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Journal articles on the topic 'Knot polynomials'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

Takioka, Hideo. "Infinitely many knots with the trivial (2,1)-cable Γ-polynomial". Journal of Knot Theory and Its Ramifications 27, № 02 (2018): 1850013. http://dx.doi.org/10.1142/s021821651850013x.

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For coprime integers [Formula: see text] and [Formula: see text], the [Formula: see text]-cable [Formula: see text]-polynomial of a knot is the [Formula: see text]-polynomial of the [Formula: see text]-cable knot of the knot, where the [Formula: see text]-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. In this paper, we show that there exist infinitely many knots with the trivial [Formula: see text]-cable [Formula: see text]-polynomial, that is, the [Formula: see text]-cable [Formula: see text]-polynomial of the trivial knot. Moreover, we see th

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2

Mellor, Blake. "Alexander and writhe polynomials for virtual knots." Journal of Knot Theory and Its Ramifications 25, no. 08 (2016): 1650050. http://dx.doi.org/10.1142/s0218216516500504.

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We give a new interpretation of the Alexander polynomial [Formula: see text] for virtual knots due to Sawollek [On Alexander–Conway polynomials for virtual knots and Links, preprint (2001), arXiv:math/9912173] and Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications 12 (2003) 987–1000], and use it to show that, for any virtual knot, [Formula: see text] determines the writhe polynomial of Cheng and Gao [A polynomial invariant of virtual links, J. Knot Theory Ramifications 22(12) (2013), Article ID: 1341002, 33pp.] (equivalently, Kauffman’s affine index polyn

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3

JEONG, MYEONG-JU, and CHAN-YOUNG PARK. "LENS KNOTS, PERIODIC LINKS AND VASSILIEV INVARIANTS." Journal of Knot Theory and Its Ramifications 13, no. 08 (2004): 1041–56. http://dx.doi.org/10.1142/s0218216504003615.

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In this paper, we study lens knots and periodic knots by using integral Vassiliev invariants. Knot polynomials such as the Jones, HOMFLY, Kauffman polynomials give infinitely many integral Vassiliev invariants and we get some necessary conditions for a link to be a lens knot or a periodic link by using these polynomial invariants.

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4

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 (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, w

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5

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 (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 whi

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6

Stoimenow, A. "On Cabled Knots and Vassiliev Invariants (Not) Contained in Knot Polynomials." Canadian Journal of Mathematics 59, no. 2 (2007): 418–48. http://dx.doi.org/10.4153/cjm-2007-018-0.

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AbstractIt is known that the Brandt–Lickorish–Millett–Ho polynomial Q contains Casson's knot invariant. Whether there are (essentially) other Vassiliev knot invariants obtainable from Q is an open problem. We show that this is not so up to degree 9. We also give the (apparently) first examples of knots not distinguished by 2-cable HOMFLY polynomials which are not mutants. Our calculations provide evidence of a negative answer to the question whether Vassiliev knot invariants of degree d ≤ 10 are determined by the HOMFLY and Kauffman polynomials and their 2-cables, and for the existence of alge

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7

Berest, Yuri, and Peter Samuelson. "Double affine Hecke algebras and generalized Jones polynomials." Compositio Mathematica 152, no. 7 (2016): 1333–84. http://dx.doi.org/10.1112/s0010437x16007314.

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In this paper we propose and discuss implications of a general conjecture that there is a natural action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K\subset S^{3}$. We prove this in a number of nontrivial cases, including all $(2,2p+1)$ torus knots, the figure eight knot, and all 2-bridge knots (when $q=\pm 1$). As the main application of the conjecture, we construct three-variable polynomial knot invariants that specialize to the classical colored Jones polynomials introduced by Reshetikhin and Turaev. We also deduce some new prop

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8

KIM, TAEHEE, TAKAHIRO KITAYAMA, and TAKAYUKI MORIFUJI. "TWISTED ALEXANDER POLYNOMIALS ON CURVES IN CHARACTER VARIETIES OF KNOT GROUPS." International Journal of Mathematics 24, no. 03 (2013): 1350022. http://dx.doi.org/10.1142/s0129167x13500225.

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For a fibered knot in the 3-sphere the twisted Alexander polynomial associated to an SL(2, ℂ)-character is known to be monic. It is conjectured that for a nonfibered knot there is a curve component of the SL(2, ℂ)-character variety containing only finitely many characters whose twisted Alexander polynomials are monic, i.e. finiteness of such characters detects fiberedness of knots. In this paper, we discuss the existence of a certain curve component which relates to the conjecture when knots have nonmonic Alexander polynomials. We also discuss the similar problem of detecting the knot genus.

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9

LU, BIN, and JIANYUAN K. ZHONG. "THE KAUFFMAN POLYNOMIALS OF PRETZEL KNOTS." Journal of Knot Theory and Its Ramifications 17, no. 02 (2008): 157–69. http://dx.doi.org/10.1142/s0218216508006026.

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Let ℚ(α, s) be the field of rational functions in α, s. We compute the Kauffman polynomials of pretzel knots [1,6] using the Kauffman skein theory and linear algebra tools. We give a formula for the Kauffman polynomial of a pretzel knot such that after inputting the sequence notation of the pretzel knot, the output is its Kauffman polynomial. Our calculation can be implemented in Mathematica, Maple, Mathcad, etc.

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10

Takioka, Hideo. "A characterization of the Γ-polynomials of knots with clasp number at most two". Journal of Knot Theory and Its Ramifications 26, № 04 (2017): 1750013. http://dx.doi.org/10.1142/s0218216517500134.

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It is known that every knot bounds a singular disk with only clasp singularities, which is called a clasp disk. The clasp number of a knot is the minimum number of clasp singularities among all clasp disks of the knot. It is known that the Conway polynomials of knots with clasp number at most two are characterized. In this paper, we focus on the common zeroth coefficient polynomial of both the HOMFLYPT and Kauffman polynomials, which is called the [Formula: see text]-polynomial. As a result, we characterize the [Formula: see text]-polynomials of knots with clasp number at most two. Moreover, i

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11

Altıntaş, İsmet, and Kemal Taşköprü. "Unoriented knot polynomials of torus links as Fibonacci-type polynomials." Asian-European Journal of Mathematics 12, no. 04 (2019): 1950053. http://dx.doi.org/10.1142/s1793557119500530.

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The focus of this paper is to study the two-variable Kauffman polynomials [Formula: see text] and [Formula: see text], and the one-variable BLM/Ho polynomial [Formula: see text] of [Formula: see text]-torus link as the Fibonacci-type polynomials and to express the Kauffman polynomials in terms of the BLM/Ho polynomial. For this purpose, we prove that each of the examined polynomials of [Formula: see text]-torus link can be determined by a third-order recurrence relation and give the recursive properties of them. We correlate these polynomials with the Fibonacci-type polynomials. By using the r

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12

He, Rachael, Austin Ho, Dorian Kalir, Jacob Miller, and Matthew Zevenbergen. "Polynomial Generalizations of Knot Colorings." PUMP Journal of Undergraduate Research 5 (January 1, 2022): 1–23. http://dx.doi.org/10.46787/pump.v5i0.2616.

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In the field of knot theory, knot invariants are properties preserved across all embeddings and projections of the same knot. Fox n-coloring is a classical knot invariant which associates to each knot projection a system of linear equations. We generalize Fox’s n-coloring by using two, not necessarily distinct, polynomials over a field F, which we say form a (g,f)F coloring. We introduce a sufficient condition, called strong, for a pair of polynomials to form a (g,f)F coloring. We confirm a family of pairs of linear polynomials each of which form a (g,f)F coloring. We prove that there are no s

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13

Ja'faruddin, Ja'faruddin. "Unveiling The Hidden Mathematics In Traditional Indonesian Culinary Art: An Exploration of Knot Theory And Alexander Polynomial in Ketupat Telur." Proximal: Jurnal Penelitian Matematika dan Pendidikan Matematika 7, no. 2 (2024): 584–92. http://dx.doi.org/10.30605/proximal.v7i2.3755.

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This research delves into the intriguing relationship between mathematical concepts and traditional Indonesian cultural heritage, particularly in the context of ketupat telur. By applying the theory of knots (knot theory) as a field of topology, the study aims to identify geometric patterns that underlie the making of ketupat telur, as well as to understand the role of symmetry in the formation of knots. The main focus of this research is the ketupat telur knot diagram with 13 points of intersection and Alexander polynomial calculations, as an alternative method for obtaining knot polynomials.

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14

Im, Young Ho, Kyoung Il Park, and Mi Hwa Shin. "Parities and polynomial invariants for virtual links." Journal of Knot Theory and Its Ramifications 23, no. 12 (2014): 1450066. http://dx.doi.org/10.1142/s0218216514500667.

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We introduce the odd Jones–Kauffman polynomial and odd Miyazawa polynomials of virtual link diagrams by using the parity of virtual link diagrams given in [Y. H. Im and K. I. Park, A parity and a multi-variable polynomial invariant for virtual links, J. Knot Theory Ramifications22(13) (2013), Article ID: 1350073, 18pp.], which are different from the original Jones–Kauffman and Miyazawa polynomials. Also, we give a family of parities and odd polynomials for virtual knots so that many virtual knots can be distinguished.

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15

Takioka, Hideo. "The (2,1)-cable Γ-polynomials of knots up to ten crossings". Journal of Knot Theory and Its Ramifications 27, № 04 (2018): 1850028. http://dx.doi.org/10.1142/s0218216518500281.

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For coprime integers [Formula: see text] and [Formula: see text], the [Formula: see text]-cable [Formula: see text]-polynomial of a knot is the [Formula: see text]-polynomial of the [Formula: see text]-cable knot of the knot, where the [Formula: see text]-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. Since it is known that the [Formula: see text]-polynomial is computable in polynomial time, the [Formula: see text]-cable [Formula: see text]-polynomial is also computable in polynomial time. In this paper, we show that the [Formula: see text]-cab

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16

Mironov, A., A. Morozov, A. Morozov, P. Ramadevi, Vivek Kumar Singh, and A. Sleptsov. "Tabulating knot polynomials for arborescent knots." Journal of Physics A: Mathematical and Theoretical 50, no. 8 (2017): 085201. http://dx.doi.org/10.1088/1751-8121/aa5574.

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17

Gill, Amrendra, Maxim Ivanov, Madeti Prabhakar, and Andrei Vesnin. "Recurrent Generalization of F-Polynomials for Virtual Knots and Links." Symmetry 14, no. 1 (2021): 15. http://dx.doi.org/10.3390/sym14010015.

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F-polynomials for virtual knots were defined by Kaur, Prabhakar and Vesnin in 2018 using flat virtual knot invariants. These polynomials naturally generalize Kauffman’s affine index polynomial and use smoothing in the classical crossing of a virtual knot diagram. In this paper, we introduce weight functions for ordered orientable virtual and flat virtual links. A flat virtual link is an equivalence class of virtual links with respect to a local symmetry changing a type of classical crossing in a diagram. By considering three types of smoothing in classical crossings of a virtual link diagram a

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18

LEE, KYEONGHUI, and YOUNG HO IM. "GENERALIZED INDEX POLYNOMIALS FOR VIRTUAL LINKS." Journal of Knot Theory and Its Ramifications 21, no. 14 (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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19

Ja’faruddin and Chen When Haw. "Topology And Tradition: The Knot Polynomials of Ketupat Nabi." ITM Web of Conferences 58 (2024): 01009. http://dx.doi.org/10.1051/itmconf/20245801009.

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This study explores the intersection of mathematics and culinary traditions, focusing on “Ketupat Nabi,” a dish from South Sulawesi’s Bugis community. By applying knot theory, it seeks to understand the mathematical properties of the dish’s knot diagrams. The research began with selecting traditional foods characterized by knots or ties, essential for framing the study’s focus. Photographs and cultural histories of these foods were then collected to provide context. The analysis involved comparing these culinary knots with established knot theory literature, leading to the creation of graphica

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20

Higa, Ryuji. "An estimation for the ascending numbers of knots by Γ-polynomials". Journal of Knot Theory and Its Ramifications 29, № 01 (2020): 1950096. http://dx.doi.org/10.1142/s0218216519500962.

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For a knot, the ascending number is the minimum number of crossing changes which are needed to obtain an descending diagram. We study the [Formula: see text]-polynomial of knots with a given ascending number. We give a lower bound of the ascending numbers by using [Formula: see text]-polynomials. We estimate the ascending numbers for 65 prime knots up to 10 crossings by using [Formula: see text]-polynomials, Conway polynomials, and the determinants.

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21

Fukuhara, Shinji. "Explicit formulae for two-bridge knot polynomials." Journal of the Australian Mathematical Society 78, no. 2 (2005): 149–66. http://dx.doi.org/10.1017/s1446788700008004.

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AbstractA two-bridge knot (or link) can be characterized by the so-called Schubert normal formKp, qwherepandqare positive coprime integers. Associated toKp, qthere are the Conway polynomial ▽kp, q(z)and the normalized Alexander polynomial Δkp, q(t). However, it has been open problem how ▽kp, q(z) and Δkp, q(t) are expressed in terms ofpandq. In this note, we will give explicit formulae for the Conway polynomials and the normalized Alexander polynomials in the case of two-bridge knots and links. This is done using elementary number theoretical functions inpandq.

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22

Tian, Wei, Xue Lei, Louis H. Kauffman, and Jie Liang. "A Knot Polynomial Invariant for Analysis of Topology of RNA Stems and Protein Disulfide Bonds." Computational and Mathematical Biophysics 5, no. 1 (2017): 21–30. http://dx.doi.org/10.1515/mlbmb-2017-0002.

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Abstract Knot polynomials have been used to detect and classify knots in biomolecules. Computation of knot polynomials in DNA and protein molecules have revealed the existence of knotted structures, and provided important insight into their topological structures. However, conventional knot polynomials are not well suited to study RNA molecules, as RNA structures are determined by stem regions which are not taken into account in conventional knot polynomials. In this study, we develop a new class of knot polynomials specifically designed to study RNA molecules, which considers stem regions. We

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23

KIM, SE-GOO. "ALEXANDER POLYNOMIALS AND ORDERS OF HOMOLOGY GROUPS OF BRANCHED COVERS OF KNOTS." Journal of Knot Theory and Its Ramifications 18, no. 07 (2009): 973–84. http://dx.doi.org/10.1142/s0218216509007300.

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Fox showed that the order of homology of a cyclic branched cover of a knot is determined by its Alexander polynomial. We find examples of knots with relatively prime Alexander polynomials such that the first homology groups of their q-fold cyclic branched covers are of the same order for every prime power q. Furthermore, we show that these knots are linearly independent in the knot concordance group using the polynomial splitting property of the Casson–Gordon–Gilmer invariants.

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Jeong, Myeong-Ju. "Delta moves and Kauffman polynomials of virtual knots." Journal of Knot Theory and Its Ramifications 23, no. 10 (2014): 1450053. http://dx.doi.org/10.1142/s0218216514500539.

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In 1990, Okada showed that the second coefficients of the Conway polynomials of two knots differ by 1 if the two knots are related by a single Δ-move. We extend the Okada's result for virtual knots by using a Vassiliev invariant v2 of virtual knots of degree 2 which is induced from the Kauffman polynomial of a virtual knot. We show that v2(K1) - v2(K2) = ±48, if K2 is a virtual knot obtained from a virtual knot K1 by applying a Δ-move. From this we have a lower bound [Formula: see text] for the number of Δ-moves if two virtual knots K1 and K2 are related by a sequence of Δ-moves.

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Ivanov, Maxim, and Andrei Vesnin. "F-polynomials of tabulated virtual knots." Journal of Knot Theory and Its Ramifications 29, no. 08 (2020): 2050054. http://dx.doi.org/10.1142/s0218216520500546.

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A sequence of [Formula: see text]-polynomials [Formula: see text] of virtual knots [Formula: see text] was defined by Kaur et al. in 2018. These polynomials have been expressed in terms of index value of crossing and [Formula: see text]-writhe of [Formula: see text]. By the construction, [Formula: see text]-polynomials are generalizations of Kauffman’s Affine Index Polynomial, and are invariants of virtual knot [Formula: see text]. We present values of [Formula: see text]-polynomials of oriented virtual knots having at most four classical crossings in a diagram.

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26

GELCA, RĂZVAN, and JEREMY SAIN. "THE NONCOMMUTATIVE A-IDEAL OF A (2, 2p + 1)-TORUS KNOT DETERMINES ITS JONES POLYNOMIAL." Journal of Knot Theory and Its Ramifications 12, no. 02 (2003): 187–201. http://dx.doi.org/10.1142/s021821650300238x.

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The noncommutative A-ideal of a knot is a generalization of the A-polynomial, defined using Kauffman bracket skein modules. In this paper we show that any knot that has the same noncommutative A-ideal as the (2,2p + 1)-torus knot has the same colored Jones polynomials. This is a consequence of the orthogonality relation, which yields a recursive relation for computing all colored Jones polynomials of the knot.

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27

Eisermann, Michael. "Knot colouring polynomials." Pacific Journal of Mathematics 231, no. 2 (2007): 305–36. http://dx.doi.org/10.2140/pjm.2007.231.305.

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BAO, YUANYUAN. "ON THE KNOT FLOER HOMOLOGY OF A CLASS OF SATELLITE KNOTS." Journal of Knot Theory and Its Ramifications 21, no. 04 (2012): 1250030. http://dx.doi.org/10.1142/s0218216511009807.

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Knot Floer homology is an invariant for knots in the three-sphere for which the Euler characteristic is the Alexander–Conway polynomial of the knot. The aim of this paper is to study this homology for a class of satellite knots, so as to see how a certain relation between the Alexander–Conway polynomials of the satellite, companion and pattern is generalized on the level of the knot Floer homology. We also use our observations to study a classical geometric invariant, the Seifert genus, of our satellite knots.

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STOIMENOW, ALEXANDER. "SOME INEQUALITIES BETWEEN KNOT INVARIANTS." International Journal of Mathematics 13, no. 04 (2002): 373–93. http://dx.doi.org/10.1142/s0129167x02001290.

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We study the existence of relations between the degrees of the knot polynomials and some classical knot invariants, partially confirming and extending the question of Morton on the skein polynomial and a recent question of Ferrand.

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BARTHOLOMEW, ANDREW, and ROGER FENN. "QUATERNIONIC INVARIANTS OF VIRTUAL KNOTS AND LINKS." Journal of Knot Theory and Its Ramifications 17, no. 02 (2008): 231–51. http://dx.doi.org/10.1142/s021821650800604x.

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In this paper, we define and give examples of a family of polynomial invariants of virtual knots and links. They arise by considering certain 2 × 2 matrices with entries in a possibly non-commutative ring, for example, the quaternions. These polynomials are sufficiently powerful to distinguish the Kishino knot from any classical knot, including the unknot.

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HOSTE, JIM, and PATRICK D. SHANAHAN. "TWISTED ALEXANDER POLYNOMIALS OF 2-BRIDGE KNOTS." Journal of Knot Theory and Its Ramifications 22, no. 01 (2013): 1250138. http://dx.doi.org/10.1142/s0218216512501386.

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We investigate the twisted Alexander polynomial of a 2-bridge knot associated to a Fox coloring. For several families of 2-bridge knots, including but not limited to, torus knots and genus-one knots, we derive formulae for these twisted Alexander polynomials. We use these formulae to confirm a conjecture of Hirasawa and Murasugi for these knots.

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齐, 园园. "Knot Polynomials and Integral Coefficient Polynomials." Advances in Applied Mathematics 12, no. 01 (2023): 443–50. http://dx.doi.org/10.12677/aam.2023.121047.

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HSIEH, CHUN-CHUNG. "GENERALIZED CONWAY POLYNOMIALS." Journal of Knot Theory and Its Ramifications 10, no. 06 (2001): 923–29. http://dx.doi.org/10.1142/s0218216501001232.

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TAKIOKA, HIDEO. "THE ZEROTH COEFFICIENT HOMFLYPT POLYNOMIAL OF A 2-CABLE KNOT." Journal of Knot Theory and Its Ramifications 22, no. 02 (2013): 1350001. http://dx.doi.org/10.1142/s0218216513500016.

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The zeroth coefficient polynomial is a one variable polynomial contained in the HOMFLYPT polynomial. In this paper, we give a basic computation of the zeroth coefficient polynomial of a 2-cable knot. In particular, we compute the zeroth coefficient polynomials of the 2-cable knots of the Kanenobu knots. It is known that the Kanenobu knots have the same HOMFLYPT polynomial and the same Khovanov–Rozansky homology. As a result, we distinguish the Kanenobu knots completely. Moreover, we estimate the braid indices of the Kanenobu knots.

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Pavlyuk, A. M. "Generalized Equidistant Chebyshev Polynomials and Alexander Knot Invariants." Ukrainian Journal of Physics 63, no. 6 (2018): 488. http://dx.doi.org/10.15407/ujpe63.6.488.

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We introduce the generalized equidistant Chebyshev polynomials T(k,h) of kind k of hyperkind h, where k, h are positive integers. They are obtained by a generalization of standard and monic Chebyshev polynomials of the first and second kinds. This generalization is fulfilled in two directions. The horizontal generalization is made by introducing hyperkind ℎ and expanding it to infinity. The vertical generalization proposes expanding kind k to infinity with the help of the method of equidistant coefficients. Some connections of these polynomials with the Alexander knot and link polynomial invar

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36

Hayashi, M. "Calculation of Knot Polynomials for Unknotted Knots." Progress of Theoretical Physics 90, no. 1 (1993): 263–68. http://dx.doi.org/10.1143/ptp/90.1.263.

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GAROUFALIDIS, STAVROS, and XINYU SUN. "THE NON-COMMUTATIVE A-POLYNOMIAL OF TWIST KNOTS." Journal of Knot Theory and Its Ramifications 19, no. 12 (2010): 1571–95. http://dx.doi.org/10.1142/s021821651000856x.

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The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative A-polynomial of twist knots. Our multivariable creative telescoping method allows us to compute linear recursions for sums of the form [Formula: see text] given a recursion relation for [Formula: see text] and the hypergeometric kernel c(n, k). As an application of our method, we explicitly compute the non-commutative A-polynomial for twist knots with -15 and 15 crossings. The non-commutative A-polynomial

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Kaur, Kirandeep, Madeti Prabhakar, and Andrei Vesnin. "Two-variable polynomial invariants of virtual knots arising from flat virtual knot invariants." Journal of Knot Theory and Its Ramifications 27, no. 13 (2018): 1842015. http://dx.doi.org/10.1142/s0218216518420154.

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We introduce two sequences of two-variable polynomials [Formula: see text] and [Formula: see text], expressed in terms of index value of a crossing and [Formula: see text]-dwrithe value of a virtual knot [Formula: see text], where [Formula: see text] and [Formula: see text] are variables. Basing on the fact that [Formula: see text]-dwrithe is a flat virtual knot invariant, we prove that [Formula: see text] and [Formula: see text] are virtual knot invariants containing Kauffman affine index polynomial as a particular case. Using [Formula: see text] we give sufficient conditions when virtual kno

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39

Morton, H. R., and H. B. Short. "The 2-variable polynomial of cable knots." Mathematical Proceedings of the Cambridge Philosophical Society 101, no. 2 (1987): 267–78. http://dx.doi.org/10.1017/s0305004100066627.

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AbstractThe 2-variable polynomial PK of a satellite K is shown not to satisfy any formula, relating it to the polynomial of its companion and of the pattern, which is at all similar to the formulae for Alexander polynomials. Examples are given of various pairs of knots which can be distinguished by calculating P for 2-strand cables about them even though the knots themselves share the same P. Properties of a given knot such as braid index and amphicheirality, which may not be apparent from the knot's polynomial P, are shown in certain cases to be detectable from the polynomial of a 2-cable abo

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40

KIM, TAEHEE, and TAKAYUKI MORIFUJI. "TWISTED ALEXANDER POLYNOMIALS AND CHARACTER VARIETIES OF 2-BRIDGE KNOT GROUPS." International Journal of Mathematics 23, no. 06 (2012): 1250022. http://dx.doi.org/10.1142/s0129167x11007653.

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We study the twisted Alexander polynomial from the viewpoint of the SL (2, ℂ)-character variety of nonabelian representations of a knot group. It is known that if a knot is fibered, then the twisted Alexander polynomials associated with nonabelian SL (2, ℂ)-representations are all monic. In this paper, we show that for a 2-bridge knot there exists a curve component in the SL (2, ℂ)-character variety such that if the knot is not fibered then there are only finitely many characters in the component for which the associated twisted Alexander polynomials are monic. We also show that for a 2-bridge

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41

FUJII, HIROZUMI. "FIRST COMMON TERMS OF THE HOMFLY AND KAUFFMAN POLYNOMIALS, AND THE CONWAY POLYNOMIAL OF A KNOT." Journal of Knot Theory and Its Ramifications 08, no. 04 (1999): 447–62. http://dx.doi.org/10.1142/s0218216599000316.

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We study the first common terms of the HOMFLY and Kauffman polynomials of a knot, which we call the ϒ-polynomial, and the Conway polynomial for the 2-bridge knots and a class of 3-bridge knots. We characterize the ϒ-polynomial using the 2-bridge knots. Then we give some relations between the two polynomial invariants, and as an application, we consider the space of the Vassiliev invariant of order four.

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42

MORTON, HUGH R., and PETER R. CROMWELL. "DISTINGUISHING MUTANTS BY KNOT POLYNOMIALS." Journal of Knot Theory and Its Ramifications 05, no. 02 (1996): 225–38. http://dx.doi.org/10.1142/s0218216596000163.

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We consider the problem of distinguishing mutant knots using invariants of their satellites. We show, by explicit calculation, that the Homfly polynomial of the 3-parallel (and hence the related quantum invariants) will distinguish some mutant pairs. Having established a condition on the colouring module which forces a quantum invariant to agree on mutants, we explain several features of the difference between the Homfly polynomials of satellites constructed from mutants using more general patterns. We illustrate this by our calculations; from these we isolate some simple quantum invariants, a

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43

DIAO, Y., G. HETYEI, and K. HINSON. "TUTTE POLYNOMIALS OF TENSOR PRODUCTS OF SIGNED GRAPHS AND THEIR APPLICATIONS IN KNOT THEORY." Journal of Knot Theory and Its Ramifications 18, no. 05 (2009): 561–89. http://dx.doi.org/10.1142/s0218216509007075.

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It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman's Tutte polynomial of signed graphs for which describing the effect of taking a signed tensor product of signed graphs is very simple. We show that this Tutte polynomial of a signed tensor product of signed graphs may be expressed in terms of the Tutte polynomials of the original signed graphs by using a simple substitution rule. Ou

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44

Owczarek, Robert. "Remarks on Chebyshev polynomials, Fibonacci polynomials and Kauffman bracket skein modules." Journal of Knot Theory and Its Ramifications 27, no. 07 (2018): 1841007. http://dx.doi.org/10.1142/s0218216518410079.

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The Chebyshev polynomials appear somewhat mysteriously in the theory of the skein modules. A generalization of the Chebyshev polynomials is proposed so that it includes both Chebyshev and Fibonacci and Lucas polynomials as special cases. Then, since it requires relaxation of a condition for traces of matrix powers and matrix representations, similar relaxation leads to a generalization of the Jones polynomial via reinterpretation of the Kauffman bracket construction. Moreover, the Witten’s approach via counting solutions of the Kapustin–Witten equation to get the Jones polynomial is simplified

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45

Tao, Zhi-Xiong. "2-Adjacency between knots." Journal of Knot Theory and Its Ramifications 24, no. 11 (2015): 1550054. http://dx.doi.org/10.1142/s0218216515500546.

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We study the properties of a knot K to be 2-adjacent to another knot W by analyzing their Conway polynomials, Jones polynomials and Homfly polynomials and give some very useful conditions. We discuss whether each pair of knots can be 2-adjacent to each other, i.e. whether 2-adjacency is a symmetric relation. We discuss also whether the trivial knot, the trefoil knot and the figure-eight knot can be 2-adjacent to any knot in Rolfsen's table and the opposite cases, except for 934 it is not decided whether it is 2-adjacent to 41. Finally, we give some examples to answer I. Torisu's problem partly

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Panagiotou, Eleni, and Louis H. Kauffman. "Knot polynomials of open and closed curves." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2240 (2020): 20200124. http://dx.doi.org/10.1098/rspa.2020.0124.

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In this manuscript, we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves. We define the bracket polynomial of curves in 3-space and show that it has real coefficients and is a continuous function of the curve coordinates. This is used to define the Jones polynomial in a way that it is applicable to both open and closed curves in 3-space. For open curves, the Jones polynomial has real coefficients and it is a continuous function of the curve coordinates and as the endpoints of the curve tend to coincide, the Jones

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Jeong, Myeong-Ju. "Reidemeister moves and parity polynomials of virtual knot diagrams." Journal of Knot Theory and Its Ramifications 26, no. 10 (2017): 1750051. http://dx.doi.org/10.1142/s0218216517500511.

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When two virtual knot diagrams are virtually isotopic, there is a sequence of Reidemeister moves and virtual moves relating them. I introduced a polynomial [Formula: see text] of a virtual knot diagram [Formula: see text] and gave lower bounds for the number of Reidemeister moves in deformation of virtually isotopic knot diagrams by using [Formula: see text]. In this paper, I introduce bridge diagrams and polynomials of virtual knot diagrams based on parity of crossings, and show that the polynomials give lower bounds for the number of the third Reidemeister moves. I give an example which show

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Tuzun, Robert E., and Adam S. Sikora. "Verification of the Jones unknot conjecture up to 22 crossings." Journal of Knot Theory and Its Ramifications 27, no. 03 (2018): 1840009. http://dx.doi.org/10.1142/s0218216518400096.

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We proved by computer enumeration that the Jones polynomial distinguishes the unknot for knots up to 22 crossings. Following an approach of Yamada, we generated knot diagrams by inserting algebraic tangles into Conway polyhedra, computed their Jones polynomials by a divide-and-conquer method, and tested those with trivial Jones polynomials for unknottedness with the computer program SnapPy. We employed numerous novel strategies for reducing the computation time per knot diagram and the number of knot diagrams to be considered. That made computations up to 21 crossings possible on a single proc

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49

Jiménez Pascual, Adrián. "On lassos and the Jones polynomial of satellite knots." Journal of Knot Theory and Its Ramifications 25, no. 02 (2016): 1650011. http://dx.doi.org/10.1142/s0218216516500115.

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In this paper, I present a new family of knots in the solid torus called lassos, and their properties. Given a knot [Formula: see text] with Alexander polynomial [Formula: see text], I then use these lassos as patterns to construct families of satellite knots that have Alexander polynomial [Formula: see text] where [Formula: see text]. In particular, I prove that if [Formula: see text] these satellite knots have different Jones polynomials.

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50

Ganzell, Sandy. "Local moves and restrictions on the Jones polynomial." Journal of Knot Theory and Its Ramifications 23, no. 02 (2014): 1450011. http://dx.doi.org/10.1142/s0218216514500114.

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We analyze various local moves on knot diagrams to show that Jones polynomials must have certain algebraic properties. In particular, we show that the Jones polynomial of a knot cannot be a nontrivial monomial.

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