knots and so on ([C. C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots (American Mathematical Society, Providence, RI, 2004); A. Kawauchi, K. Kishimoto and A. Shimizu, Knot theory and game (in Japanese) (Asakura Publishing, Tokyo, 2013); L. Rudolph, Qualitative Mathematics for the Social Sciences (Routledge, London, 2013); K. Murasugi, Knot Theory and Its Applications, Translated from the 1993 Japanese original by Bohdan Kupita (Birkhauser, Boston, MA, 1996)], etc.). In this paper, we see how knots can be unknotted (and therefore how they are knotted) by considering various unknotting operations and their associated unknotting numbers.

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34

Meliani, Zakaria, and Olivier Hervet. "Knots in Relativistic Transverse Stratified Jets." Galaxies 5, no. 3 (September 5, 2017): 50. http://dx.doi.org/10.3390/galaxies5030050.

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35

Dynnikov, I. A. "Transverse-Legendrian links." Sibirskie Elektronnye Matematicheskie Izvestiya 16 (December 24, 2019): 1960–80. http://dx.doi.org/10.33048/semi.2019.16.141.

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36

RANKIN, STUART, and ORTHO FLINT. "ENUMERATING THE PRIME ALTERNATING LINKS." Journal of Knot Theory and Its Ramifications 13, no. 01 (February 2004): 151–73. http://dx.doi.org/10.1142/s0218216504003068.

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In [5], four knot operators were introduced and used to construct all prime alternating knots of a given crossing size. An efficient implementation of this construction was made possible by the notion of the master array of an alternating knot. The master array and an implementation of the construction appeared in [6]. The basic scheme (as described in [5]) is to apply two of the operators, D and ROTS, to the prime alternating knots of minimal crossing size n-1, which results in a large set of prime alternating knots of minimal crossing size n, and then the remaining two operators, T and OTS, are applied to these n crossing knots to complete the production of the set of prime alternating knots of minimal crossing size n. In this paper, we show how to obtain all prime alternating links of a given minimal crossing size. More precisely, we shall establish that given any two prime alternating links of minimal crossing size n, there is a finite sequence of T and OTS operations that transforms one of the links into the other. Consequently, one may select any prime alternating link of minimal crossing size n (which is then called the seed link), and repeatedly apply only the operators T and OTS to obtain all prime alternating links of minimal crossing size n from the chosen seed link. The process may be standardized by specifying the seed link to be (in the parlance of [5]) the unique link of n crossings with group number 1, the (n, 2) torus link.

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37

OZAWA, MAKOTO. "TANGLE DECOMPOSITIONS OF DOUBLE TORUS KNOTS AND LINKS." Journal of Knot Theory and Its Ramifications 08, no. 07 (November 1999): 931–39. http://dx.doi.org/10.1142/s0218216599000584.

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38

AFANASIEV, DENIS MIKHAILOVICH, and VASSILY OLEGOVICH MANTUROV. "ON VIRTUAL CROSSING NUMBER ESTIMATES FOR VIRTUAL LINKS." Journal of Knot Theory and Its Ramifications 18, no. 06 (June 2009): 757–72. http://dx.doi.org/10.1142/s021821650900718x.

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Considering extremal properties of one polynomial of virtual knots, we establish estimates for virtual crossing numbers of virtual knots from a given class. This yields minimality of certain diagrams of virtual knots with respect to the virtual crossing number. Infinite series of pairwise distinct minimal virtual knot diagrams are constructed and their properties are discussed.

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Pongtanapaisan, Puttipong. "Wirtinger numbers for virtual links." Journal of Knot Theory and Its Ramifications 28, no. 14 (December 2019): 1950086. http://dx.doi.org/10.1142/s021821651950086x.

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The Wirtinger number of a virtual link is the minimum number of generators of the link group over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a virtual link equals its virtual bridge number. Since the Wirtinger number is algorithmically computable, it gives a more effective way to calculate an upper bound for the virtual bridge number from a virtual link diagram. As an application, we compute upper bounds for the virtual bridge numbers and the quandle counting invariants of virtual knots with 6 or fewer crossings. In particular, we found new examples of nontrivial virtual bridge number one knots, and by applying Satoh’s Tube map to these knots we can obtain nontrivial weakly superslice links.

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40

OZAWA, MAKOTO. "ASCENDING NUMBER OF KNOTS AND LINKS." Journal of Knot Theory and Its Ramifications 19, no. 01 (January 2010): 15–25. http://dx.doi.org/10.1142/s0218216510007723.

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We introduce a new numerical invariant of knots and links from the descending diagrams. It is considered to live between the unknotting number and the bridge number. Some fundamental results and an incomplete table of the invariant for knots with 8-crossings or less are given.

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Nguyen, Hoang-An, and Anh T. Tran. "Adjoint twisted Alexander polynomial of twisted Whitehead links." Journal of Knot Theory and Its Ramifications 27, no. 04 (April 2018): 1850026. http://dx.doi.org/10.1142/s0218216518500268.

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The adjoint twisted Alexander polynomial has been computed for twist knots [A. Tran, Twisted Alexander polynomials with the adjoint action for some classes of knots, J. Knot Theory Ramifications 23(10) (2014) 1450051], genus one two-bridge knots [A. Tran, Adjoint twisted Alexander polynomials of genus one two-bridge knots, J. Knot Theory Ramifications 25(10) (2016) 1650065] and the Whitehead link [J. Dubois and Y. Yamaguchi, Twisted Alexander invariant and nonabelian Reidemeister torsion for hyperbolic three dimensional manifolds with cusps, Preprint (2009), arXiv:0906.1500 ]. In this paper, we compute the adjoint twisted Alexander polynomial and nonabelian Reidemeister torsion of twisted Whitehead links.

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IM, YOUNG HO, KYEONGHUI LEE, and SANG YOUL LEE. "SIGNATURE, NULLITY AND DETERMINANT OF CHECKERBOARD COLORABLE VIRTUAL LINKS." Journal of Knot Theory and Its Ramifications 19, no. 08 (August 2010): 1093–114. http://dx.doi.org/10.1142/s0218216510008315.

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In this paper, we present the Goeritz matrix for checkerboard colorable virtual links or, equivalently, checkerboard colorable links in thickened surfaces Sg × [0, 1], which is an extension of the Goeritz matrix for classical knots and links in ℝ3. Using this, we show that the signature, nullity and determinant of classical oriented knots and links extend to those of checkerboard colorable oriented virtual links.

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43

Conway, J. "Tight contact structures via admissible transverse surgery." Journal of Knot Theory and Its Ramifications 28, no. 04 (April 2019): 1950032. http://dx.doi.org/10.1142/s0218216519500329.

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We investigate the line between tight and overtwisted for surgeries on fibered transverse knots in contact 3-manifolds. When the contact structure [Formula: see text] is supported by the fibered knot [Formula: see text], we obtain a characterization of when negative surgeries result in a contact structure with nonvanishing Heegaard Floer contact class. To do this, we leverage information about the contact structure [Formula: see text] supported by the mirror knot [Formula: see text]. We derive several corollaries about the existence of tight contact structures, L-space knots outside [Formula: see text], nonplanar contact structures, and nonplanar Legendrian knots.

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Buck, Dorothy, and Kai Ishihara. "Coherent band pathways between knots and links." Journal of Knot Theory and Its Ramifications 24, no. 02 (February 2015): 1550006. http://dx.doi.org/10.1142/s0218216515500066.

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We categorize coherent band (aka nullification) pathways between knots and 2-component links. Additionally, we characterize the minimal coherent band pathways (with intermediates) between any two knots or 2-component links with small crossing number. We demonstrate these band surgeries for knots and links with small crossing number. We apply these results to place lower bounds on the minimum number of recombinant events separating DNA configurations, restrict the recombination pathways and determine chirality and/or orientation of the resulting recombinant DNA molecules.

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Kim, Dongseok, and Jaeun Lee. "Some invariants of Pretzel links." Bulletin of the Australian Mathematical Society 75, no. 2 (April 2007): 253–71. http://dx.doi.org/10.1017/s0004972700039198.

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We show that nontrivial classical pretzel knots L (p, q, r) are hyperbolic with eight exceptions which are torus knots. We find Conway polynomials of n-pretzel links using a new computation tree. As applications, we compute the genera of n-pretzel links using these polynomials and find the basket number of pretzel links by showing that the genus and the canonical genus of a pretzel link are the same.

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46

Cromwell, Peter. "Arc presentations of knots and links." Banach Center Publications 42, no. 1 (1998): 57–64. http://dx.doi.org/10.4064/-42-1-57-64.

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Ernst, Claus, and Anthony Montemayor. "Nullification of Torus knots and links." Journal of Knot Theory and Its Ramifications 23, no. 11 (October 2014): 1450058. http://dx.doi.org/10.1142/s0218216514500588.

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It is known that a knot/link can be nullified, i.e. can be made into the trivial knot/link, by smoothing some crossings in a projection diagram of the knot/link. The minimum number of such crossings to be smoothed in order to nullify the knot/link is called the nullification number. In this paper we investigate the nullification numbers of a particular knot family, namely the family of torus knots and links.

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48

Orlandini, Enzo, Guido Polles, Davide Marenduzzo, and Cristian Micheletti. "Self-assembly of knots and links." Journal of Statistical Mechanics: Theory and Experiment 2017, no. 3 (March 23, 2017): 034003. http://dx.doi.org/10.1088/1742-5468/aa5bb5.

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CHEN, W., and S. P. BANKS. "KNOTS, LINKS AND SPUN DYNAMICAL SYSTEMS." International Journal of Bifurcation and Chaos 20, no. 04 (April 2010): 1041–48. http://dx.doi.org/10.1142/s0218127410026307.

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In this paper we consider spun dynamical systems and show that we can obtain a system on S3 × S1 which contains any finite set of knotted and linked surfaces which are invariant surfaces for the flow.

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50

Adams, Colin, Martin Hildebrand, and Jeffrey Weeks. "Hyperbolic invariants of knots and links." Transactions of the American Mathematical Society 326, no. 1 (January 1, 1991): 1–56. http://dx.doi.org/10.1090/s0002-9947-1991-0994161-2.

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