Hugh M. Hilden

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 (set of zeroes of a polynomial r(x, z)) is associated to p/q parametrizing the representations of in PSL . The coordinates x, z, are trace(A2)=x, trace(AB)=z where A and B are the images of canonical generators a, b of . Let (xn, zn) be the point of 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 . 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 universal groups and three-manifolds

Let P be a regular dodecahedron in the hyperbolic 3-space H3with the dihedral angles 90∘. Choose 6 mutually disjoint edgesE1,E2,⋯,E6 of P such that each face of P intersects E1∪E2∪⋯∪E6 in one edge and the opposite vertex. Let U be the group of orientation-preserving isometries of H3 generated by 90∘-rotations about E1,⋯,E6. It was observed by W. Thurston that H3/U=S3 and that the projection H3→H3/U is a covering branched over the Borromean rings with branching indices 4. The main result of the paper is the following universality of U. Theorem: For every closed, oriented 3-manifold M there exists a subgroup G of U of finite index such that M=H3/G. In other words M is a hyperbolic orbifold finitely covering the hyperbolic orbifold H3/U. The main ingredient of the proof is the strict form of the universality of the Borromean rings (earlier obtained by the first three authors): each closed, oriented 3-manifold is shown to be a covering of S3 branched over the Borromean rings with indices 1, 2, 4. This theorem offers a new approach to the Poincare conjecture: If M=H3/G as above and π1(M)=1 then G is generated by elements of finite order. The authors start off an algebraic investigation of U⊂PSL2(C) by constructing three generators of U which are 2×2 matrices over the ring of algebraic integers in the field Q(2√,3√,5√,1√+5√,−1−−−√).

VOLUMES AND CHERN-SIMONS INVARIANTS OF CYCLIC COVERINGS OVER RATIONAL KNOTS

In this paper, the authors compute the volumes and Chern-Simons invariants for a class of hyperbolic 3-manifolds, namely, the n-fold branched covers of S3 along the 2-bridge knots p/q. The computation is based on the formula of Schlaffli. In a 1-parameter family of polytopes in a space of constant curvature K, KdV=(1/2)∑lidαi, where V is the volume, and the sum is taken over all edges, li is the length of the ith edge and αi is its dihedral angle. Thus the volume of a 1-parameter family of cone-manifolds can be computed in terms of an initial volume and an integration involving length and cone angle of the singular curves. Similarly, the Chern-Simons invariant can be expressed in terms of an initial value and an integration involving the jump and the angle, based on earlier work of the authors. The 1-parameter family of cone-manifolds arises from the following. It is well-known that these 2-bridge knots have hyperbolic complements, which can be considered as hyperbolic cone-manifold structures on S3 with cone-angle 0 around the knot. It is also well-known that the 2-fold branched cover of S3 along p/q is the lens space Lp,q, which has spherical geometry, which induces a spherical cone-manifold structure on S3 with cone-angle π around the knot. These two structures are members of the family of cone-manifold structures on S3 having the 2-bridge knot p/q as a singular curve with angle α (0≤α≤π). There is an angle αh such that the cone structure is hyperbolic when 0≤α<αh, Euclidean when α=αh, and spherical when αh<α≤π. The authors choose the parameter to be x=2cosα, where α is the cone angle around the knot. They compute the functions jump, β(x), and length, δ(x), from the excellent component of the curve of representations of the knot group into SL(2,C). This allows them to compute the volumes and Chern-Simons invariants of the cone manifolds in terms of explicit integrals. The computation of the covering manifolds follows from the multiplicity of these invariants. Examples of numerical computations are shown at the end.

On the character variety of periodic knots and links

A link L of the 3-sphere S3 is said to be g-periodic (g≥2 an integer) if there exists an orientation preserving auto-homeomorphism h of S3 such that h(L)=L, h is of order g and the set of fixed points of h is a circle disjoint from L. A knot is called periodic with rational quotient if it is obtained as the preimage of one component of a 2-bridge link by a g-fold cyclic covering branched on the other component. In this paper the authors introduce a method to compute the excellent component of the character variety of periodic knots (note that for hyperbolic knots the excellent component of the character curve contains the complete hyperbolic structure). Among other examples, this method is applied to the seven hyperbolic periodic knots with rational quotient in Rolfsens table and with bridge number greater than 2.

TUNNEL NUMBER ONE KNOTS HAVE PALINDROME PRESENTATIONS

We show that any tunnel number one knot group has a two generator one relator presentation in which the relator is a palindrome in the generators. We use this fact to compute the character variety for this knot groups and we show that it is an affine algebraic set .

CHARACTER VARIETIES AND PERIPHERAL POLYNOMIALS OF A CLASS OF KNOTS

The representation space or character variety of a finitely generated group is easy to define but difficult to do explicit computations with. The fundamental group of a knot can have two interesting representations into PSL2(C) coming from oppositely oriented complete hyperbolic structures. These two representations lift to give four excellent SL2(C) representations. The excellent curves of a knot are the components of the SL2(C) character variety containing the excellent representations. It is possible to compute geometric invariants of hyperbolic cone manifolds from suitable descriptions of the excellent curve. In this paper, Hilden, Lozano and Montesinos describe a method for analyzing the character varieties of a large class of knots. The main ingredients in this method are a non-obvious, but convenient parametrization of 2×2 complex matrices and an explicit computation relating the holonomies of the four punctures of a four punctured sphere. In order to qualify when their method will work, Hilden, Lozano and Montesinos introduce the notion of a 2n-net. A 2n-net is an interesting generalization of a 2n-plat. Recall that a 2n-plat is obtained by separately closing the top and the bottom of a 2n-strand braid. A 2n-net is the generalization obtained by allowing rational tangles at the crossing points. The given method to analyze the character variety works for any knot with a 4-net description. The method is remarkably robust. For example, it works for essentially every knot in the table in D. Rolfsens book [Knots and links, Publish or Perish, Berkeley, Calif., 1976

On the character variety of tunnel number 1 knots

Given a hyperbolic knot K in S3, the SL2(C) characters ofπ1(S3−K) form an algebraic variety Cˆ(K). The algebraic component containing the character of the complete hyperbolic structure of S3−K is an algebraic curve CˆE(K). The desingularization of the projective curve corresponding to CˆE(K) is a Riemann surface Σ(K), and the trace function corresponding to the meridian of K induces a map p:Σ(K)→C. The pair (Σ(K),p) contains a great deal of information about the knot K and its hyperbolic structure. It can be described by a polynomial rE[K](y,z). There is an algebraic number yh which is a particular critical point of p in the interval (−2,2). It defines an angle 0<αh<2π with yh=2cos(αh/2), called the limit of hyperbolicity. The minimal polynomial hK(y) of yh is called the h-polynomial of K. The calculation of these invariants is in general quite complicated. In this paper the authors develop a method to calculate rE[K](y,z) and hK(y) for any tunnel number one knot, and they apply the method to the knots 10139 and 10161.

All three-manifolds are pullbacks of a branched covering ³ to ³

This paper establishes two new ways of representing all closed orientable 3-manifolds. (1) Let F,N be a pair of disjoint bounded orientable surfaces in the 3-sphere S3. Let (Sk,Fk,Nk), k=1,2,3, be 3 copies of the triplet (S,F,N). Split S1 along F1; S2 along F2 and N2; S3 along N3. Glue F1 to F2, N2 to N3 to obtain a closed orientable 3-manifold. Then every closed orientable 3-manifold can be obtained in this way. (2) Let q:S→S be any 3-fold irregular branched covering of the 3-sphere S over itself. Let M be any 3-manifold. Then there is a 3-fold irregular branched covering p:M→S and a smooth map f:S→S such that f is transverse to the branch set of q and p is the pullback of q and f.

Lifting surgeries to branched covering spaces

Long ago J. W. Alexander showed that any closed, orientable, triangulated n-manifold can be expressed as a branched covering of the n-sphere [Bull. Amer. Math. Soc. 26 (1919/20), 370–372; Jbuch 47, 529]. In general, the branch set is not a manifold and no useful information is given about the degree of the branched covering. When n=3, however, he did indicate that the branch set could be arranged to be a link. Much more recently, the first author [Amer. J. Math. 98 (1976), no. 4, 989–997], U. Hirsch [Math. Z. 140 (1974), 203–230] and the second author [Quart. J. Math. Ser. (2) 27 (1976), no. 105, 85–94] showed that when n=3 the branched covering can be constructed to have degree 3 and a knot as branch set. Of course, these branched coverings are highly irregular. The authors here address similar questions in higher dimensions. Starting with a branched covering Mn→Sn, the authors give some technical, sufficient conditions for a manifold obtained from Mn by a single surgery to be a branched covering of Sn of the same degree and with a branch set easily described in terms of the initial branch set. The nicest corollary of the general technique is that if Mn→Sn is a branched covering of degree d, then there is a branched covering Mn×Sk→Sn+k of degree d+1. The new branch set is an orientable and/or locally flat submanifold if and only if the original branch set is. In particular, the n-torus is an n-fold branched covering of the n-sphere, branched along a locally flat, orientable submanifold. (For known cohomological reasons, n is the smallest possible degree of such a branched covering.)

ON REPRESENTATIONS OF 2-BRIDGE KNOT GROUPS IN QUATERNION ALGEBRAS

Representations of two bridge knot groups in the isometry group of some complete Riemannian 3-manifolds as