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Journal articles on the topic 'Torus knot'

Author: Grafiati

Published: 9 March 2023

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1

Lee, Sangyop. "Knot types of twisted torus knots." Journal of Knot Theory and Its Ramifications 26, no. 12 (October 2017): 1750074. http://dx.doi.org/10.1142/s0218216517500742.

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Dean introduced twisted torus knots, which are obtained from a torus knot and a torus link by splicing them together along a number of adjacent strands of each of them. We study the knot types of these knots.

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2

Lee, Sangyop. "Twisted torus knots T(p,q,p − kq,−1) which are torus knots." Journal of Knot Theory and Its Ramifications 29, no. 09 (August 2020): 2050068. http://dx.doi.org/10.1142/s0218216520500686.

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A twisted torus knot is a torus knot with some consecutive strands twisted. More precisely, a twisted torus knot [Formula: see text] is a torus knot [Formula: see text] with [Formula: see text] consecutive strands [Formula: see text] times fully twisted. We determine which twisted torus knots [Formula: see text] are a torus knot.

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3

Lee, Sangyop. "Composite Knots Obtained by Twisting Torus Knots." International Mathematics Research Notices 2019, no. 18 (December 9, 2017): 5744–76. http://dx.doi.org/10.1093/imrn/rnx282.

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Abstract A twisted torus knot is obtained from a torus knot by performing full twists on some adjacent strands of the torus knot. Morimoto constructed infinitely many twisted torus knots which are composite knots and he conjectured that his knots are all of composite twisted torus knots. We show that his conjecture is almost true by giving a complete list of such twisted torus knots.

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4

OZAWA, MAKOTO. "SATELLITE DOUBLE TORUS KNOTS." Journal of Knot Theory and Its Ramifications 10, no. 01 (February 2001): 133–42. http://dx.doi.org/10.1142/s0218216501000779.

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We characterize satellite double torus knots. Especially, if a satellite double torus knot is not a cable knot, then it has a torus knot companion. This answers Question 12 (a) raised by Hill and Murasugi in [4].

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5

Amoranto, Evan, Brandy Doleshal, and Matt Rathbun. "Additional cases of positive twisted torus knots." Journal of Knot Theory and Its Ramifications 26, no. 12 (October 2017): 1750078. http://dx.doi.org/10.1142/s021821651750078x.

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A twisted torus knot is a knot obtained from a torus knot by twisting adjacent strands by full twists. The twisted torus knots lie in [Formula: see text], the genus 2 Heegaard surface for [Formula: see text]. Primitive/primitive and primitive/Seifert knots lie in [Formula: see text] in a particular way. Dean gives sufficient conditions for the parameters of the twisted torus knots to ensure they are primitive/primitive or primitive/Seifert. Using Dean’s conditions, Doleshal shows that there are infinitely many twisted torus knots that are fibered and that there are twisted torus knots with distinct primitive/Seifert representatives with the same slope in [Formula: see text]. In this paper, we extend Doleshal’s results to show there is a four parameter family of positive twisted torus knots. Additionally, we provide new examples of twisted torus knots with distinct representatives with the same surface slope in [Formula: see text].

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6

SATOH, SHIN. "VIRTUAL KNOT PRESENTATION OF RIBBON TORUS-KNOTS." Journal of Knot Theory and Its Ramifications 09, no. 04 (June 2000): 531–42. http://dx.doi.org/10.1142/s0218216500000293.

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Any ribbon torus-knot in 4-space is naturally associated with a virtual knot. We investigate a relationship between ribbon torus-knots and virtual knots from this viewpoint. We also give a new example of a non-classical virtual knots which can not be detected by the group and the Z-polynomial.

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7

BAADER, SEBASTIAN. "Unknotting sequences for torus knots." Mathematical Proceedings of the Cambridge Philosophical Society 148, no. 1 (July 6, 2009): 111–16. http://dx.doi.org/10.1017/s0305004109990156.

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AbstractThe unknotting number of a knot is bounded from below by its slice genus. It is a well-known fact that the genera and unknotting numbers of torus knots coincide. In this paper we characterize quasipositive knots for which the genus bound is sharp: the slice genus of a quasipositive knot equals its unknotting number, if and only if the given knot appears in an unknotting sequence of a torus knot.

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8

NAKAMURA, INASA. "BRAIDING SURFACE LINKS WHICH ARE COVERINGS OVER THE STANDARD TORUS." Journal of Knot Theory and Its Ramifications 21, no. 01 (January 2012): 1250011. http://dx.doi.org/10.1142/s0218216511009650.

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We consider a surface link in the 4-space which can be presented by a simple branched covering over the standard torus, which we call a torus-covering link. Torus-covering links include spun T2-knots and turned spun T2-knots. In this paper we braid a torus-covering link over the standard 2-sphere. This gives an upper estimate of the braid index of a torus-covering link. In particular we show that the turned spun T2-knot of the torus (2, p)-knot has the braid index four.

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9

Tran, Anh T. "The strong AJ conjecture for cables of torus knots." Journal of Knot Theory and Its Ramifications 24, no. 14 (December 2015): 1550072. http://dx.doi.org/10.1142/s0218216515500728.

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The AJ conjecture, formulated by Garoufalidis, relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. It has been confirmed for all torus knots, some classes of two-bridge knots and pretzel knots, and most cable knots over torus knots. The strong AJ conjecture, formulated by Sikora, relates the A-ideal and the colored Jones polynomial of a knot. It was confirmed for all torus knots. In this paper we confirm the strong AJ conjecture for most cable knots over torus knots.

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10

ABE, TETSUYA. "AN ESTIMATION OF THE ALTERNATION NUMBER OF A TORUS KNOT." Journal of Knot Theory and Its Ramifications 18, no. 03 (March 2009): 363–79. http://dx.doi.org/10.1142/s021821650900694x.

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We give a lower bound for the alternation number of a knot by using the Rasmussen s-invariant and the signature of a knot. Then, we determine the torus knots with alternation number one and show that many torus knots are "far" from the alternating knots. As an application, we determine the almost alternating torus knots, solving a conjecture due to Adams et al.

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11

HILL, PETER, and KUNIO MURASUGI. "ON DOUBLE-TORUS KNOTS (II)." Journal of Knot Theory and Its Ramifications 09, no. 05 (August 2000): 617–67. http://dx.doi.org/10.1142/s0218216500000359.

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A double-torus knot is a knot embedded in a genus two Heegaard surface [Formula: see text] in S3. We consider double-torus knots L such that [Formula: see text] is connected, and consider fibred knots in various classes.

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12

GILLAM, WILLIAM D. "KNOT HOMOLOGY OF (3, m) TORUS KNOTS." Journal of Knot Theory and Its Ramifications 21, no. 08 (May 10, 2012): 1250072. http://dx.doi.org/10.1142/s0218216512500721.

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We give a direct computation of the Khovanov knot homology of the (3, m) torus knots/links. Our computation yields complete results with ℤ[½] coefficients, though we leave a slight ambiguity concerning 2-torsion when integer coefficients are used. Our computation uses only the basic long exact sequence in knot homology and Rasmussen's result on the triviality of the embedded surface invariant.

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13

Ruppe, Dennis, and Xingru Zhang. "The AJ-Conjecture and cabled knots over torus knots." Journal of Knot Theory and Its Ramifications 24, no. 09 (August 2015): 1550051. http://dx.doi.org/10.1142/s0218216515500510.

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We show that most cabled knots over torus knots in S3 satisfy the AJ-conjecture, namely each (r, s)-cabled knot over each (p, q)-torus knot satisfies the AJ-conjecture if r is not a number between 0 and pqs.

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14

Choi, Doo Ho, and Ki Hyoung Ko. "Parameterizations of 1-Bridge Torus Knots." Journal of Knot Theory and Its Ramifications 12, no. 04 (June 2003): 463–91. http://dx.doi.org/10.1142/s0218216503002445.

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A 1-bridge torus knot in a 3-manifold of genus ≤ 1 is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are similar to the Schubert's normal form and the Conway's normal form for 2-bridge knots. For a given Schubert's normal form we give algorithms to determine the number of components and to compute the fundamental group of the complement when the normal form determines a knot. We also give a description of the double branched cover of an ambient 3-manifold branched along a 1-bridge torus knot by using its Conway's normal form and obtain an explicit formula for the first homology of the double cover.

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15

HILL, PETER. "ON DOUBLE-TORUS KNOTS (I)." Journal of Knot Theory and Its Ramifications 08, no. 08 (December 1999): 1009–48. http://dx.doi.org/10.1142/s0218216599000651.

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A double-torus knot is knot embedded in a genus two Heegaard surface [Formula: see text] in S3. After giving a notation for these knots, we consider double-torus knots L such that [Formula: see text] is not connected, and give a criterion for such knots to be non-trivial. Various new types of non-trivial knots with trivial Alexander polynomial are found.

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16

Lee, Sangyop. "Positively twisted torus knots which are torus knots." Journal of Knot Theory and Its Ramifications 28, no. 03 (March 2019): 1950023. http://dx.doi.org/10.1142/s0218216519500238.

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A twisted torus knot [Formula: see text] is obtained from a torus knot [Formula: see text] by twisting [Formula: see text] adjacent strands of [Formula: see text] fully [Formula: see text] times. In this paper, we determine the parameters [Formula: see text] for which [Formula: see text] is a torus knot with [Formula: see text].

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17

Elliot, Ross, and Sergei Gukov. "Exceptional knot homology." Journal of Knot Theory and Its Ramifications 25, no. 03 (March 2016): 1640003. http://dx.doi.org/10.1142/s0218216516400034.

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The goal of this paper is twofold. First, we find a natural home for the double affine Hecke algebras (DAHA) in the physics of BPS states. Second, we introduce new invariants of torus knots and links called hyperpolynomials that address the “problem of negative coefficients” often encountered in DAHA-based approaches to homological invariants of torus knots and links. Furthermore, from the physics of BPS states and the spectra of singularities associated with Landau–Ginzburg potentials, we also describe a rich structure of differentials that act on homological knot invariants for exceptional groups and uniquely determine the latter for torus knots.

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18

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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19

Hayashi, Chuichiro. "Trivial Double-Torus Knot." Journal of Knot Theory and Its Ramifications 12, no. 05 (August 2003): 579–88. http://dx.doi.org/10.1142/s0218216503002652.

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A knot K in a closed connected orientable 3-manifold M is called a double-torus knot, if it is in a genus two Heegaard splitting surface H of M. We give a necessary and sufficient condition for a double-torus knot to be the trivial knot in words of meridian disks of genus two handlebodies obtained by splitting M along H.

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20

ICHIMORI, ATSUSHI, and TAIZO KANENOBU. "RIBBON TORUS KNOTS PRESENTED BY VIRTUAL KNOTS WITH UP TO FOUR CROSSINGS." Journal of Knot Theory and Its Ramifications 21, no. 13 (October 24, 2012): 1240005. http://dx.doi.org/10.1142/s0218216512400056.

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A ribbon torus knot embedded in the 4-space is presented by a welded virtual knot through the tube operation due to Shin Satoh. We make an attempt of classification of ribbon torus knots presented by virtual knots with up to four crossings, where we use the list of virtual knots enumerated by Jeremy Green. We mainly investigate the groups of virtual knots.

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21

OZAWA, MAKOTO. "Nonminimal bridge positions of torus knots are stabilized." Mathematical Proceedings of the Cambridge Philosophical Society 151, no. 2 (May 4, 2011): 307–17. http://dx.doi.org/10.1017/s0305004111000235.

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AbstractWe show that any nonminimal bridge decomposition of a torus knot is stabilized and that n-bridge decompositions of a torus knot are unique for any integer n. This implies that a knot in a bridge position is a torus knot if and only if there exists a torus containing the knot such that it intersects the bridge sphere in two essential loops.

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22

NAKAMURA, INASA. "UNKNOTTING NUMBERS AND TRIPLE POINT CANCELLING NUMBERS OF TORUS-COVERING KNOTS." Journal of Knot Theory and Its Ramifications 22, no. 03 (March 2013): 1350010. http://dx.doi.org/10.1142/s0218216513500107.

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It is known that any surface knot can be transformed to an unknotted surface knot or a surface knot which has a diagram with no triple points by a finite number of 1-handle additions. The minimum number of such 1-handles is called the unknotting number or the triple point cancelling number, respectively. In this paper, we give upper bounds and lower bounds of unknotting numbers and triple point cancelling numbers of torus-covering knots, which are surface knots in the form of coverings over the standard torus T. Upper bounds are given by using m-charts on T presenting torus-covering knots, and lower bounds are given by using quandle colorings and quandle cocycle invariants.

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23

Ishikawa, Masaharu, and Hirokazu Yanagi. "Virtual unknotting numbers of certain virtual torus knots." Journal of Knot Theory and Its Ramifications 26, no. 11 (October 2017): 1750070. http://dx.doi.org/10.1142/s0218216517500705.

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The virtual unknotting number of a virtual knot is the minimal number of crossing changes that makes the virtual knot to be the unknot, which is defined only for virtual knots virtually homotopic to the unknot. We focus on the virtual knot obtained from the standard [Formula: see text]-torus knot diagram by replacing all crossings on one overstrand into virtual crossings and prove that its virtual unknotting number is equal to the unknotting number of the [Formula: see text]-torus knot, that is, [Formula: see text].

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24

LEE, SANGYOP. "TWISTED TORUS KNOTS WITH ESSENTIAL TORI IN THEIR COMPLEMENTS." Journal of Knot Theory and Its Ramifications 22, no. 08 (July 2013): 1350041. http://dx.doi.org/10.1142/s0218216513500417.

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A twisted torus knot is a torus knot with a number of full-twists on some adjacent strands. In this paper, we show that if a twisted torus knot is a satellite knot then the number of full-twists is generically at most two.

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25

ASHIHARA, SOSUKE. "FUNDAMENTAL BIQUANDLES OF RIBBON 2-KNOTS AND RIBBON TORUS-KNOTS WITH ISOMORPHIC FUNDAMENTAL QUANDLES." Journal of Knot Theory and Its Ramifications 23, no. 01 (January 2014): 1450001. http://dx.doi.org/10.1142/s0218216514500011.

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The fundamental quandles and biquandles are invariants of classical knots and surface knots. It is unknown whether there exist classical or surface knots which have isomorphic fundamental quandles and distinct fundamental biquandles. We show that ribbon 2-knots or ribbon torus-knots with isomorphic fundamental quandles have isomorphic fundamental biquandles. For this purpose, we give a method for obtaining a presentation of the fundamental biquandle of a ribbon 2-knot/torus-knot from its fundamental quandle.

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26

Kumar, S. Vinoth, and A. R. Harish. "Trefoil Torus Knot Monopole Antenna." IEEE Antennas and Wireless Propagation Letters 15 (2016): 464–67. http://dx.doi.org/10.1109/lawp.2015.2453198.

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27

Hikami, Kazuhiro, and Anatol N. Kirillov. "Torus knot and minimal model." Physics Letters B 575, no. 3-4 (November 2003): 343–48. http://dx.doi.org/10.1016/j.physletb.2003.09.007.

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28

Rassai, Rassa, George Syrmos, and Robert W. Newcomb. "Solid holed torus knot oscillators." Circuits Systems and Signal Processing 9, no. 2 (June 1990): 135–45. http://dx.doi.org/10.1007/bf01236447.

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29

NAKAMURA, K., Y. NAKANISHI, and Y. UCHIDA. "DELTA-UNKNOTTING NUMBER FOR KNOTS." Journal of Knot Theory and Its Ramifications 07, no. 05 (August 1998): 639–50. http://dx.doi.org/10.1142/s0218216598000334.

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The Δ-unknotting number for a knot is defined to be the minimum number of Δ-unknotting operations which deform the knot into the trivial knot. We determine the Δ-unknotting numbers for torus knots, positive pretzel knots, and positive closed 3-braids.

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30

ADAMS, COLIN, and TODD SHAYLER. "THE PROJECTION STICK INDEX OF KNOTS." Journal of Knot Theory and Its Ramifications 18, no. 07 (July 2009): 889–99. http://dx.doi.org/10.1142/s0218216509007294.

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The stick index of a knot K is defined to be the least number of line segments needed to construct a polygonal embedding of K. We define the projection stick index of K to be the least number of line segments in any projection of a polygonal embedding of K. In this paper, we establish bounds on the projection stick index for various torus knots. We then show that the stick index of a (p, 2p + 1)-torus knot is 4p, and the projection stick index is 2p + 1. This provides examples of knots such that the projection stick index is one greater than half the stick index. We show that for all other torus knots for which the stick index is known, the projection stick index is larger than this. We conjecture that a projection stick index of half the stick index is unattainable for any knot.

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31

GOVINDARAJAN, T. R. "KNOT SOLITONS." Modern Physics Letters A 13, no. 39 (December 21, 1998): 3179–84. http://dx.doi.org/10.1142/s0217732398003387.

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The existence of ring-like and knotted solitons in O(3) nonlinear σ-model is analyzed. The role of isotopy of knots/links in classifying such solitons is pointed out. Appearance of torus knot solitons is seen.

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32

YAMAGUCHI, YOSHIKAZU. "Higher even dimensional Reidemeister torsion for torus knot exteriors." Mathematical Proceedings of the Cambridge Philosophical Society 155, no. 2 (April 25, 2013): 297–305. http://dx.doi.org/10.1017/s0305004113000248.

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AbstractWe study the asymptotics of the higher dimensional Reidemeister torsion for torus knot exteriors, which is related to the results by W. Müller and P. Menal–Ferrer and J. Porti on the asymptotics of the Reidemeister torsion and the hyperbolic volumes for hyperbolic 3-manifolds. We show that the sequence of 1/(2N)2) log | Tor(EK; ρ2N)| converges to zero when N goes to infinity where TorEK; ρ2N is the higher dimensional Reidemeister torsion of a torus knot exterior and an acyclic SL2N(ℂ)-representation of the torus knot group. We also give a classification for SL2(ℂ)-representations of torus knot groups, which induce acyclic SL2N(ℂ)-representations.

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33

WINTER, BLAKE. "THE CLASSIFICATION OF SPUN TORUS KNOTS." Journal of Knot Theory and Its Ramifications 18, no. 09 (September 2009): 1287–98. http://dx.doi.org/10.1142/s0218216509007476.

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Satoh has defined a construction to obtain a ribbon torus knot given a welded knot. This construction is known to be surjective. We show that it is not injective. Using the invariant of the peripheral structure, it is possible to provide a restriction on this failure of injectivity. In particular we also provide an algebraic classification of the construction when restricted to classical knots, where it is equivalent to the torus spinning construction.

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34

HOSTE, JIM. "TORUS KNOTS ARE FOURIER-(1,1,2) KNOTS." Journal of Knot Theory and Its Ramifications 18, no. 02 (February 2009): 265–70. http://dx.doi.org/10.1142/s0218216509006926.

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Every torus knot can be represented as a Fourier-(1,1,2) knot which is the simplest possible Fourier representation for such a knot. This answers a question of Kauffman and confirms the conjecture made by Boocher, Daigle, Hoste and Zheng. In particular, the torus knot Tp,q can be parameterized as [Formula: see text]

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35

BOYLE, JEFFREY. "THE TURNED TORUS KNOT IN S4." Journal of Knot Theory and Its Ramifications 02, no. 03 (September 1993): 239–49. http://dx.doi.org/10.1142/s0218216593000155.

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A spun knotted torus in the 4-sphere is formed by rigidly sweeping a knotted curve along a circle. Alternately, as the knotted curve is swept along the circle we could give it a number of full turns (Dehn twists). We show the resulting knotted torus depends only on the knotted curve and whether the number of turns is even or odd. The even and odd turned spun tori have nondiffeomorphic complements. This is generalized in some cases to include twist spun turned torus knots.

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36

Matignon, Daniel. "Distance to longitude." Journal of Knot Theory and Its Ramifications 28, no. 01 (January 2019): 1950011. http://dx.doi.org/10.1142/s0218216519500111.

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Let [Formula: see text] be a hyperbolic knot in the [Formula: see text]-sphere. If a [Formula: see text]-Dehn surgery on [Formula: see text] produces manifold with an embedded Klein bottle or essential [Formula: see text]-torus, then we prove that [Formula: see text], where [Formula: see text] is the genus of [Formula: see text]. We obtain different upper bounds according to the production of a Klein bottle, a non-separating [Formula: see text]-torus, or an essential and separating [Formula: see text]-torus. The well known examples which are the figure eight knot and the pretzel knot [Formula: see text] reach the given upper bounds. We study this problem considering null-homologous hyperbolic knots in compact, orientable and closed [Formula: see text]-manifolds.

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37

Keener, J. P. "Knotted vortex filaments in an ideal fluid." Journal of Fluid Mechanics 211 (February 1990): 629–51. http://dx.doi.org/10.1017/s0022112090001732.

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Knotted closed-curve solutions of the equation of self-induced vortex motion are studied. It is shown that there are invariant torus knots which translate and rotate as rigid bodies. The general motion of ‘small-amplitude’ torus knots and iterated (cabled) torus knots is described and found to be almost periodic in time, and for some, but not all, initial data, the topology of the knot is shown to be invariant.

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38

STOŠIĆ, MARKO. "ON CONJECTURES ABOUT POSITIVE BRAID KNOTS AND ALMOST ALTERNATING TORUS KNOTS." Journal of Knot Theory and Its Ramifications 19, no. 11 (November 2010): 1471–86. http://dx.doi.org/10.1142/s0218216510008509.

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In this paper we resolve some conjectures concerning positive braid knots and almost alternating torus knots. Namely, we prove that the first Khovanov homology group of positive braid knot is trivial, as conjectured by Khovanov. Also, we generalize this result to show that the same is true in the case of Khovanov–Rozansky homology (sl(n) link homology) for any positive integer n. Moreover, by using the Khovanov homology theory, we prove the classical knot theory conjecture by Adams, that the only almost alternating torus knots are T3, 4 and T3, 5.

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39

Tsau, Chichen M. "Incompressible surfaces in the knot manifolds of torus knots." Topology 33, no. 1 (January 1994): 197–201. http://dx.doi.org/10.1016/0040-9383(94)90042-6.

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40

Gordon, Cameron, and Tye Lidman. "Knot contact homology detects cabled, composite, and torus knots." Proceedings of the American Mathematical Society 145, no. 12 (June 16, 2017): 5405–12. http://dx.doi.org/10.1090/proc/13643.

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41

Vafaee, Faramarz. "On the Knot Floer Homology of Twisted Torus Knots." International Mathematics Research Notices 2015, no. 15 (August 21, 2014): 6516–37. http://dx.doi.org/10.1093/imrn/rnu130.

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42

Lee, Sangyop. "Twisted torus knots T(mn + m + 1,mn + 1,mn + m + 2,−1) and T(n + 1,n,2n − 1,−1) are torus knots." Journal of Knot Theory and Its Ramifications 30, no. 03 (March 2021): 2150016. http://dx.doi.org/10.1142/s0218216521500164.

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A twisted torus knot [Formula: see text] is a torus knot [Formula: see text] with [Formula: see text] adjacent strands twisted fully [Formula: see text] times. In this paper, we determine the braid index of the knot [Formula: see text] when the parameters [Formula: see text] satisfy [Formula: see text]. If the last parameter [Formula: see text] additionally satisfies [Formula: see text], then we also determine the parameters [Formula: see text] for which [Formula: see text] is a torus knot.

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43

Jabuka, Stanislav, and Cornelia A. Van Cott. "Comparing nonorientable three genus and nonorientable four genus of torus knots." Journal of Knot Theory and Its Ramifications 29, no. 03 (March 2020): 2050013. http://dx.doi.org/10.1142/s0218216520500133.

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We compare the values of the nonorientable three genus (or, crosscap number) and the nonorientable four genus of torus knots. In particular, let [Formula: see text] be any torus knot with [Formula: see text] even and [Formula: see text] odd. The difference between these two invariants on [Formula: see text] is at least [Formula: see text], where [Formula: see text] and [Formula: see text] and [Formula: see text]. Hence, the difference between the two invariants on torus knots [Formula: see text] grows arbitrarily large for any fixed odd [Formula: see text], as [Formula: see text] ranges over values of a fixed congruence class modulo [Formula: see text]. This contrasts with the orientable setting. Seifert proved that the orientable three genus of the torus knot [Formula: see text] is [Formula: see text], and Kronheimer and Mrowka later proved that the orientable four genus of [Formula: see text] is also this same value.

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44

Oberti, Chiara, and Renzo L. Ricca. "On torus knots and unknots." Journal of Knot Theory and Its Ramifications 25, no. 06 (May 2016): 1650036. http://dx.doi.org/10.1142/s021821651650036x.

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A comprehensive study of geometric and topological properties of torus knots and unknots is presented. Torus knots/unknots are particularly symmetric, closed, space curves, that wrap the surface of a mathematical torus a number of times in the longitudinal and meridian direction. By using a standard parametrization, new results on local and global properties are found. In particular, we demonstrate the existence of inflection points for a given critical aspect ratio, determine the location and prescribe the regularization condition to remove the local singularity associated with torsion. Since to first approximation total length grows linearly with the number of coils, its nondimensional counterpart is proportional to the topological crossing number of the knot type. We analyze several global geometric quantities, such as total curvature, writhing number, total torsion, and geometric ‘energies’ given by total squared curvature and torsion, in relation to knot complexity measured by the winding number. We conclude with a brief presentation of research topics, where geometric and topological information on torus knots/unknots finds useful application.

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45

Motegi, Kimihiko, and Kazushige Tohki. "Hyperbolic L-space knots and exceptional Dehn surgeries." Journal of Knot Theory and Its Ramifications 23, no. 14 (December 2014): 1450079. http://dx.doi.org/10.1142/s0218216514500795.

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A knot in the 3-sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space. Like torus knots and Berge knots, many known L-space knots admit a Seifert fibered L-space surgery. We give a concrete example of a hyperbolic L-space knot which has no exceptional surgeries, in particular, no Seifert fibered surgeries.

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46

Barthel, Senja. "On chirality of toroidal embeddings of polyhedral graphs." Journal of Knot Theory and Its Ramifications 26, no. 08 (May 22, 2017): 1750050. http://dx.doi.org/10.1142/s021821651750050x.

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We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [2, 3]. Building on this and using the chirality of torus knots and links [9, 10], we prove that the nontrivial embeddings of simple 3-connected planar graphs in the standard torus are chiral. For the case that the spatial graph contains a nontrivial knot, the statement was shown by Castle et al. [5]. We give an alternative proof using minors instead of the Euler characteristic. To prove the case in which the graph embedding contains a nonsplit link, we show the chirality of Hopf ladders with at least three rungs, thus generalizing a theorem of Simon [12].

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47

Ichihara, Kazuhiro, and Yuki Temma. "Non-left-orderable surgeries and generalized Baumslag–Solitar relators." Journal of Knot Theory and Its Ramifications 24, no. 01 (January 2015): 1550003. http://dx.doi.org/10.1142/s0218216515500030.

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We show that a knot has a non-left-orderable surgery if the knot group admits a generalized Baumslag–Solitar relator and satisfies certain conditions on a longitude of the knot. As an application, it is shown that certain positively twisted torus knots admit non-left-orderable surgeries.

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48

MILLETT, KENNETH C., MICHAEL PIATEK, and ERIC J. RAWDON. "POLYGONAL KNOT SPACE NEAR ROPELENGTH-MINIMIZED KNOTS." Journal of Knot Theory and Its Ramifications 17, no. 05 (May 2008): 601–31. http://dx.doi.org/10.1142/s0218216508006282.

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For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, is an embedded torus. Given a thick configuration K, perturbations of size r < R(K) define satellite structures, or local knotting. We explore knotting within these tubes both theoretically and numerically. We provide bounds on perturbation radii for which one can obtain small trefoil and figure-eight summands and use Monte Carlo simulations to estimate the relative probabilities of these structures as a function of the number of edges.

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Lozano, María Teresa, and José María Montesinos-Amilibia. "On continuous families of geometric Seifert conemanifold structures." Journal of Knot Theory and Its Ramifications 25, no. 14 (December 2016): 1650083. http://dx.doi.org/10.1142/s0218216516500838.

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In this paper, dedicated to Prof. Lou Kauffman, we determine the Thurston’s geometry possesed by any Seifert fibered conemanifold structure in a Seifert manifold with orbit space [Formula: see text] and no more than three exceptional fibers, whose singular set, composed by fibers, has at most three components which can include exceptional or general fibers (the total number of exceptional and singular fibers is less than or equal to three). We also give the method to obtain the holonomy of that structure. We apply these results to three families of Seifert manifolds, namely, spherical, Nil manifolds and manifolds obtained by Dehn surgery on a torus knot [Formula: see text]. As a consequence we generalize to all torus knots the results obtained in [Geometric conemanifolds structures on [Formula: see text], the result of [Formula: see text] surgery in the left-handed trefoil knot [Formula: see text], J. Knot Theory Ramifications 24(12) (2015), Article ID: 1550057, 38pp., doi: 10.1142/S0218216515500571] for the case of the left handle trefoil knot. We associate a plot to each torus knot for the different geometries, in the spirit of Thurston.

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VIKASH, SIWACH, and MADETI PRABHAKAR. "A SHARP UPPER BOUND FOR REGION UNKNOTTING NUMBER OF TORUS KNOTS." Journal of Knot Theory and Its Ramifications 22, no. 05 (April 2013): 1350019. http://dx.doi.org/10.1142/s0218216513500193.

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Region crossing change for a knot or a proper link is an unknotting operation. In this paper, we provide a sharp upper bound on the region unknotting number for a large class of torus knots and proper links. Also, we discuss conditions on torus links to be proper.

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