2-bridge Knots Research Articles

Essential laminations and Dehn surgery on 2-bridge knots

In this paper we show that any nontrivial Dehn surgery on a nontorus 2-bridge knot produces a manifold which is covered by R 3. In particular, this manifold is irreducible and has infinite fundamental group. (As a consequence, it is also clear that 2-bridge knots satisfy property P, although this was shown previously by Takahashi (1981) using more algebraic techniques.) The result is a consequence of showing that such a manifold is laminar, that is it contains an essential lamination. We accomplish this by constructing in the exterior of each nontorus 2-bridge knot an essential lamination which remains essential in all manifolds produced by nontrivial Dehn filling. We call an essential lamination with this property persistent. The examples of essential laminations produced in this way are of interest since they tend support the conjecture that “most” manifolds with infinite fundamental group contain essential laminations.

ON THE ARITHMETIC 2-BRIDGE KNOTS AND LINK ORBIFOLDS AND A NEW KNOT INVARIANT

Let (p/q, n) denote the orbifold with singular set the two bridge knot or link p/q and isotropy group cyclic of orden n. An algebraic curve [Formula: see text] (set of zeroes of a polynomial r(x, z)) is associated to p/q parametrizing the representations of [Formula: see text] in PSL [Formula: see text]. The coordinates x, z, are trace(A2)=x, trace(AB)=z where A and B[Formula: see text] are the images of canonical generators a, b of [Formula: see text]. Let (xn, zn) be the point of [Formula: see text] corresponding to the hyperbolic orbifold (p/q, n). We prove the following result: The (orbifold) fundamental group of (p/q, n) is arithmetic if and only if the field Q(xn, zn) has exactly one complex place and ϕ(xn)<ϕ(zn)<2 for every real embedding [Formula: see text]. Consider the angle α for which the cone-manifold (p/q, α) is euclidean. We prove that 2cosα is an algebraic number. Its minimal polynomial (called the h-polynomial) is then a knot invariant. We indicate how to generalize this h-polynomial invariant for any hyperbolic knot. Finally, we compute h-polynomials and arithmeticity of (p/q, n) with p≦40, and (p/q, n) with p≦99q2≡1 mod p. We finish the paper with some open problems.

On composite tunnel number one links

In this paper, we show that if a composite link has tunnel number one, then it is a connected sum of a 2-bridge knot and a Hopf link.

VARIETIES OF REPRESENTATIONS AND THEIR COHOMOLOGY-JUMP SUBVARIETIES FOR KNOT GROUPS

This paper studies the spaces of representations of knot groups into a linear group , their categorical factor-spaces (i.e., the spaces of all characters of the representations), and their cohomology-jump subspaces. Connections are established between the latter and the spaces of representations of dimension one greater. A complete description is given of these spaces for 2-bridge knots. Bibliography: 12 titles.

SURGERY AND FOLIATIONS OF KNOT COMPLEMENTS

An interesting class of knots have complement with a remarkably simple topological description. This class includes all the arborescent knots with only even weights hence, in particular, the two bridge knots and many knots of ten or fewer crossings. For these knots, there are choices of minimal genus Seifert surfaces S such that all taut, depth one foliations of the knot complement, having S as sole compact leaf, can be classified up to isotopy. These foliations correspond exactly to the lattice points over the open faces of the unit ball in a Thurston-like norm on the relative homology of the complement of S.

On the additivity of tunnel number of knots

Let K 1 and K 2 be nontrivial knots in the 3-sphere S 3. In this paper, we show that if the tunnel number of K 1# K 2 is two, then either both tunnel numbers of K 1 and K 2 are one, or one of K 1 and K 2 is a 2-bridge knot and the other's tunnel number is at most two.

UNIVERSAL 2-BRIDGE KNOT AND LINK ORBIFOLDS

Let (p/q, n) be the orbifold with cyclic isotropy of order n and with singular set the 2-bridge knot or link p/q where p and q are relatively prime numbers, q is odd, q is less than p, and q is not congruent to ±1 mod p (i.e. p/q is any non toroidal 2-bridge knot or link). We show that the orbifold fundamental group π1(p/q, n) is universal for n any multiple of 12. This means that if Γ is any such group, it can be thought of as a discrete group of hyperbolic isometries of hyperbolic 3-space ℍ3, and then, given any closed, oriented 3-manifold M, there exists a subgroup of finite index G of Γ such that M is homeomorphic to G\ℍ3. Since we have shown elsewhere that the group π1(5/3, 12) is an arithmetic group, it follows that there exists an orbifold, namely (5/3, 12), whose singular set is a knot, the figure eight, and whose fundamental group is both arithmetic and universal.

Polynomial Invariants of 2-Bridge Knots through 22 Crossings

Polynomial Invariants of 2-Bridge Knots through 22 Crossings

Supplement to Polynomial Invariants of 2-Bridge Knots through 22 Crossings

Supplement to Polynomial Invariants of 2-Bridge Knots through 22 Crossings

Polynomial invariants of 2-bridge knots through 22 crossings

We calculate the homfly, Kauffman, Jones, Q, and Conway polynomials of 2-bridge knots through 22 crossings and list all the pairs sharing the same polynomial invariants.

Algebra for Heckoid Groups

We introduce an infinite collection of (Laurent) polynomials associated with a $2$-bridge knot or link normal form $K = (\alpha ,\beta )$. Experimental evidence suggests that these "Heckoid polynomials" define the affine representation variety of certain groups, the Heckoid groups, for $K$ . We discuss relations which hold in the image of the generic representation for each polynomial. We show that, with a certain change of variable, each Heckoid polynomial divides the nonabelian representation polynomial of $L$ , where $L$ belongs to an infinite collection of $2$-bridge knots/links determined by $K$ and the Heckoid polynomial. Finally, we introduce a "precusp polynomial" for each $2$-bridge knot normal form, and show it is the product of two (possibly reducible) non-constant polynomials. We are preparing a sequel on the Heckoid groups and the evidence for some of the geometrical assertions stated in the introduction.

CLOSED 3–MANIFOLDS UNCHANGED BY DEHN SURGERY

If X is the space of the trefoil knot, we can find, for every integer e, two slopes on ∂X, ρe, γe, such that the Dehn fillings X (ρe) and X (γe) are homeomorphic 3-manifolds. The cores of the surgeries are inequivalent knots with homeomorphic complements. Analogous results are available if X is the knotspace of a 2-bridge torus knot. In an atoroidal homology sphere knots are determined by their complement if and only if hyperbolic knots are.

Genus and Kauffman polynomial of a 2-bridge knot

Genus and Kauffman polynomial of a 2-bridge knot

Jones and Q Polynomials for 2-Bridge Knots and Links

Jones and Q Polynomials for 2-Bridge Knots and Links

Pseudo-Anosov maps and surgery on fibred 2-bridge knots

We study fibred knots by computing invariant laminations of the monodromy of the fibration and then computing the degeneracy type of the knot. We show that nontrivial surgery on a nontorus fibred 2-bridge knot yields a manifold whose universal cover is R 3.

Branched covers over strongly amphicheiral links

THIS PAPER deals with the question: which 3-manifolds have fundamental groups which are virtually Z-representable-have a subgroup of finite index which maps onto the infinite cyclic group, Z. A compact, orientable, irreducible 3-manifold with this property is finitely covered by a Haken manifold (is virtually Haken). The Haken manifolds are quite well understood: they are determined by homotopy type (of the pair (M, 8M)) [19], they have a solvable classification problem [4], [6], they have residually finite fundamental groups [lo], and they have a unique torus-annulus decomposition [15], [16] into geometric pieces [lS]. There is hope in extending much of these results for Haken manifolds to virtually Haken manifolds. It has been conjectured that for a 3-manifold, M, z,(M) is infinite if and only if xl(M) is virtually Z-representable (see [7], [8] for more discussion). In this paper we will consider (closed, orientable) 3-manifolds which are described as branched covers of the 3-sphere, branched over a fixed link, L c S3. There are many known examples of universal links-for which every 3-manifold can be represented as a cover of S3 branched over the link. Each 2-bridge knot or link, with the exception of those which are also torus knots or links, is universal [14]. For any link, L, in a 3-manifold, M, the collection of coverings of M branched over L forms a lattice. We observe (Lemma 2.3) that whenever an element of this lattice has virtually Z-representable fundamental group then so does every element above it in the lattice. We combine this with an analysis of the upper bound of a pair of elements (Theorem 2.1) to study how the property of virtual Z-representability proliferates throughout the lattice. When the link is strongly amphicheiral (invariant under an orientation reversing involution of S3), the involution theorem (section 3) will apply to show that certain coverings branched over the link will have virtually Z-representable fundamental group. Others will inherit this property by comparison. This includes those whose branching indices satisfy certain divisibility conditions (Theorem 4.1). We apply this to some of the known strongly amphicheiral universal links such as the figure eight knot and the Borromean rings. For example, any cover of S3 branched over the figure eight knot all of whose branching indices are divisible by some integer q > 2 has virtually Z-representable fundamental group (Corollary 4.3). This improves on results of [1], [12]. Throughout this paper we will consider only closed, orientable 3-manifolds. A map

Jones and 𝑄 polynomials for 2-bridge knots and links

It is known that the Q Q polynomial of a 2 2 -bridge knot or link can be obtained from the Jones polynomial. We construct arbitrarily many 2 2 -bridge knots or links with the same Q Q polynomial but distinct Jones polynomials.

Fibered 2-knots and lens spaces

and pi>qi>0 (cf. [10], [13]). The punctured manifold M° means the space obtained from M by removing an open 3-ball. We consider fibered 2-knoΐs in S with fiber M°, which we call M°-fibered 2-knots. For example, a connected sum of 2-twist-spun knots of 2-bridge knots is such a fibered 2-knot. We shall determine possible M°-fibered 2-knots in S (or more generally, a homology 4-sρhere Σ) for all M. There are two reasons for selecting these fibered 2-knots: (1) Any 2-knot with Seifert manifold a connected sum of lens spaces is determined by its exterior [3], [4]. In particular, any fibered 2-knot with fiber a connected sum of lens spaces is determined by its monodromy (precisely, the class of its monodromy in the diίfeotopy group of the fiber). (2) The diffeotopy groups of all lens spaces are computed by Bonahon [1] and Hodgson-Rubinstein [7]. In section 3, we prove that any fibered 2-knot with fiber a punctured lens space L(p, q)° is the 2-twist-spun knot of K(p} q). In general, we show in section 4 that any M°-fibered 2-knot with cyclic monodromy is a cable knot about the 2-twist-spun knot of a 2-bridge knot. In section 5, we consider branched covering spaces of cable knots, and observe, for each odd r>l, the existence of a 2-knot which is the same fixed point set of many inequivalent semi-free ^-actions on S (cf. [2], [5], [14]). I would like to express my gratitude to Professor Akio Kawauchi for suggesting the problem, Professor Fujitsugu Hosokawa and Professor Yasutaka Nakanishi for leading me throughout.

Boundary slopes for Montesinos knots

FOR A KNOT K c S3, let S(K) c Q u {CQ} be the set of slopes of boundary curves of incompressible, %incompressible orientable surfaces in the knot exterior, slopes being normalized in the standard way so that a longitude has slope 0, a meridian slope co. These sets S(K) of %slopes are of special interest because of their relation with Dehn surgery and character varieties; see e.g., [2]. The only general results known so far are that S(K) is always finite [S] and contains at least two elements [3], including of course 0 (coming from a minimal genus Seifert surface). Only for special classes of knots has S(K) been determined exactly. For the (p, q) torus knot, S(K) = (0, pq}. For 2-bridge knots, S(K) is an arbitrarily large set of even integers computable via continued fractions [7], A non-integer s for the motivating example (- 2,3,7) we find S(K) = (0, 16, 18; .20). For somewhat more complicated cases, a small computer can do the work rather quickly. Some examples of these computer calculations are given in the last section of the paper. These include the Montesinos knots of I 10 crossings, plus a few other random examples of greater complexity.

Deforming twist spun $2$-bridge knots of genus one

Deforming twist spun $2$-bridge knots of genus one