2-bridge Knots Research Articles

On 2-bridge knots with differing smooth and topological slice genera

We give infinitely many examples of 2-bridge knots for which the topological and smooth slice genera differ. The smallest of these is the 12-crossing knot $12a255$. These also provide the first known examples of alternating knots for which the smooth and topological genera differ.

The volume of hyperbolic cone-manifolds of the knot with Conway’s notation C(2n,3)

Let [Formula: see text] be the family of two bridge knots of slope [Formula: see text]. We calculate the volumes of the [Formula: see text] cone-manifolds using the Schläfli formula. We present the concrete and explicit formula of them. We apply the general instructions of Hilden, Lozano and Montesinos-Amilibia and extend the Ham, Mednykh and Petrov’s methods. As an application, we give the volumes of the cyclic coverings over those knots. For the fundamental group of [Formula: see text], we take and tailor Hoste and Shanahan’s. As a byproduct, we give an affirmative answer for their question whether their presentation is actually derived from Schubert’s canonical two-bridge diagram or not.

Parabolic generating pairs of genus-one 2-bridge knot groups

We show that any parabolic generating pair of a genus-one hyperbolic 2-bridge knot group is equivalent to the upper or lower meridian pair. As an application, we obtain a complete classification of the epimorphisms from 2-bridge knot groups to genus-one hyperbolic 2-bridge knot groups.

Double affine Hecke algebras and generalized Jones polynomials

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 properties of the classical Jones polynomials and prove that these hold for all knots (independently of the conjecture). We furthermore conjecture that the skein module of the unknot is a submodule of the skein module of an arbitrary knot. We confirm this for the same example knots, and we show that this implies that the colored Jones polynomials of $K$ satisfy an inhomogeneous recursion relation.

On the HOMFLY polynomial of 4-plat presentations of knots

Abstract. In this paper, I give a method to calculate the HOMFLY polynomials of twobridge knots by using a representation of the braid group B 4 into a group of 3×3 matrices.Also, I will give examples of a 2-bridge knot and a 3-bridge knot that have the same Jonespolynomial, but different HOMFLY polynomials. 1. IntroductionIn 1985, Hoste-Ocneanu-Millett-Freyd-Lickorish-Yetter [3] had discovered the HOMFLYpolynomial which is a 2-variable oriented link polynomial P L (a,m) motivated by the Jonespolynomial. Also, Prztycki and Traczyk [7] independently discovered the HOMFLY polyno-mial. The calculation of the HOMFLY polynomial is based on the HOMFLY skein relationsas follows.(1) P(L) is an isotopy invariant.(2) P(unknot)=1.(3) a·P(L + ) +a −1 ·P(L − )+m·P(L 0 ) = 0.L L _ L+ 0 Figure 1.Actually, Lickorish and Millett [6] gave a formula to calculate the HOMFLY polynomials ofrational knots by using a representation of the continued fraction of a rational knot into agroup of 2× 2 matrices. (See Proposition 14.)In this paper, I use the Kauffman blacket polynomial and the plat presentation of knots tocalculate the HOMFLY polonomials of rational knots by using a representation of the braidgroup B

Stein fillings of contact 3-manifolds obtained as Legendrian surgeries

In this note, we classify Stein fillings of an infinite family of contact 3-manifolds up to diffeomorphism. Some contact 3-manifolds in this family can be obtained by Legendrian surgeries on $(S^3,\xi_{std})$ along certain Legendrian 2-bridge knots. We also classify Stein fillings, up to symplectic deformation, of an infinite family of contact 3-manifolds which can be obtained by Legendrian surgeries on $(S^3,\xi_{std})$ along certain Legendrian twist knots. As a corollary, we obtain a classification of Stein fillings of an infinite family of contact hyperbolic 3-manifolds up to symplectic deformation.

Repeated boundary slopes for 2-bridge knots

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.

A remark on 2-adjacency relation between Montesinos knots and 2-bridge knots

We show that for every 2-bridge knot K, there is an infinite family of Montesinos knots {Kq} such that each Kq is 2-adjacent to K. We also give a new example of 2-adjacency relation between 2-bridge knots.

Region unknotting number of 2-bridge knots

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 notations are C(m, n), C(m, 2, m), C(m, 2, m ± 1) and C(2, m, 2, n). By generalizing, we also provide a sharp upper bound for all the remaining classes of 2-bridge knots.

Lower bound of the unknotting number of prime knots with up to 12 crossings

We list prime knots with up to 12 crossings having the property: the lower bound of the unknotting number cannot be decided by the signature or non-triviality but is given by either: (i) the condition of a 2-bridge knot with unknotting number one, (ii) the criteria using special values of the Jones, Q, or HOMFLYPT polynomials, or (iii) Wendt's formula. Then we can give new information to the table of unknotting numbers in "KnotInfo", a web-based table of knot invariants.

The Goeritz matrix and signature of a two bridge knot

According to a formula by Gordon and Litherland \cite{GordonLitherland}, the signature $\sigma (K)$ of a knot $K$ can be computed as $\sigma (K) = \sigma (G) - \mu$, where $G$ is the Goeritz matrix of a projection $D$ of $K$ and $\mu$ is a ``correction term,'' read off from the projection $D$. In this article, we consider the family of two bridge knots $K_{p/q}$ and compute the signature of the Goeritz matrices of their ``standard projections,'' $D_{p/q}$, by explicitly diagonalizing the Goertiz matrix over the rationals. We give applications to signature computations and concordance questions.

CYCLIC BRANCHED COVERS OF ALTERNATING KNOTS AND L-SPACES

Abstract. For any alternating knot, it is known that the double branch-ed cover of the 3-sphere branched over the knot is an L-space. We showthat the three-fold cyclic branched cover is also an L-space for any genusone alternating knot. 1. IntroductionAn L-space M is a rational homology 3-sphere whose Heegaard Floer ho-mology HFd(M) is a free abelian group of rank equal to |H 1 (M;Z)| ([12]).Prototypical examples of L-spaces are lens spaces. In recent years, it is recog-nized that L-spaces form an important class of 3-manifolds. For example, see[2, 12].We considerthe problem when cyclic branched coversof the 3-spherebranch-ed over a knot or link is an L-space. Toward this direction, Ozsv´ath and Szabo´[13] first showed that the double branched cover of any non-split alternatinglink (more generally, quasi-alternating link) is an L-space. Peters [15] verifiedthat for a genus one, 2-bridge knot C[2m,2n] (m,n > 0) in Conway’s notation,the d-fold cyclic branched cover is an L-space for any d ≥ 2, and that forC[2m,−2n] (m,n > 0), so is the 3-fold cyclic branched cover. For the latter,the same conclusion still holds for the cases d = 4 ([17]) and d = 5 ([9]), but itwould be false for sufficiently large d ([10, 18]).In this paper, we restrictourselvesto alternatingknots. As mentioned above,the double branched cover of any alternating knot is an L-space. Then, is the3-fold cyclic branched cover an L-space? The answer is positive for genus one,2-bridge knots. However, it is negative, in general. Let Σ

On the degeneration of tunnel numbers under a connected sum

We show that, for any integer $n\ge 3$, there is a prime knot $k$ such that (1) $k$ is not meridionally primitive, and (2) for every $m$-bridge knot $k'$ with $m\leq n$, the tunnel numbers satisfy $t(k\# k')\le t(k)$. This gives counterexamples to a conjecture of Morimoto and Moriah on tunnel number under connected sum and meridionally primitive knots.

Minimum lattice length and ropelength of 2-bridge knots and links

Knots are commonly found in molecular chains such as DNA and proteins, and they have been considered to be useful models for structural analysis of these molecules. One interested quantity is the minimum number of monomers necessary to realize a molecular knot. The minimum lattice length Len(K) of a knot K indicates the minimum length necessary to construct K in the cubic lattice. Another important quantity in physical knot theory is the ropelength which is one of the knot energies measuring the complexity of knot conformation. The minimum ropelength Rop(K) is the minimum length of an ideally flexible rope necessary to tie a given knot K. Much effort has been invested in the research project for finding upper bounds on both quantities in terms of the minimum crossing number c(K) of the knot. It is known that Len(K) and Rop(K) lie between \documentclass[12pt]{minimal}\begin{document}$\mbox{O}(c(K)^{\frac{3}{4}})$\end{document}O(c(K)34) and O(c(K)[ln (c(K))]5), but unknown yet whether any family of knots has superlinear growth. In this paper, we focus on 2-bridge knots and links. Linear growth upper bounds on the minimum lattice length and minimum ropelength for nontrivial 2-bridge knots or links are presented as Len(K) ⩽ 8c(K) + 2 and Rop(K) ⩽ 11.39c(K) + 12.37.

Left-orderable fundamental groups and Dehn surgery on genus one 2–bridge knots

For any hyperbolic genus one 2-bridge knot in the 3-sphere, we show that the resulting manifold by $r$-surgery on the knot has left-orderable fundamental group if the slope $r$ lies in some range which depends on the knot.

Left-orderable fundamental groups and Dehn surgery on genus one 2–bridge knots

Left-orderable fundamental groups and Dehn surgery on genus one 2–bridge knots

Systems of 3-braid equations

The tangle model is a useful topological tool in the study of the mechanism of action of certain enzymes on DNA molecules. In particular, the model proves helpful to determine the topological structure of the DNA molecules resulting from those reactions. Roughly speaking, the tangle model consists in solving a system of three equations in which the unknowns on the left-hand side of each equation are tangles, whereas the known data on the right-hand sides are 2-bridge knots in many cases. Initially [6], the model was successfully applied to study the Tn3 enzyme acting on rational 2-tangles, for which a complete classification exists [4]. By contrast, the Gin enzyme is known [8] to act on 3-tangles and, since no complete classification is known for general 3-tangles, the tangle model was used to study the mechanism of action of Gin under the assumption [2] that the 3-tangles involved were in fact 3-braids, a particular class of 3-tangles. Some questions derived from the application of the tangle model are of mathematical interest in themselves, e.g., given a system of equations that admits a solution, what kinds of \(2\)-bridge knots may appear on the right-hand sides of the equations so that a (nonempty) solution is guaranteed to exist? In this paper, we address and solve this question by showing that, while a system of two equations always admits a solution for any selection of \(2\)-bridge knots, adding a third equation reduces the number of possible knots to only 6, 9 or 18, the exact value depending on the relationships satisfied by the knots in the first two equations. If a fourth equation is adjoined, however, exactly one \(2\)-bridge knot may appear in its the right-hand side for the system to admit a solution. Furthermore, a new simple method that exploits an unexpected cyclic behavior of the solutions is presented and used to construct the proofs. The method relies on the continued fractions associated with \(2\)-bridge knots and their behavior under the concatenation of 3-braids.

Knot contact homology and representations of knot groups

We study certain linear representations of the knot group that induce augmentations of knot contact homology. This perspective on augmentations enhances our understanding of the relationship between the augmentation polynomial and the A-polynomial of the knot. For example, we show that for 2-bridge knots the polynomials agree and that this is never the case for (non-2-bridge) torus knots, nor for a family of 3-bridge pretzel knots. In addition, we obtain a lower bound on the meridional rank of the knot. As a consequence, our results give another proof that torus knots and a family of pretzel knots have meridional rank equal to their bridge number.

Twisted Alexander polynomials of 2-bridge knots for parabolic representations

In this paper we show that the twisted Alexander polynomial associated to a parabolic representation determines fiberedness and genus of a wide class of 2-bridge knots. As a corollary we give an affirmative answer to a conjecture of Dunfield, Friedl and Jackson for infinitely many hyperbolic knots.

The pillowcase and perturbations of traceless representations of knot groups

We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowka's singular instanton knot homology non-degenerate. The mechanism for this study is a (Lagrangian) intersection diagram which arises, through restriction of representations, from a tangle decomposition of a knot. When one of the tangles is trivial, our perturbations allow us to study isolated intersections of two Lagrangians to produce minimal generating sets for singular instanton knot homology. The (symplectic) manifold where this intersection occurs corresponds to the traceless character variety of the four-punctured 2-sphere, which we identify with the familiar pillowcase. We investigate the image in this pillowcase of the traceless representations of tangles obtained by removing a trivial tangle from 2-bridge knots and torus knots. Using this, we compute the singular instanton homology of a variety of torus knots. In many cases, our computations allow us to understand non-trivial differentials in the spectral sequence from Khovanov homology to singular instanton homology.