We show that every two-bridge knot $K$ of crossing number $N$ admits a polynomial parametrization $x=T_3(t), y = T_b(t), z =C(t)$ where $T_k(t)$ are the Chebyshev polynomials and $b+°C = 3N$. If $C (t)= T_c(t)$ is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots for $a \le 3.$ Most results are derived from continued fractions and their matrix representations.
A Chebyshev knot is a knot which admits a parametrization of the form $ x(t)=T_a(t); \ y(t)=T_b(t) ; \ z(t)= T_c(t + ϕ), $ where $a,b,c$ are pairwise coprime, $T_n(t)$ is the Chebyshev polynomial of degree $n,$ and $ϕ\in \RR .$ Chebyshev knots are non compact analogues of the classical Lissajous knots. We show that there are infinitely many Chebyshev knots with $ϕ= 0.$ We also show that every knot is a Chebyshev knot.
A Chebyshev knot ${\cal C}(a,b,c,ϕ)$ is a knot which has a parametrization of the form $ x(t)=T_a(t); y(t)=T_b(t) ; z(t)= T_c(t + ϕ), $ where $a,b,c$ are integers, $T_n(t)$ is the Chebyshev polynomial of degree $n$ and $ϕ\in \R.$ We show that any two-bridge knot is a Chebyshev knot with $a=3$ and also with $a=4$. For every $a,b,c$ integers ($a=3, 4$ and $a$, $b$ coprime), we describe an algorithm that gives all Chebyshev knots $\cC(a,b,c,ϕ)$. We deduce a list of minimal Chebyshev representations of two-bridge knots with small crossing number.
We show that every rational knot $K$ of crossing number $N$ admits a polynomial parametrization $x=T_a(t), y = T_b(t), z = C(t)$ where $T_k(t)$ are the Chebyshev polynomials, $a=3$ and $b+ °C = 3N.$ We show that every rational knot also admits a polynomial parametrization with $a=4$. If $C (t)= T_c(t)$ is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots for $a \le 4.$
A Chebyshev curve C(a,b,c,ϕ) has a parametrization of the form x(t)=Ta(t); y(t)=T_b(t) ; z(t)= Tc(t + ϕ), where a,b,c are integers, Tn(t) is the Chebyshev polynomial of degree n and ϕ\in \RR. When C(a,b,c,ϕ) has no double points, it defines a polynomial knot. We determine all possible knots when a, b and c are given.
This work presents formulas for the Kauffman bracket and Jones polynomials of 3-bridge knots using the structure of Chebyshev knots and their billiard table diagrams. In particular, these give far fewer terms than in the Skein relation expansion. The subject is introduced by considering the easier case of 2-bridge knots, where some geometric interpretation is provided, as well, via combinatorial tiling problems.
The Alexander polynomials Δ_{n,3}(t) and Δ_{n,4}(t) are presented as a sum of the Alexander polynomials Δ_{k,2}(t). These polynomials are also expressed in the form of a sum of Chebyshev polynomials of the second kind. These expansions allow one to introduce the "coordinates" in corresponding bases, which are proposed to be the numerical invariants characterizing links and knots.
We use the Chebyshev knot diagram model of Koseleff and Pecker in order to introduce a random knot diagram model by assigning the crossings to be positive or negative uniformly at random. We give a formula for the probability of choosing a knot at random among all knots with bridge index at most 2. Restricted to this class, we define internal and external reduction moves that decrease the number of crossings of the diagram. We make calculations based on our formula, showing the numerics in graphs and providing evidence for our conjecture that the probability of any knot appearing in this model…
In the previous paper, we showed that the Riley polynomial $\mathcal{R}_K(λ)$ of each 2-bridge knot $K$ is split into $\mathcal{R}_K(-u^2)=\pm g(u)g(-u)$, for some integral coefficient polynomial $g(u)\in \mathbb Z[u]$. In this paper, we study this splitting property of the Riley polynomial. We show that the Riley polynomial can be expressed by `$ε$-Chebyshev polynomials', which is a generalization of Chebyshev polynomials containing the information of $ε_i$-sequence $(ε_i=(-1)^{[i\fracβα]})$ of the 2-bridge knot $K=S(α,β)$, and then we give an explicit formula for the splitting polynomial $g(…
We give the bridge indices for 11-crossing prime knots and give a minimal bridge projection for each of these knots. The results on the indices may be easily summarized: all of these knots that are not rational knots or Montesinos knots have bridge index three.
Any 2-bridge knot in the 3-sphere has a bridge sphere from which any other bridge surface can be obtained by stabilization, meridional stabilization, perturbation and proper isotopy.
For any given number of crossings $c$, there exists a formula to determine the number of 2-bridge knots of $c$ crossings, and indeed it is a simple matter to actually construct presentations of these knots. However, the determination of whether a given (prime) knot is a 2-bridge knot remains a nontrivial exercise, and we have no procedure to determine bridge numbers more generally. Herein, we identify the 2-bridge knots within the Hoste--Thistlethwaite--Weeks tables of prime knots of up to 16 crossings by an exhaustive search of a larger set of 2-bridge knots. As the unknot is the only knot wi…
We show that any non-minimal bridge decomposition of a torus knot is stabilized and that $n$-bridge decompositions of a torus knot are unique for any integer $n$. This implies that a knot in a bridge position is a torus knot if and only if there exists a torus containing the knot such that it intersects the bridge sphere in two essential loops.
Let $K,K'$ be two-bridge knots of genus $n,k$ respectively. We show the necessary and sufficient condition of $n$ in terms of $k$ that there exists an epimorphism from the knot group of $K$ onto that of $K'$.
We have the generating function which determines the number of $2$-bridge knot groups admitting epimorphisms onto the knot group of a given $2$-bridge knot, in terms of crossing number. In this paper, we will refine this formula by taking account into genus as well as crossing number. Next, we determine the number of epimorphisms between fibered $2$-bridge knot groups. Moreover, we discuss degree one maps and $2$-bridge knots uknotting number one.
The 2-bridge knots are a family of knots with bridge number 2. In this paper, we compute the Kauffman polynomials of 2-bridge knots using the Kauffman skein theory and linear algebra techniques. Our calculation can be easily carried out using Mathematica, Maple, Mathcad, etc.
Suppose that there exists an epimorphism from the knot group of a $2$-bridge knot $K$ onto that of another knot $K'$. In this paper, we study the relationship between their crossing numbers $c(K)$ and $c(K')$. Especially it is shown that $c(K)$ is greater than or equal to $3 c(K')$ and we estimate how many knot groups a $2$-bridge knot group maps onto. Moreover, we formulate the generating function which determines the number of $2$-bridge knot groups admitting epimorphisms onto the knot group of a given $2$-bridge knot.
The Meridional Rank Conjecture asks whether the bridge number of a knot in $S^3$ is equal to the minimal number of meridians needed to generate the fundamental group of its complement. In this paper we investigate the analogous conjecture for knotted surfaces in $S^4$. Towards this end, we give a construction to produce classical knots with quotients sending meridians to elements of any finite order and which detect their meridional ranks. We establish the equality of bridge number and meridional rank for these knots and knotted spheres obtained from them by twist-spinning. On the other hand, …
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.
It is known that there are only finitely many knots with super bridge index 3. Jin and Jeon have provided a list of possible such candidates. However, they conjectured that the only knots with super bridge index 3 are trefoil and the figure eight knot. In this paper, we prove that the $5_2$ knot and the $6_2$ knot are also 3-super bridge knots by providing a polynomial representation of these knots in degree $6.$ This also answers a question asked by Durfee and O'Shea in their paper on polynomial knots: is there any 5-crossing knot in degree 6?
A knot is called minimal if its knot group admits epimorphisms onto the knot groups of only the trivial knot and itself. In this paper, we determine which two-bridge knot $\mathfrak{b}(p,q)$ is minimal where $q \leq 6$ or $p \leq 100$.
In this short note we show the existence of an epimorphism between groups of $2$-bridge knots by means of an elementary argument using the Riley polynomial. As a corollary, we give a classification of $2$-bridge knots by Riley polynomials.
In this paper, we characterize closed incompressible surfaces of genus two in the complements of 3-bridge knots and links. This characterization includes that of essential 2-string tangle decompositions for 3-bridge knots and links.
We present a practical algorithm to determine the minimal genus of non-orientable spanning surfaces for 2-bridge knots, called the crosscap numbers. We will exhibit a table of crosscap numbers of 2-bridge knots up to 12crossings (all 362 of them).
We give an alternative proof of a result of Kobayashi and Saeki that every genus one $1$-bridge position of a non-trivial $2$-bridge knot is a stabilization.
We show that there exists an infinite family of knots, each of which has, for each integer k>=0, a destabilized (2k+5)-bridge sphere. We also show that, for each integer n>=4, there exists a knot with a destabilized 3-bridge sphere and a destabilized n-bridge sphere.
We compute the genus zero bridge numbers and give lower bounds on the genus one bridge numbers for a large class of sufficiently generic hyperbolic twisted torus knots. As a result, the bridge spectra of these knots have two gaps which can be chosen to be arbitrarily large, providing the first known examples of hyperbolic knots exhibiting this property. In addition, we show that there are Berge and Dean knots with arbitrarily large genus one bridge numbers, and as a result, we give solutions to problems of Eudave-Muñoz concerning tunnel number one knots.
We show that a hyperbolic 2-bridge knot complement is the unique knot complement in its commensurability class. We also discuss constructions of commensurable hyperbolic knot complements and put forth a conjecture on the number of hyperbolic knot complements in a commensurability class.
Morifuji computed the twisted Alexander polynomial of twist knots for nonabelian representations. In this paper we compute the twisted Alexander polynomial and the Reidemeister torsion of genus one two-bridge knots, a class of knots which includes twist knots. As an application, we give a formula for the Reidemeister torsion of the 3-manifold obtained by a Dehn surgery on a genus one two-bridge knot.
We use twisted Alexander polynomials to show that certain algebraically slice 2-bridge knots are not topologically slice, even though all prime power Casson-Gordon signatures vanish. We also provide some computations indicating the efficacy of Casson-Gordon signatures in obstructing the smooth sliceness of 2-bridge knots.
We give a locally minimal, but not globally minimal bridge position of a knot, that is, an unstabilized, nonminimal bridge position of a knot. It implies that a bridge position cannot always be simplified so that the bridge number monotonically decreases to the minimal.
We define and compare several natural ways to compute the bridge number of a knot diagram. We study bridge numbers of crossing number minimizing diagrams, as well as the behavior of diagrammatic bridge numbers under the connected sum operation. For each notion of diagrammatic bridge number considered, we find crossing number minimizing knot diagrams which fail to minimize bridge number. Furthermore, we construct a family of minimal crossing diagrams for which the difference between diagrammatic bridge number and the actual bridge number of the knot grows to infinity.
We determine the set of all genus g bridge numbers of many iterated torus knots, listing these numbers in a sequence called the bridge spectrum. In addition, we prove a structural lemma about the decomposition of a strongly irreducible bridge surface induced by cutting along a collection of essential surfaces.
In this paper, we discuss the region unknotting number of different classes of 2-bridge knots. In particular, we provide region unknotting number for the classes of $2$-bridge knots whose Conway notation is $C(m,\ n), C(m,\ 2,\ m),$ $ C(m,\ 2,\ m\pm1)$ and $C(2,\ m,\ 2,\ n)$. By generalizing, we also provide a sharp upper bound for all the remaining classes of $2$-bridge knots.
We exhibit an algorithm to determine the bridge number of a hyperbolic knot in the 3-sphere. The proof uses adaptations of almost normal surface theory for compact surfaces with boundary in ideally triangulated knot exteriors.
In this paper, we consider generalizations of the Alexander polynomial and signature of 2-bridge knots by considering the Gordon-Litherland bilinear forms associated to essential state surfaces of the 2-bridge knots. We show that the resulting invariants are well-defined and explore properties of these invariants. Finally we realize the boundary slopes of the essential surfaces as a difference of signatures of the knot.
In this article we study a partial ordering on knots in the 3-sphere where K_1 is greater than or equal to K_2 if there is an epimorphism from the knot group of K_1 onto the knot group of K_2 which preserves peripheral structure. If K_1 is a 2-bridge knot and K_1 > K_2, then it is known that K_2 must also be 2-bridge. Furthermore, Ohtsuki, Riley, and Sakuma give a construction which, for a given 2-bridge knot K_{p/q}, produces infinitely 2-bridge knots K_{p'/q'} with K_{p'/q'}>K_{p/q}. After characterizing all 2-bridge knots with 4 or less distinct boundary slopes, we use this to prove that in…
The harmonic knot $\H(a,b,c)$ is parametrized as $K(t)= (T_a(t) ,T_b (t), T_c (t))$ where $a$, $b$ and $c$ are pairwise coprime integers and $T_n$ is the degree $n$ Chebyshev polynomial of the first kind. We classify the harmonic knots $\H(a,b,c)$ for $ a \le 4. $ We study the knots $\H (2n-1, 2n, 2n+1),$ the knots $\H(5,n,n+1),$ and give a table of the simplest harmonic knots.
A 1-bridge torus knot in a 3-manifold of genus $\le 1$ is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are similar to the Schubert's normal form and the Conway's normal form for 2-bridge knots. For a given Schubert's normal form we give algorithms to determine the number of components and to compute the fundamental group of the complement when the normal form determines a knot. We also give a description of the double branched cover of an ambient 3-manifold branched along a 1-bridge torus knot by usi…
We extend techniques due to Pardon to show that there is a lower bound on the distortion of a knot in $\mathbb{R}^3$ proportional to the minimum of the bridge distance and the bridge number of the knot. We also exhibit an infinite family of knots for which the minimum of the bridge distance and the bridge number is unbounded and Pardon's lower bound is constant.
We will classify all exceptional Dehn surgeries on 2-bridge knots according to whether they produce reducible, toroidal, or small Seifert fibered manifolds.
Suppose M is a closed irreducible orientable 3-manifold, K is a knot in M, P and Q are bridge surfaces for K and K is not removable with respect to Q. We show that either Q is equivalent to P or $d(K,P) \leq 2-χ(Q-K)$. If K is not a two bridge knot, then the result holds even if K is removable with respect to Q. As a corollary we show that if a knot in the 3-sphere has high distance with respect to some bridge sphere and low bridge number, then the knot has a unique minimal bridge position.
In this paper, we study the Riley polynomial of double twist knots with higher genus. Using the root of the Riley polynomial, we compute the range of rational slope $r$ such that $r$-filling of the knot complement has left-orderable fundamental group. Further more, we make a conjecture about left-orderable surgery slopes of two-bridge knots.
We modify an approach of Johnson to define the distance of a bridge splitting of a knot in a 3-manifold using the dual curve complex and pants complex of the bridge surface. This distance can be used to determine a complexity, which becomes constant after a sufficient number of stabilizations and perturbations, yielding an invariant of the manifold-knot pair. We also give evidence toward the relationship between the pants distance of a bridge splitting and the hyperbolic volume of the exterior of a knot.
Negami found an upper bound on the stick number $s(K)$ of a nontrivial knot $K$ in terms of the minimal crossing number $c(K)$ of the knot which is $s(K) \leq 2 c(K)$. Furthermore McCabe proved $s(K) \leq c(K) + 3$ for a $2$-bridge knot or link, except in the case of the unlink and the Hopf link. In this paper we construct any $2$-bridge knot or link $K$ of at least six crossings by using only $c(K)+2$ straight sticks. This gives a new upper bound on stick numbers of $2$-bridge knots and links in terms of crossing numbers.
We investigate the question of when distinct branched surfaces in the complement of a 2-bridge knot support essential surfaces with identical boundary slopes. We determine all instances in which this occurs and identify an infinite family of knots for which no boundary slopes are repeated.
In this paper, we determine the average genus of all the $2$-bridge knots with a given crossing number. As a consequence, we obtain the oblique asymptote of this value as the crossing number grows.
Using Gauss diagrams, one can define the virtual bridge number ${\rm vb}(K)$ and the welded bridge number ${\rm wb}(K),$ invariants of virtual and welded knots with ${\rm wb}(K) \leq {\rm vb}(K).$ If $K$ is a classical knot, Chernov and Manturov showed that ${\rm vb}(K) = {\rm br}(K),$ the bridge number as a classical knot, and we ask whether the same thing is true for welded knots. The welded bridge number is bounded below by the meridional rank of the knot group $G_K$, and we use this to relate this question to a conjecture of Cappell and Shaneson. We show how to use other virtual and welded…
In this paper, we show that any unknotting tunnel for a two bridge knot is isotopic to either one of known ones. This together with Morimoto-Sakuma's result gives the complete classification of unknotting tunnels for two bridge knots up to isotopies and homeomorphisms.
In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually $\bf S^3$), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of $\bf S^3$ branched over a knot. Moreover, we show that all branched cyclic coverings of a 2-bridge knot belong to this subclass; this implies that the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.
We provide a new proof of the following results of H. Schubert: If K is a satellite knot with companion J and pattern L that lies in a solid torus T in which it has index k, then the bridge numbers satisfy the following: 1) The bridge number of K is greater than or equal to the product of k and the bridge number of J; 2) If K is a composite knot (this is the case k = 1), then the bridge number of K is one less than the sum of the bridge numbers of J and L.
