2-bridge Knots Research Articles

Braid group and leveling of a knot

Any knot [Formula: see text] in genus-1 1-bridge position can be moved by isotopy to lie in a union of [Formula: see text] parallel tori tubed by [Formula: see text] tubes so that [Formula: see text] intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal [Formula: see text] for which this is possible is an invariant of the position, called the level number. In this work, we describe the leveling by the braid group on two points in the torus, which yields a numerical invariant of the position, called the (1, 1)-length. We show that the (1, 1)-length equals the level number. We then find braid descriptions for (1,1)-positions of all 2-bridge knots providing upper bounds for their level numbers and also show that the (-2, 3, 7)-pretzel knot has level number two.

The special rank of virtual knot groups

In this paper we introduce the special rank for virtual knots and some properties of this number are studied. Although we do not know if it can be considered as a nontrivial extension of the meridional rank given by [H. U. Boden and A. I. Gaudreau, Bridge number for virtual and welded knots, J. Knot Theory Ramifications 24 (2015), Article ID: 1550008] and by [M. Boileau and B. Zimmermann, The [Formula: see text]-orbifold group of a link, Math. Z. 200 (1989) 187–208], we prove that classical knots with special rank [Formula: see text] are [Formula: see text]-bridge knots. Therefore, a modified version of the so called Cappell and Shaneson conjecture could be considered.

Rational cobordisms and integral homology

We consider the question of when a rational homology $3$-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology group injects in the first homology group of any other element in the same class. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite-rank cokernels. Further consequences include a divisibility condition between the determinants of a connected sum of $2$-bridge knots and any other knot in the same concordance class. Lastly, we use knot Floer homology combined with our main result to obstruct Dehn surgeries on knots from being rationally cobordant to lens spaces.

Symplectic quandles and parabolic representations of 2-bridge knots and links

In this paper, we study the parabolic representations of 2-bridge links by finiding arc coloring vectors on the Conway diagram. The method we use is to convert the system of conjugation quandle equations to that of symplectic quandle equations. In this approach, we have an integer coefficient monic polynomial [Formula: see text] for each 2-bridge link [Formula: see text], and each zero of this polynomial gives a set of arc coloring vectors on the diagram of [Formula: see text] satisfying the system of symplectic quandle equations, which gives an explicit formula for a parabolic representation of [Formula: see text]. We then explain how these arc coloring vectors give us the closed form formulas of the complex volume and the cusp shape of the representation. As other applications of this method, we show some interesting arithmetic properties of the Riley polynomial and of the trace field, and also describe a necessary and sufficient condition for the existence of epimorphisms between 2-bridge link groups in terms of divisibility of the corresponding Riley polynomials.

Meander diagrams of knots and spatial graphs: Proofs of generalized Jablan–Radović conjectures

We study decomposition into simple arcs (i.e., arcs without self-intersections) for diagrams of knots and spatial graphs. In this paper, it is proved in particular that if no edge of a finite spatial graph G is a knotted loop, then there exists a plane diagram D of G such that (i) each edge of G is represented by a simple arc of D and (ii) each vertex of G is represented by a point on the boundary of the convex hull of D. This generalizes the conjecture of S. Jablan and L. Radović stating that each knot has a meander diagram, i.e., a diagram composed of two simple arcs whose common endpoints lie on the boundary of the convex hull of the diagram. Also, we prove another conjecture of Jablan and Radović stating that each 2-bridge knot has a semi-meander minimal diagram, i.e., a minimal diagram composed of two simple arcs.

Cluster variables, ancestral triangles and Alexander polynomials

In this paper, we show that Alexander polynomials for any 2-bridge knots are specializations of cluster variables. A key tool is an ancestral triangle which appeared in both quantum topology and hyperbolic geometry in different ways.

Cluster algebras and Jones polynomials

We present a new and very concrete connection between cluster algebras and knot theory. This connection is being made via continued fractions and snake graphs. It is known that the class of 2-bridge knots and links is parametrized by continued fractions, and it has recently been shown that one can associate to each continued fraction a snake graph, and hence a cluster variable in a cluster algebra. We show that up to normalization by the leading term the Jones polynomial of the 2-bridge link is equal to the specialization of this cluster variable obtained by setting all initial cluster variables to 1 and specializing the initial principal coefficients of the cluster algebra as follows $$y_1=t^{-2}$$ and $$ y_i=-t^{-1}$$ , for all $$i> 1$$ . As a consequence we obtain a direct formula for the Jones polynomial of a 2-bridge link as the numerator of a continued fraction of Laurent polynomials in $$q=-t^{-1}$$ . We also obtain formulas for the degree and the width of the Jones polynomial, as well as for the first three and the last three coefficients. Along the way, we also develop some basic facts about even continued fractions and construct their snake graphs. We show that the snake graph of an even continued fraction is isomorphic to the snake graph of a positive continued fraction if the continued fractions have the same value. We also give recursive formulas for the Jones polynomials.

On 2-knots and connected sums with projective planes

In this paper, we generalize a result of Satoh to show that for any odd natural n, the connected sum of the n-twist spun sphere of a knot K and an unknotted projective plane in the 4-sphere is equivalent to the same unknotted projective plane. We additionally provide a fix to a small error in Satoh’s proof of the case that K is a 2-bridge knot.

A note on Alexander polynomials of 2-bridge links

A formula for the Alexander polynomial of a 2-bridge knot or link given by Hartley and also by Minkus has a beautiful interpretation as a walk on the integers. We extend this to the 2-variable Alexander polynomial of a 2-component, 2-bridge link, obtaining a formula that corresponds to a walk on the 2-dimensional integer lattice.

Remarks on Suzuki’s knot epimorphism number

A partial order on prime knots can be defined by declaring [Formula: see text], if there exists an epimorphism from the knot group of [Formula: see text] onto the knot group of [Formula: see text]. Suppose that [Formula: see text] is a 2-bridge knot that is strictly greater than [Formula: see text] distinct, nontrivial knots. In this paper, we determine a lower bound on the crossing number of [Formula: see text] in terms of [Formula: see text]. Using this bound, we answer a question of Suzuki regarding the 2-bridge epimorphism number [Formula: see text] which is the maximum number of nontrivial knots which are strictly smaller than some 2-bridge knot with crossing number [Formula: see text]. We establish our results using techniques associated with parsings of a continued fraction expansion of the defining fraction of a 2-bridge knot.

Derivation of Schubert normal forms of 2-bridge knots from (1,1)-diagrams

In this paper, we show that the dual [Formula: see text]-diagram of a [Formula: see text]-diagram (a.k.a. a two-pointed genus one Heegaard diagram) [Formula: see text] with [Formula: see text] and [Formula: see text] is given by [Formula: see text] where [Formula: see text] is the multiplicative inverse of [Formula: see text] modulo [Formula: see text] with [Formula: see text]. We also present explicitly how to derive a Schubert normal form of a [Formula: see text]-bridge knot from the dual [Formula: see text]-diagram of [Formula: see text] using weakly K-reducibility of [Formula: see text]-decompositions. This gives an alternative proof of Grasselli and Mulazzani’s result asserting that [Formula: see text] is a [Formula: see text]-diagram of the [Formula: see text]-bridge knot with a Schubert normal form [Formula: see text].

Circuit complexity of knot states in Chern-Simons theory

We compute an upper bound on the circuit complexity of quantum states in 3d Chern-Simons theory corresponding to certain classes of knots. Specifically, we deal with states in the torus Hilbert space of Chern-Simons that are the knot complements on the 3-sphere of arbitrary torus knots. These can be constructed from the unknot state by using the Hilbert space representation of the S and T modular transformations of the torus as fundamental gates. The upper bound is saturated in the semiclassical limit of Chern-Simons theory. The results are then generalized for a family of multi-component links that are obtained by “Hopf-linking” different torus knots. We also use the braid word presentation of knots to discuss states on the punctured sphere Hilbert space associated with 2-bridge knots and links. The calculations present interesting number theoretic features related with continued fraction representations of rational numbers. In particular, we show that the minimization procedure defining the complexity naturally leads to regular continued fractions, allowing a geometric interpretation of the results in the Farey tesselation of the upper-half plane. Finally, we relate our discussion to the framework of path integral optimization by generalizing the original argument to non-trivial topologies.

Varieties via a filtration of the KBSM and knot contact homology

This paper explains a research on the character varieties and the Kauffman bracket skein module (KBSM) of knot exteriors, which has been done by the author in his stay at the University of California, Riverside, 2004-2006. As a main consequence, for any 2-bridge knot, we gave a representation theoretical proof to the conjecture that degree 0 abelian knot contact homology HC0ab(K) of a knot K in the 3-space R3 is isomorphic to the SL2(C)-character ring of the fundamental group of the 2-fold branched cover of the 3-sphere S3 branched along K. (This result was first shown by L. Ng [23] by using cord formalism.) Based on this result, deep studies of the conjecture has been done in [18–20].

Infinite families of prime knots with alt(K) = 1 and their Alexander polynomials

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.

Hoste’s conjecture for generalized Fibonacci polynomials

One very long-standing theme in the theory of knots and links in S3 is the description of Alexander polynomials of alternating links. Hoste, based on computer verification, made the following conjecture: If is a root of the Alexander polynomial of an alternating knot, then . For 15 years Hoste’s problem has remained open for any moderately noteworthy subclass of alternating knots. We prove that for every sequence of polynomials , for integers ai, the complex roots of satisfy . This confirms the conjecture of Hoste for 2-bridge knots. The resolution even for 2-bridge knots requires us to bring up a substantial new technical framework. Our approach will be to work numerically and fundamentally rely on the capacity of MATHEMATICA. It draws on methods and motivation from several fields (it proves numerically an algebraic statement with knot theory background.)

Hoste’s Conjecture and Roots of Link Polynomials

In relation to a conjecture of Hoste on the roots of the Alexander polynomial of alternating knots, we prove that any root z of the Alexander polynomial of a 2-bridge (rational) knot or link satisfies $${| z^{1/2}-z^{-1/2} | < 2}$$ . We extend our result to properties of zeros for some Montesinos knots, and to an analogous statement about the skein polynomial. A similar estimate is derived for alternating 3-braid links.

The disk complex and 2-bridge knots

We give a purely combinatorial proof of a result of Kobayashi and Saeki that every genus one 1-bridge position of a nontrivial 2-bridge knot is a stabilization.

State invariants of 2-bridge knots

In this paper, we consider generalizations of the Alexander polynomial and signature of 2-bridge knots by considering the Gordon–Litherland bilinear forms associated with 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 essential surfaces as differences of signatures of the knot.

Bridge spectra of cables of 2-bridge knots

We compute the bridge spectra of cables of 2-bridge knots. We also give some results about bridge spectra and distance of Montesinos knots.

Twisted Alexander polynomials of 2-bridge knots associated to dihedral representations

Let [Formula: see text] be a non-abelian semi-direct product of a cyclic group [Formula: see text] and an elementary abelian [Formula: see text]-group [Formula: see text] of order [Formula: see text], [Formula: see text] being a prime and [Formula: see text]. Suppose that the knot group [Formula: see text] of a knot [Formula: see text] in the [Formula: see text]-sphere is represented on [Formula: see text]. Then we conjectured (and later proved) that the twisted Alexander polynomial [Formula: see text] associated to [Formula: see text] is of the form: [Formula: see text], where [Formula: see text] is the Alexander polynomial of [Formula: see text] and [Formula: see text] is an integer polynomial in [Formula: see text]. In this paper, we present a proof of the following. For a [Formula: see text]-bridge knot [Formula: see text] in [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text] is written as [Formula: see text], where [Formula: see text] is the set of [Formula: see text]-bridge knots whose knot groups map on that of [Formula: see text] with [Formula: see text] odd.