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Journal articles on the topic 'Knots and links in $S^3$'

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Published: 4 June 2021

Last updated: 4 February 2022

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1

Manion, Andrew. "The rational Khovanov homology of 3-strand pretzel links." Journal of Knot Theory and Its Ramifications 23, no. 08 (July 2014): 1450040. http://dx.doi.org/10.1142/s0218216514500400.

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The 3-strand pretzel knots and links are a well-studied source of examples in knot theory. However, while there have been computations of the Khovanov homology and Rasmussen s-invariants of some sub-families of 3-strand pretzel knots, no general formula has been given for all of them. We give a formula for the unreduced Khovanov homology, over the rational numbers, of all 3-strand pretzel links. We also compute generalized s-invariants of these links.

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2

MULAZZANI, MICHELE. "ALL LINS-MANDEL SPACES ARE BRANCHED CYCLIC COVERINGS OF S3." Journal of Knot Theory and Its Ramifications 05, no. 02 (April 1996): 239–63. http://dx.doi.org/10.1142/s0218216596000175.

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In this paper we show that all Lins-Mandel spaces S (b, l, t, c) are branched cyclic coverings of the 3-sphere. When the space is a 3-manifold, the branching set of the covering is a two-bridge knot or link of type (l, t) and otherwise is a graph with two vertices joined by three edges (a θ-graph). In the latter case the singular set of the space is always composed by two points with homeomorphic links. The first homology groups of the Lins-Mandel manifolds are computed when t=1 and when the branching set is a knot of genus one. Furthermore the family of spaces has been extended in order to contain all branched cyclic coverings of two-bridge knots or links.

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3

KODA, YUYA. "LINKS AND SPINES." Journal of Knot Theory and Its Ramifications 21, no. 03 (March 2012): 1250027. http://dx.doi.org/10.1142/s0218216511009674.

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We define two kinds of invariants of links in closed 3-manifolds, the s-complexity(s ∈ ℕ) and the block number, by considering decompositions of links in closed orientable 3-manifolds by spines. The first one is a generalization of the complexity of links defined by Pervova and Petronio. After providing properties of these invariants, we construct special spines of strongly-cyclic coverings branched over generalized twist knots in lens spaces, including S3 and ℝP3, which provide upper bounds for the invariants.

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4

Planat, Michel, Raymond Aschheim, Marcelo Amaral, and Klee Irwin. "Universal Quantum Computing and Three-Manifolds." Symmetry 10, no. 12 (December 19, 2018): 773. http://dx.doi.org/10.3390/sym10120773.

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A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere S 3 . Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognizes to be a 3-manifold M 3 . More precisely, the d-dimensional POVMs defined from subgroups of finite index of the modular group P S L ( 2 , Z ) correspond to d-fold M 3 - coverings over the trefoil knot. In this paper, we also investigate quantum information on a few ‘universal’ knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on the software SnapPy. Further connections between POVMs based UQC and M 3 ’s obtained from Dehn fillings are explored.

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5

GE, MO-LIN, LU-YU WANG, KANG XUE, and YONG-SHI WU. "AKUTZU-WADATI LINK POLYNOMIALS FROM FEYNMAN-KAUFFMAN DIAGRAMS." International Journal of Modern Physics A 04, no. 13 (August 10, 1989): 3351–73. http://dx.doi.org/10.1142/s0217751x89001370.

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By employing techniques familiar to particle physicists, we develop Kauffman’s state model for the Jones polynomial, which uses diagrams looking like Feynman diagrams for scattering, into a systematic, diagrammatic approach to new link polynomials. We systematize the ansatz for S matrix by symmetry considerations and find a natural interpretation for CPT symmetry in the context of knot theory. The invariance under Reidemeister moves of type III, II and I can be imposed diagrammatically step by step, and one obtains successively braid group representations, regular isotopy and ambient isotopy invariants from Kauffman’s bracket polynomials. This procedure is explicitiy carried out for the N=3 and 4 cases. N being the number of particle labels (or charges). With appropriate symmetry ansatz and with annihilation and creation included in the S matrix, we have obtained link polynomials which generalize the definition of the Akutzu-Wadati polynomials from closed braids to any oriented knots or links with explicit invariance under Reidemeister moves.

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6

RANKIN, STUART, ORTHO FLINT, and JOHN SCHERMANN. "ENUMERATING THE PRIME ALTERNATING KNOTS, PART I." Journal of Knot Theory and Its Ramifications 13, no. 01 (February 2004): 57–100. http://dx.doi.org/10.1142/s0218216504003044.

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The enumeration of prime knots has a long and storied history, beginning with the work of T. P. Kirkman [9,10], C. N. Little [14], and P. G. Tait [19] in the late 1800's, and continuing through to the present day, with significant progress and related results provided along the way by J. H. Conway [3], K. A. Perko [17, 18], M. B. Thistlethwaite [6, 8, 15, 16, 20], C. H. Dowker [6], J. Hoste [1, 8], J. Calvo [2], W. Menasco [15, 16], W. B. R. Lickorish [12, 13], J. Weeks [8] and many others. Additionally, there have been many efforts to establish bounds on the number of prime knots and links, as described in the works of O. Dasbach and S. Hougardy [4], D. J. A. Welsh [22], C. Ernst and D. W. Sumners [7], and C. Sundberg and M. Thistlethwaite [21] and others. In this paper, we provide a solution to part of the enumeration problem, in that we describe an efficient inductive scheme which uses a total of four operators to generate all prime alternating knots of a given minimal crossing size, and we prove that the procedure does in fact produce them all. The process proceeds in two steps, where in the first step, two of the four operators are applied to the prime alternating knots of minimal crossing size n to produce approximately 98% of the prime alternating knots of minimal crossing size n+1, while in the second step, the remaining two operators are applied to these newly constructed knots, thereby producing the remaining prime alternating knots of crossing size n+1. The process begins with the prime alternating knot of four crossings, the figure eight knot. In the sequel, we provide an actual implementation of our procedure, wherein we spend considerable effort to make the procedure efficient. One very important aspect of the implementation is a new way of encoding a knot. We are able to assign an integer array (called the master array) to a prime alternating knot in such a way that each regular projection, or plane configuration, of the knot can be constructed from the data in the array, and moreover, two knots are equivalent if and only if their master arrays are identical. A fringe benefit of this scheme is a candidate for the so-called ideal configuration of a prime alternating knot. We have used this generation scheme to enumerate the prime alternating knots up to and including those of 19 crossings. The knots up to and including 17 crossings produced by our generation scheme concurred with those found by M. Thistlethwaite, J. Hoste and J. Weeks (see [8]). The current implementation of the algorithms involved in the generation scheme allowed us to produce the 1,769,979 prime alternating knots of 17 crossings on a five node beowulf cluster in approximately 2.3 hours, while the time to produce the prime alternating knots up to and including those of 16 crossings totalled approximately 45 minutes. The prime alternating knots at 18 and 19 crossings were enumerated using the 48 node Compaq ES-40 beowulf cluster at the University of Western Ontario (we also received generous support from Compaq at the SC 99 conference). The cluster was shared with other users and so an accurate estimate of the running time is not available, but the generation of the 8,400,285 knots at 18 crossings was completed in 17 hours, and the generation of the 40,619,385 prime alternating knots at 19 crossings took approximately 72 hours. With the improvements that are described in the sequel, we anticipate that the knots at 19 crossings will be generated in not more than 10 hours on a current Pentium III personal computer equipped with 256 megabytes of main memory.

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7

OHTSUKI, TOMOTADA. "INVARIANTS OF KNOTS DERIVED FROM EQUIVARIANT LINKING MATRICES OF THEIR SURGERY PRESENTATIONS." International Journal of Mathematics 20, no. 07 (July 2009): 883–913. http://dx.doi.org/10.1142/s0129167x09005583.

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The quantum U(1) invariant of a closed 3-manifold M is defined from the linking matrix of a framed link of a surgery presentation of M. As an equivariant version of it, we formulate an invariant of a knot K from the equivariant linking matrix of a lift of a framed link of a surgery presentation of K. We show that this invariant is determined by the Blanchfield pairing of K, or equivalently, determined by the S-equivalent class of a Seifert matrix of K, and that the "product" of this invariant and its complex conjugation is presented by the Alexander module of K. We present some values of this invariant of some classes of knots concretely.

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8

Cautis, Sabin, Aaron D. Lauda, and Joshua Sussan. "Curved Rickard complexes and link homologies." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 769 (December 1, 2020): 87–119. http://dx.doi.org/10.1515/crelle-2019-0044.

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AbstractRickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

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9

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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10

Lassen, Augusto E., Rogerio Riffel, Ana L. Chies-Santos, Evelyn Johnston, Boris Häußler, Gabriel M. Azevedo, Daniel Ruschel-Dutra, and Rogemar A. Riffel. "The metal-poor dwarf irregular galaxy candidate next to Mrk 1172." Monthly Notices of the Royal Astronomical Society 506, no. 3 (July 1, 2021): 3527–39. http://dx.doi.org/10.1093/mnras/stab1838.

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ABSTRACT In this work, we characterize the properties of the object SDSS J020536.84−081424.7, an extended nebular region with projected extension of 14 × 14 kpc2 in the line of sight of the ETG Mrk 1172, using unprecedented spectroscopic data from MUSE. We perform a spatially resolved stellar population synthesis and estimate the stellar mass for both Mrk 1172 (1 × 1011 M⊙) and our object of study (3 × 109 M⊙). While the stellar content of Mrk 1172 is dominated by an old (∼10 Gyr) stellar population, the extended nebular emission has its light dominated by young to intermediate age populations (from ∼100 Myr to ∼1 Gyr) and presents strong emission lines such as H β; [O iii] λλ4959, 5007 Å; H α; [N ii] λλ6549, 6585 Å; and [S ii] λλ6717, 6732 Å. Using these emission lines, we find that it is metal poor (with Z ∼ 1/3 Z⊙, comparable to the LMC) and is actively forming stars (0.70 M⊙ yr−1), especially in a few bright clumpy knots that are readily visible in H α. The object has an ionized gas mass ≥3.8 × 105 M⊙. Moreover, the motion of the gas is well described by a gas in circular orbit in the plane of a disc and is being affected by interaction wtih Mrk 1172. We conclude that SDSS J020536.84−081424.7 is most likely a dwarf irregular galaxy (the dIGal).

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11

Dudarev, S. L., V. A. Berezhnaya, and S. P. Kolkova. "ABOUT NEW FINDS OF THE HORSE HARNESS OF SCYTHIAN-SARMATIAN AGE FROM ZAKUBANYE." Archaeology and Early History of Ukraine 31, no. 2 (June 25, 2019): 287–93. http://dx.doi.org/10.37445/adiu.2019.02.20.

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The article introduces into the scientific circulation the objects of the horse harness of the Scythian and Sarmatian time, found at different times by local residents and local historians of the village of Voznesenskaya, Labinsky district, Krasnodar region of the Russian Federation, and located in the local children’s library-museum. A notable feature of some of the published iron loopy bridle bits are the large sizes of their links (up to 30 cm). This probably indicates a dilution in the second half of the 1st millennium BC in Zakubanye of a large-breed horse breed, which was the forerunner of the famous Kabardian breed of these animals. Its origins, the eminent archeologist-кavkazologyst, E. I. Krupnov traced to the beginning of the early Iron Age. The subject of consideration of the authors of the article are also the features of the cross-shaped nozzles with spikes, which are available on a number of published looped rods, or on their individual links. Most of the presented «cross-shaped nozzles» on the bits (Fig. 1: 2, 4—5; 2: 2) may be ultimately described, following I. I. Marchenko, as a psalm in the form of a small cross with flattened sharp curved spikes. Functionally, «strict» cheek-pieces and nozzles performed the same role. As shown by E. I. Savchenko, they were located at the outer rings of the rods and when the reins were tensioned, they pressed on the toothless edges of the horse’s jaw. Three separate types of specimens can be distinguished from those who are separately from the angled duplicates: 1. bipods (Fig. 3: 4, 5); 2. rod short straight two types — a specimen tapering towards the ends (Fig. 3: 6) and a sample with cylindrical grooved processes extending from the holes (Fig. 3: 7); 3. S-visible with knobs on the ends (Fig. 3: 1—3). The published bits and cheek-pieces belong to the types common in the Northern Black Sea Region and the Northern Caucasus dating back to the 5th—1st cc. BC. At the same time, most of them may have a narrower dating. For the bits with one broken off outer ring (Fig. 2: 1), the date should be marked — the end of IV—III c. BC. For bits in Fig. 2: 2, as well as links of samples like them depicted in Fig. 1: 2, 4, 5, one can accept the date of I. I. Marchenko — IV — first half of the 3rd century BC. Link fished with a hat may be attributed to the IV BC, as having a parallel in the Melitopol mound. Date of duplicated cheek-pieces with two lobes (Fig. 3: 4) — III — first half of I c. BC. Rod duplicated cheek-pieces (Fig. 3: 6, 7) may date to the 4th—3rd centuries BC, S-visible, most likely, the early period of this time period. The items presented in the article characterize the occupation of the local Meotian population by horse breeding, which since Pre-Scythian time has been one of the most important economic branches of the autochthons of Zakubanye.

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12

BIRMAN, JOAN S., and WILLIAM W. MENASCO. "A NOTE ON CLOSED 3-BRAIDS." Communications in Contemporary Mathematics 10, supp01 (November 2008): 1033–47. http://dx.doi.org/10.1142/s0219199708003150.

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This is a review article about knots and links of braid index 3. Its goal is to gather together, in one place, some of the tools that are special to knots and links of braid index 3, in a form that could be useful for those who have a need to calculate, and need to know precisely all the exceptional cases.

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13

LEE, KYEONGHUI, and YOUNG HO IM. "GENERALIZED INDEX POLYNOMIALS FOR VIRTUAL LINKS." Journal of Knot Theory and Its Ramifications 21, no. 14 (December 2012): 1250128. http://dx.doi.org/10.1142/s0218216512501283.

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We construct some polynomial invariants for virtual links by the recursive method, which are different from the index polynomial invariant defined in [Y. H. Im, K. Lee and S. Y. Lee, Index polynomial invariant of virtual links, J. Knot Theory Ramifications19(5) (2010) 709–725]. We show that these polynomials can distinguish whether virtual knots can be invertible or not although the index polynomial cannot distinguish the invertibility of virtual knots.

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14

GEE, CAROL GWOSDZ. "STRONG S-EQUIVALENCE OF ORDERED LINKS." Journal of Knot Theory and Its Ramifications 17, no. 08 (August 2008): 961–81. http://dx.doi.org/10.1142/s0218216508006476.

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In 2003, Naik and Stanford proved that two S-equivalent knots are related by a finite sequence of doubled-delta moves on their knot diagrams. We show that classical S-equivalence is not sufficient to extend their result to ordered links. We define a new algebraic relation on Seifert matrices, called strong S-equivalence, and prove that two oriented, ordered links L and L′ are related by a sequence of doubled-delta moves if and only if they are strongly S-equivalent. We also show that this is equivalent to the fact that L′ can be obtained from L through a sequence of Y-clasper surgeries, where each clasper leaf has total linking number zero with L.

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15

TORISU, ICHIRO. "3-ADJACENCY RELATION OF TWO-BRIDGE KNOTS AND LINKS." Journal of Knot Theory and Its Ramifications 21, no. 06 (April 7, 2012): 1250051. http://dx.doi.org/10.1142/s0218216511010036.

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We show that there is no non-trivial 3-adjacency relation between two-bridge knots or links. As an application of our proof argument, we also show that if a knot K is 3-adjacent to a two-bridge knot or fibered knot K′, then K and K′ have the same determinant.

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16

CASASAYAS, J., J. MARTINEZ ALFARO, and A. NUNES. "KNOTS AND LINKS IN INTEGRABLE HAMILTONIAN SYSTEMS." Journal of Knot Theory and Its Ramifications 07, no. 02 (March 1998): 123–53. http://dx.doi.org/10.1142/s0218216598000097.

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The main purpose of this paper is to prove that Bott integrable Hamiltonian flows and non-singular Morse-Smale flows are closely related. As a consequence, we obtain a classification of the knots and links formed by periodic orbits of Bott integrable Hamiltonians on the 3-sphere and on the solid torus. We also show that most of Fomenko's theory on the topology of the energy levels of Bott integrable Hamiltonians can be derived from Morgan's results on 3-manifolds that admit non-singular Morse-Smale flows.

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17

Teragaito, Masakazu. "Links with surgery yielding the 3-sphere." Journal of Knot Theory and Its Ramifications 11, no. 01 (February 2002): 105–8. http://dx.doi.org/10.1142/s0218216502001494.

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For any n≥2, we give infinitely many unsplittable links of n components in the 3-sphere which admit non-trivial surgery yielding the 3-sphere again and whose components are mutually distinct hyperbolic knots. In particular, our links have tunnel number n-1. We can also give infinitely many 2-component hyperbolic links with such properties.

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18

TORISU, ICHIRO. "ON 2-ADJACENCY RELATION OF TWO-BRIDGE KNOTS AND LINKS." Journal of the Australian Mathematical Society 84, no. 1 (February 2008): 139–44. http://dx.doi.org/10.1017/s1446788708000116.

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AbstractWe give a necessary condition for a two-bridge knot or link S(p,q) to be 2-adjacent to another two-bridge knot or link S(r,s). In particular, we show that if the trivial knot or link is 2-adjacent to S(p,q), then S(p,q) is trivial, that if S(p,q) is 2-adjacent to its mirror image, then S(p,q) is amphicheiral, and that for a prime integer p, if S(p,q) is 2-adjacent to S(r,s), then S(p,q)=S(r,s) or S(r,s)=S(1,0).

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19

Soma, Teruhiko. "On Preimage Knots in S 3." Proceedings of the American Mathematical Society 100, no. 3 (July 1987): 589. http://dx.doi.org/10.2307/2046453.

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20

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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21

Freyd, P., D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, and A. Ocneanu. "A new polynomial invariant of knots and links." Bulletin of the American Mathematical Society 12, no. 2 (April 1, 1985): 239–47. http://dx.doi.org/10.1090/s0273-0979-1985-15361-3.

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22

Huh, Youngsik, Sungjong No, and Seungsang Oh. "Stick numbers of $2$-bridge knots and links." Proceedings of the American Mathematical Society 139, no. 11 (November 1, 2011): 4143–52. http://dx.doi.org/10.1090/s0002-9939-2011-10832-3.

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23

Tao, Zhi-Xiong. "2-Adjacency between some special links and pretzel links." Journal of Knot Theory and Its Ramifications 27, no. 12 (October 2018): 1850062. http://dx.doi.org/10.1142/s0218216518500621.

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This paper studies 2-adjacency between a 3-strand pretzel link and one of the Hopf link, the Solomon’s link and the Whitehead link by using the results that have been obtained about 2-adjacency between knots or links and their polynomials and etc. This paper shows that of all 3-strand pretzel links, only ordinary pretzel links are 2-adjacent to the Hopf link or the Solomon’s link or the Whitehead link. Conversely, these special links are not 2-adjacent to any other 3-strand pretzel links, except for themselves, respectively.

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24

LEE, SANG YOUL, and MYOUNGSOO SEO. "FORMULAS FOR THE CASSON INVARIANT OF CERTAIN INTEGRAL HOMOLOGY 3-SPHERES." Journal of Knot Theory and Its Ramifications 18, no. 11 (November 2009): 1551–76. http://dx.doi.org/10.1142/s0218216509007610.

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In this paper, we introduce a representation of knots and links in S3 by integral matrices and then give an explicit formula for the Casson invariant for integral homology 3-spheres obtained from S3 by Dehn surgery along the knots and links represented by the integral matrices in which either all entries are even or the entries of each row are the same odd number. As applications, we study the preimage of the Casson invariant for a given integer and also give formulas for the Casson invariants of some special classes of integral homology 3-spheres.

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25

Taylor, Scott, and Maggy Tomova. "Thin position for knots, links, and graphs in 3–manifolds." Algebraic & Geometric Topology 18, no. 3 (April 3, 2018): 1361–409. http://dx.doi.org/10.2140/agt.2018.18.1361.

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26

Jeong, Myeong-Ju, and Dae Gy Hong. "DNA and the SU(3) Invariant of Knots and Links." Kyungpook mathematical journal 53, no. 3 (September 23, 2013): 385–95. http://dx.doi.org/10.5666/kmj.2013.53.3.385.

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27

Taylor, Scott, and Maggy Tomova. "Additive invariants for knots, links and graphs in 3–manifolds." Geometry & Topology 22, no. 6 (September 23, 2018): 3235–86. http://dx.doi.org/10.2140/gt.2018.22.3235.

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28

KANENOBU, TAIZO. "BAND SURGERY ON KNOTS AND LINKS, II." Journal of Knot Theory and Its Ramifications 21, no. 09 (May 16, 2012): 1250086. http://dx.doi.org/10.1142/s0218216512500861.

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An oriented 2-component link is called band-trivializable, if it can be unknotted by a single band surgery. We consider whether a given 2-component link is band-trivializable or not. Then we can completely determine the band-trivializability for the prime links with up to 9 crossings. We use the signature, the Jones and Q polynomials, and the Arf invariant. Since a band-trivializable link has 4-ball genus zero, we also give a table for the 4-ball genus of the prime links with up to 9 crossings. Furthermore, we give an additional answer to the problem of whether a (2n + 1)-crossing 2-bridge knot is related to a (2, 2n) torus link or not by a band surgery for n = 3, 4, which comes from the study of a DNA site-specific recombination.

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29

Mikhalkin, Grigory, and Stepan Orevkov. "Real algebraic knots and links of small degree." Journal of Knot Theory and Its Ramifications 25, no. 12 (October 2016): 1642010. http://dx.doi.org/10.1142/s0218216516420104.

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The paper gives a topological as well as rigid isotopy classification of smooth irreducible algebraic curves in the real projective 3-space for the case when the degree of the curve is at most six and its genus is at most one.

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30

ADAMS, COLIN, MICHELLE CHU, THOMAS CRAWFORD, STEPHANIE JENSEN, KYLER SIEGEL, and LIYANG ZHANG. "STICK INDEX OF KNOTS AND LINKS IN THE CUBIC LATTICE." Journal of Knot Theory and Its Ramifications 21, no. 05 (April 2012): 1250041. http://dx.doi.org/10.1142/s0218216511009935.

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The cubic lattice stick index of a knot type is the least number of sticks glued end-to-end that are necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p, p + 1)-torus knots, and show how composing and taking satellites can be used to obtain the cubic lattice stick index for a relatively large infinite class of knots. Additionally, we present several bounds relating cubic lattice stick index to other known invariants.

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31

Soma, Teruhiko. "On preimage knots in $S\sp 3$." Proceedings of the American Mathematical Society 100, no. 3 (March 1, 1987): 589. http://dx.doi.org/10.1090/s0002-9939-1987-0891169-7.

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32

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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33

Guevara Hernández, María de los Angeles, and Hugo Cabrera Ibarra. "Infinite families of prime knots with alt(K) = 1 and their Alexander polynomials." Journal of Knot Theory and Its Ramifications 28, no. 02 (February 2019): 1950010. http://dx.doi.org/10.1142/s021821651950010x.

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In this paper, we construct, by using the Alexander polynomial, infinite families of nonalternating prime knots, which have alternation number equal to one. More specifically these knots after one crossing change yield a 2-bridge knot or the trivial knot. In particular, we display two infinite families of nonalternating knots and their Alexander polynomials. Moreover, we give formulae to obtain the Conway and Alexander polynomials of oriented 3-tangles and the links formed from their closure with a specific orientation. In particular, we propose a construction to form families of links for which their Alexander polynomials can be obtained by nonrecursive formulae.

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34

Manfredi, Enrico. "Lift in the 3-sphere of knots and links in lens spaces." Journal of Knot Theory and Its Ramifications 23, no. 05 (April 2014): 1450022. http://dx.doi.org/10.1142/s0218216514500229.

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An important geometric invariant of links in lens spaces is the lift in the 3-sphere of a link L in L(p, q), that is the counterimage [Formula: see text] of L under the universal covering of L(p, q). If lens spaces are defined as a lens with suitable boundary identifications, then a link in L(p, q) can be represented by a disk diagram, that is to say, a regular projection of the link on a disk. Starting from this diagram of L, we obtain a diagram of the lift [Formula: see text] in S3. Using this construction, we are able to find different knots and links in L(p, q) having equivalent lifts, that is to say, we cannot distinguish different links in lens spaces only from their lift.

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35

Kholodenko, Arkady. "Black magic session of concordance: Regge mass spectrum from Casson’s invariant." International Journal of Modern Physics A 30, no. 33 (November 26, 2015): 1550189. http://dx.doi.org/10.1142/s0217751x15501894.

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Recently, there had been a great deal of interest in obtaining and describing all kinds of knots in links in hydrodynamics, electrodynamics, non-Abelian gauge field theories and gravity. Although knots and links are observables of the Chern–Simons (CS) functional, the dynamical conditions for their generation lie outside the scope of the CS theory. The nontriviality of dynamical generation of knotted structures is caused by the fact that the complements of all knots/links, say, in S3are 3-manifolds which have positive, negative or zero curvature. The ability to curve the ambient space is thus far attributed to masses. The mass theorem of general relativity requires the ambient 3-manifolds to be of nonnegative curvature. Recently, we established that, in the absence of boundaries, complements of dynamically generated knots/links are represented by 3-manifolds of nonnegative curvature. This fact opens the possibility to discuss masses in terms of dynamically generated knotted/linked structures. The key tool is the notion of knot/link concordance. The concept of concordance is a specialization of the concept of cobordism to knots and links. The logic of implementation of the concordance concept to physical masses results in new interpretation of Casson’s surgery formula in terms of the Regge trajectories. The latest thoroughly examined Chew–Frautschi (CF) plots associated with these trajectories demonstrate that the hadron mass spectrum for both mesons and baryons is nicely described by the data on the corresponding CF plots. The physics behind Casson’s surgery formula is similar but not identical to that described purely phenomenologically by Keith Moffatt in 1990. The developed topological treatment is fully consistent with available rigorous mathematical and experimentally observed results related to physics of hadrons.

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36

Cavicchioli, Alberto, Fulvia Spaggiari, and Agnese Ilaria Telloni. "Dehn surgeries on some classical links." Proceedings of the Edinburgh Mathematical Society 54, no. 1 (January 19, 2011): 33–45. http://dx.doi.org/10.1017/s0013091509000777.

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AbstractWe consider orientable closed connected 3-manifolds obtained by performing Dehn surgery on the components of some classical links such as Borromean rings and twisted Whitehead links. We find geometric presentations of their fundamental groups and describe many of them as 2-fold branched coverings of the 3-sphere. Finally, we obtain some topological applications on the manifolds given by exceptional surgeries on hyperbolic 2-bridge knots.

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37

Cavicchioli, Alberto, Fulvia Spaggiari, and Agnese Ilaria Telloni. "Fundamental Group and Covering Properties of Hyperbolic Surgery Manifolds." Geometry 2013 (September 30, 2013): 1–8. http://dx.doi.org/10.1155/2013/484508.

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We study a family of closed connected orientable 3-manifolds obtained by Dehn surgeries with rational coefficients along the oriented components of certain links. This family contains all the manifolds obtained by surgery along the (hyperbolic) 2-bridge knots. We find geometric presentations for the fundamental group of such manifolds and represent them as branched covering spaces. As a consequence, we prove that the surgery manifolds, arising from the hyperbolic 2-bridge knots, have Heegaard genus 2 and are 2-fold coverings of the 3-sphere branched over well-specified links.

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38

KOYTCHEFF, ROBIN, BRIAN A. MUNSON, and ISMAR VOLIĆ. "CONFIGURATION SPACE INTEGRALS AND THE COHOMOLOGY OF THE SPACE OF HOMOTOPY STRING LINKS." Journal of Knot Theory and Its Ramifications 22, no. 11 (October 2013): 1350061. http://dx.doi.org/10.1142/s0218216513500612.

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Configuration space integrals have been used in recent years for studying the cohomology of spaces of (string) knots and links in ℝn for n > 3 since they provide a map from a certain differential graded algebra of diagrams to the deRham complex of differential forms on the spaces of knots and links. We refine this construction so that it now applies to the space of homotopy string links — the space of smooth maps of some number of copies of ℝ in ℝn with fixed behavior outside a compact set and such that the images of the copies of ℝ are disjoint — even for n = 3. We further study the case n = 3 in degree zero and show that our integrals represent a universal finite type invariant of the space of classical homotopy string links. As a consequence, we deduce that Milnor invariants of string links can be written in terms of configuration space integrals.

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39

BALDRIDGE, SCOTT, and ADAM M. LOWRANCE. "CUBE DIAGRAMS AND 3-DIMENSIONAL REIDEMEISTER-LIKE MOVES FOR KNOTS." Journal of Knot Theory and Its Ramifications 21, no. 05 (April 2012): 1250033. http://dx.doi.org/10.1142/s0218216511009832.

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In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under five cube diagram moves. A knot homology is constructed from cube diagrams and shown to be equivalent to knot Floer homology.

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40

Zhang, Xingru. "On Property I for Knots in S 3." Transactions of the American Mathematical Society 339, no. 2 (October 1993): 643. http://dx.doi.org/10.2307/2154291.

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41

HILDEN, HUGH M., JOSÉ M. MONTESINOS-AMILIBIA, DÉBORA M. TEJADA, and MARGARITA M. TORO. "Fox coloured knots and triangulations of $S^{3}$." Mathematical Proceedings of the Cambridge Philosophical Society 141, no. 03 (November 2006): 443. http://dx.doi.org/10.1017/s0305004106009510.

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42

Armentrout, Steve. "Knots and shellable cell partitionings of $S^{3}$." Illinois Journal of Mathematics 38, no. 3 (September 1994): 347–65. http://dx.doi.org/10.1215/ijm/1255986719.

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43

Menasco, William, and Morwen Thistlethwaite. "A geometric proof that alternating knots are non-trivial." Mathematical Proceedings of the Cambridge Philosophical Society 109, no. 3 (May 1991): 425–31. http://dx.doi.org/10.1017/s0305004100069887.

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There are many proofs in the literature of the non-triviality of alternating, classical links in the 3-sphere, but almost all use a combinatorial argument involving some algebraic invariant, namely the determinant [1], the Alexander polynomial [3], the Jones polynomial [5], and, in [6], the Q-polynomial of Brandt–Lickorish–Millett. Indeed, alternating links behave remarkably well with respect to these and other invariants, but this fact has not led to any significant geometric understanding of alternating link types. Therefore it is natural to seek purely geometric proofs of geometric properties of these links. Gabai has given in [4] a striking geometric proof of a related result, also proved earlier by algebraic means in [3], namely that the Seifert surface obtained from a reduced alternating link diagram by Seifert's algorithm has minimal genus for that link. Here, we give an elementary geometric proof of non-triviality of alternating knots, using a slight variation of the techniques set forth in [7, 8]. Note that if L is a link of more than one component and some component of L is spanned by a disk whose interior lies in the complement of L, then L is a split link, i.e. it is separated by a 2-sphere in S3\L; thus we do not consider alternating links of more than one component here, as it is proved in [7] that a connected alternating diagram cannot represent a split link.

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44

Havens, Andrew, and Robin Koytcheff. "Spaces of knots in the solid torus, knots in the thickened torus, and links in the 3-sphere." Geometriae Dedicata 214, no. 1 (June 11, 2021): 671–737. http://dx.doi.org/10.1007/s10711-021-00633-y.

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45

CAMPISI, MARION MOORE, and MATT RATHBUN. "HIGH DISTANCE KNOTS IN CLOSED 3-MANIFOLDS." Journal of Knot Theory and Its Ramifications 21, no. 02 (February 2012): 1250017. http://dx.doi.org/10.1142/s0218216511009637.

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Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance with respect to the splitting surface. This generalizes a result of Minsky, Moriah, and Schleimer for knots in S3. We also show that in the complex of curves, handlebody sets are either coarsely distinct or identical. We define the coarse mapping class group of a Heegaard splitting, and show that if (S, V, W) is a Heegaard splitting of genus ≥2, then the coarse mapping class group of (S, V, W) is isomorphic to the mapping class group of (S, V, W).

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46

LEININGER, CHRISTOPHER J. "SMALL CURVATURE SURFACES IN HYPERBOLIC 3-MANIFOLDS." Journal of Knot Theory and Its Ramifications 15, no. 03 (March 2006): 379–411. http://dx.doi.org/10.1142/s0218216506004531.

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In a paper of Menasco and Reid, it is conjectured that there exist no hyperbolic knots in S3 for which the complement contains a closed embedded totally geodesic surface. In this note, we show that one can get "as close as possible" to a counter-example. Specifically, we construct a sequence of hyperbolic knots {Kn} with complements containing closed embedded essential surfaces having principal curvatures converging to zero as n tends to infinity. We also construct a family of two-component links for which the complements contain closed embedded totally geodesic surfaces of arbitrarily large genera. In addition, we prove that a closed embedded surface with sufficiently small principal curvatures is not only quasi-Fuchsian (a result of Thurston's), but it is also either acylindrical or the boundary of a twisted I-bundle.

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47

Zhang, Xingru. "On property I for knots in $S\sp 3$." Transactions of the American Mathematical Society 339, no. 2 (February 1, 1993): 643–57. http://dx.doi.org/10.1090/s0002-9947-1993-1154545-0.

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48

Li, Eileen, and Yi Ni. "Half-integral finite surgeries on knots in S^3." Annales de la faculté des sciences de Toulouse Mathématiques 24, no. 5 (2015): 1157–78. http://dx.doi.org/10.5802/afst.1479.

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49

PRZYTYCKI, JÓZEF H., and KOUKI TANIYAMA. "ALMOST POSITIVE LINKS HAVE NEGATIVE SIGNATURE." Journal of Knot Theory and Its Ramifications 19, no. 02 (February 2010): 187–289. http://dx.doi.org/10.1142/s0218216510007838.

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We analyze properties of links which have diagrams with a small number of negative crossings. We show that if a nontrivial link has a diagram with all crossings positive except possibly one, then the signature of the link is negative. If a link diagram has two negative crossings, we show that the signature of the link is nonpositive with the exception of the left-handed Hopf link (possibly, with extra trivial components). We also characterize those links which have signature zero and diagrams with two negative crossings. In particular, we show that if a nontrivial knot has a diagram with two negative crossings then the signature of the knot is negative, unless the knot is a twist knot with negative clasp. We completely determine all trivial link diagrams with two or fewer negative crossings. For a knot diagram with three negative crossings, the signature of the knot is nonpositive except for the left-handed trefoil knot. These results generalize those of Rudolph, Cochran, Gompf, Traczyk and Przytycki, solve [27, Conjecture 5], and give a partial answer to [3, Problem 2.8] about knots dominating the trefoil knot or the trivial knot. We also describe all unknotting number one positive knots.

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50

Heusler, Stefan, and Malte Ubben. "A Haptic Model for the Quantum Phase of Fermions and Bosons in Hilbert Space Based on Knot Theory." Symmetry 11, no. 3 (March 22, 2019): 426. http://dx.doi.org/10.3390/sym11030426.

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A generalization of the famous Dirac belt trick opens up the way to a haptic model for quantum phases of fermions and bosons in Hilbert space based on knot theory. We introduce a simple paper strip model as an aid for visualization of the quantum phases before and after Hopf-mapping, which can be extended to arbitrary spin states with almost no mathematical formalism. Knot theory arises naturally, leading to the Jones polynomials derived from Artin’s braid group for fermionic knots and for bosonic links. The paper strip model explicitly illuminates the relation between these knots and links within the S U ( 2 ) -representation of spin-jstates in C 2 j + 1 before Hopf-mapping and the number p = 2 j of nodes in the stellar representation in C P 1 after Hopf mapping.

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