Torus Knots Of Type Research Articles

Signed mosaic graphs and alternating mosaic number of knots

Lomonaco and Kauffman introduced knot mosaics in their work on quantum knots. This definition is intended to represent an actual physical quantum system. A knot n-mosaic is an n×n matrix of 11 kinds of specific mosaic tiles representing a knot or a link. In this paper, we consider the alternating mosaic number of an alternating knot K which is defined as the smallest integer n for which K is representable as a reduced alternating knot n-mosaic. We define a signed mosaic graph and a diagonal grid graph and construct Hamiltonian cycles derived from the diagonal grid graphs. Using the cycles, we completely determine the alternating mosaic number of torus knots of type (2,q) for q≥2, which grows in an order of q1/2.

Petal number of torus knots of type (r,r + 2)

Let [Formula: see text] be an odd integer, [Formula: see text]. Then the petal number of the torus knot of type [Formula: see text] is equal to [Formula: see text].

Crossing number of an alternating knot and canonical genus of its whitehead double

A conjecture proposed by J. Tripp in 2002 and modified by T. Nakamura in 2006 states that the crossing number of any alternating knot coincides with the canonical genus of its Whitehead double. In the meantime, it has been established that this conjecture is true for a large class of alternating knots including (2,m) torus knots, 2-bridge knots, algebraic alternating knots, alternating pretzel knots and so on. In this paper, we prove that the conjecture is not true for any alternating 3-braid knot which is the connected sum of two torus knots of type (2,m) and (2,n) with odd integers m,n≥3. This results in a new modified conjecture that the crossing number of any prime alternating knot coincides with the canonical genus of its Whitehead double. We prove that this modified conjecture holds for all prime alternating 3-braid knots in addition to the known classes above.

TWO DIMENSIONAL ARRAYS FOR ALEXANDER POLYNOMIALS OF TORUS KNOTS

Given a pair p, q of relative prime positive integers, we have uniquely determined positive integers x, y, u and v such that vx-uy = 1, p = x + y and q = u + v. Using this property, we show that<TEX>$${\sum\limits_{1{\leq}i{\leq}x,1{\leq}j{\leq}v}}\;{t^{(i-1)q+(j-1)p}\;-\;{\sum\limits_{1{\leq}k{\leq}y,1{\leq}l{\leq}u}}\;t^{1+(k-1)q+(l-1)p}$$</TEX> is the Alexander polynomial <TEX>${\Delta}_{p,q}(t)$</TEX> of a torus knot t(p, q). Hence the number <TEX>$N_{p,q}$</TEX> of non-zero terms of <TEX>${\Delta}_{p,q}(t)$</TEX> is equal to vx + uy = 2vx - 1. Owing to well known results in knot Floer homology theory, our expanding formula of the Alexander polynomial of a torus knot provides a method of algorithmically determining the total rank of its knot Floer homology or equivalently the complexity of its (1,1)-diagram. In particular we prove (see Corollary 2.8); Let q be a positive integer> 1 and let k be a positive integer. Then we have <TEX>$$\begin{array}{rccl}(1)&N_{kq}+1,q&=&2k(q-1)+1\\(2)&N_{kq}+q-1,q&=&2(k+1)(q-1)-1\\(3)&N_{kq}+2,q&=&{\frac{1}{2}}k(q^2-1)+q\\(4)&N_{kq}+q-2,q&=&{\frac{1}{2}}(k+1)(q^2-1)-q\end{array}$$</TEX> where we further assume q is odd in formula (3) and (4). Consequently we confirm that the complexities of (1,1)-diagrams of torus knots of type t(kq + 2, q) and t(kq + q - 2, q) in [5] agree with <TEX>$N_{kq+2,q}$</TEX> and <TEX>$N_{kq+q-2,q}$</TEX> respectively.

Geometry of the SL(3, ℂ)–character variety of torus knots

Let G be the fundamental group of the complement of the torus knot of type (m,n). This has a presentation G=<x,y|x^m=y^n>. We find the geometric description of the character variety X(G) of characters of representations of G into SL(3,C), GL(3,C) and PGL(3,C).

The $SU(2)$-character varieties of torus knots

Let $G$ be the fundamental group of the complement of the torus knot of type $(m,n)$. We study the relationship between $SU(2)$ and $\sldos$-representations of this group, looking at their characters. Using the description of the character variety of $G$, $X(G)$, we give a geometric description of $Y(G)\subset X(G)$, the set of characters arising from $SU(2)$-representations.

2-BRIDGE NUMBERS OF TORUS KNOTS

The 2-bridge number of knots was introduced by Hass, Rubinstein and Thompson [Knots and k-width, Geom. Dedicata143 (2009) 7–18] as a natural generalization of the bridge number introduced by Schubert [Über eine numerisch knoteninvariante, Math. Z.61 (1954) 245–288]. We show that the 2-bridge number of a torus knot of type (p, q) is (p - 1)q + 2 if 1 &lt; p &lt; q.

THE TENEVA GAME

For each prime p &gt; 7 we obtain the expression for an upper bound on the minimum number of colors needed to non-trivially color T(2, p), the torus knot of type (2, p), modulo p. This expression is t + 2l -1 where t and l are extracted from the prime p. It is obtained from iterating the so-called Teneva transformations which we introduced in a previous article. With the aid of our estimate we show that the ratio "number of colors needed vs. number of colors available" tends to decrease with increasing modulus p. For instance as of prime 331, the number of colors needed is already one tenth of the number of colors available. Furthermore, we prove that 5 is the minimum number of colors needed to non-trivially color T(2, 11) modulo 11. Finally, as a preview of our future work, we prove that 5 is the minimum number of colors modulo 11 for two rational knots with determinant 11.

The SL(2,C)-character varieties of torus knots

Let G be the fundamental group of the complement of the torus knot of type (m, n). This has a presentation G = ‹x, y | xm&#8234;&#8234;= y&#8319;›. We find the geometric description of the character variety X(G) of characters of representations of G into SL(2,C).

MINIMAL DEGREE SEQUENCE FOR TORUS KNOTS OF TYPE (p, q)

In this paper we prove the following result: for coprime positive integers p and q with p &lt; q, if r is the least positive integer such that 2p-1 and q + r are coprime, then the minimal degree sequence for a torus knot of type (p, q) is the triple (2p - 1, q + r, d) or (q + r, 2p - 1, d) where q + r + 1 ≤ d ≤ 2q - 1.

ON POLYNOMIAL TORUS KNOTS

We show that no torus knot of type (2, n), n &gt; 3 odd, can be obtained from a polynomial embedding t ↦ (f(t), g(t), h(t)) where (deg(f), deg(g)) ≤ (3, n + 1). Eventually, we give explicit examples with minimal lexicographic degree.

On the minimum number of colors for knots

In this article we take up the calculation of the minimum number of colors needed to produce a non-trivial coloring of a knot. This is a knot invariant and we use the torus knots of type ( 2 , n ) as our case study. We calculate the minima in some cases. In other cases we estimate upper bounds for these minima leaning on the features of modular arithmetic. We introduce a sequence of transformations on colored diagrams called Teneva transformations. Each of these transformations reduces the number of colors in the diagrams by one (up to a point). This allows us to further decrease the upper bounds on these minima. We conjecture on the value of these minima. We apply these transformations to rational knots.

MINIMAL DEGREE SEQUENCE FOR TORUS KNOTS OF TYPE (p,2p-1)

In this paper, we explore the issue of minimizing the degree sequence for torus knots. We find the minimal degree sequence for torus knots of type (p, 2p-1) for any integer p ≥2. We use some results from algebraic geometry to prove our main result.

(1, 1)-knots via the mapping class group of the twice punctured torus

We develop an algebraic representation for ð1; 1Þ-knots using the mapping class group of the twice punctured torus MCG2ðTÞ. We prove that every ð1; 1Þ-knot in a lens space Lð p; qÞ can be represented by the composition of an element of a certain rank two free sub- group of MCG2ðTÞ with a standard element only depending on the ambient space. As notable examples, we obtain a representation of this type for all torus knots and for all two-bridge knots. Moreover, we give explicit cyclic presentations for the fundamental groups of the cyclic branched coverings of torus knots of type ðk; ck þ 2Þ.

Torus Knots and Dunwoody Manifolds

We obtain an explicit representation as Dunwoody manifolds of all cyclic branched coverings of torus knots of type (p,mp±1), with p > 1 and m > 0.

Strongly-cyclic branched coverings of (1, 1)-knots and cyclic presentations of groups

We study the connections among the mapping class group of the twice punctured torus, the cyclic branched coverings of (1, 1)-knots and the cyclic presentations of groups. We give the necessary and sufficient conditions for the existence and uniqueness of the n-fold strongly-cyclic branched coverings of (1, 1)-knots, through the elements of the mapping class group. We prove that every n-fold strongly-cyclic branched covering of a (1, 1)-knot admits a cyclic presentation for the fundamental group, arising from a Heegaard splitting of genus n .M oreover, we give an algorithm to produce the cyclic presentation and illustrate it in the case of cyclic branched coverings of torus knots of type (k, hk ± 1).

Minimal Degree Sequence for Torus Knots

It was observed by Arnold[6] and later proved by Shastri[5] that every non-compact knot-type can be represented by a polynomial embedding. For practical reasons it is important to construct real polynomials /(£), #(£), and h(t) such that the map t H-» (f(t),g(t),h(t)) from M to M is an embedding that represents a given knot type. If deg(f(t)) = Z, deg(g(t)) = m and deg(h(t)) = n, we say that the given knot-type can be represented by polynomials of degree /, m and n. We have tried this construction for a torus knot of type (p, q) which we denote by KPiq and obtained the following results [1],[4],

Polynomial Representation of Torus Knots of type (p, q)

Polynomial Representation of Torus Knots of type (p, q)

POLYNOMIAL REPRESENTATION OF TORUS KNOTS

Journal of Knot Theory and Its RamificationsVol. 05, No. 02, pp. 279-294 (1996) No AccessPOLYNOMIAL REPRESENTATION OF TORUS KNOTSA. RANJAN and RAMA SHUKLAA. RANJANDepartment of Mathematics, Indian Institute of Technology, Powai, Bombay, 400 076, India and RAMA SHUKLAThe Mehta Research Institute of Mathematics and Mathematical Physics, 10, Kasturba Gandhi Marg (Old Kutchery Road), Allahabad, 211 002, Indiahttps://doi.org/10.1142/S0218216596000199Cited by:9 Previous AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail FiguresReferencesRelatedDetailsCited By 9Untangling trigonal diagramsErwan Brugallé, Pierre-Vincent Koseleff, and Daniel Pecker20 June 2016 | Journal of Knot Theory and Its Ramifications, Vol. 25, No. 07On the lexicographic degree of two-bridge knotsErwan Brugallé, Pierre-Vincent Koseleff, and Daniel Pecker20 June 2016 | Journal of Knot Theory and Its Ramifications, Vol. 25, No. 07Spaces of polynomial knots in low degreeRama Mishra and Hitesh Raundal16 December 2015 | Journal of Knot Theory and Its Ramifications, Vol. 24, No. 14Polynomial Unknotting and Singularity IndexRama Mishra23 Jun 2014 | Kyungpook mathematical journal, Vol. 54, No. 2CHEBYSHEV KNOTSP.-V. KOSELEFF and D. PECKER2 May 2012 | Journal of Knot Theory and Its Ramifications, Vol. 20, No. 04Chebyshev diagrams for two-bridge knotsP. -V. Koseleff and D. Pecker23 June 2010 | Geometriae Dedicata, Vol. 150, No. 1A polynomial parametrization of torus knotsP. -V. Koseleff and D. Pecker28 August 2009 | Applicable Algebra in Engineering, Communication and Computing, Vol. 20, No. 5-6ON POLYNOMIAL TORUS KNOTSP.-V. KOSELEFF and D. PECKER21 November 2011 | Journal of Knot Theory and Its Ramifications, Vol. 17, No. 12MINIMAL DEGREE SEQUENCE FOR TORUS KNOTS OF TYPE (p,2p-1)PRABHAKAR MADETI and RAMA MISHRA21 November 2011 | Journal of Knot Theory and Its Ramifications, Vol. 15, No. 09 Recommended Vol. 05, No. 02 Metrics History Received 22 December 1994 PDF download

An estimate for the unknotting numbers of torus knots

In this note we present an estimate for the unknotting numbers of torus knots, which is an improvement of results by Murasugi (1965), Weintraub (1979), Yamamoto (1982), Boileau and Weber (1986) and Shibuya (1986). As a corollary, we show that Milnor's conjecture (1968) is true for torus knots of type (2, q), (3,4), (3,5), (3,7), (3,8), (3,10) and (4,5). For torus knots of type (2, q), (3,4) and (3,5), this has already been known.