C. Adams, M. Hildebrand, and J. Weeks. Hyperbolic invariants of knots and links. Trans. Amer. Math. Soc. 326(1991), 1-56.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....3 17 : after A ;B ;A , this link becomes 6 3 1 . Almost all the other links in Rolfsen s tables can be distinguished using their Alexander polynomials, their hyperbolic volumes, and the shapes of their cusps. The Alexander polynomials are tabulated in Rolfsen; the hyperbolic data is tabulated in [AHW]. As pointed out to us by N. Dun eld, there are some misprints in Rolfsen s tables; the Alexander polynomial for 9 2 55 should actually be the same as that for 9 2 56 , and the matrix 0 0 1 1 0 1 1 0 1 1 0 0 # gives the Alexander polynomial for 9 2 59 . There is one pair of links (L 1 ; ....

C. Adams, M. Hildebrand, and J. Weeks. Hyperbolic invariants of knots and links. Trans. Amer. Math. Soc. 326(1991), 1-56.

....3 17 : after A ;B ;A , this link becomes 6 3 1 . Almost all the other links in Rolfsen s tables can be distinguished using their Alexander polynomials, their hyperbolic volumes, and the shapes of their cusps. The Alexander polynomials are tabulated in Rolfsen; the hyperbolic data is tabulated in [AHW]. As pointed out to us by N. Dunfield, there are some misprints in Rolfsen s tables; the Alexander polynomial for 9 2 55 should actually be the same as that for 9 2 56 , and the matrix 0 0 1 Gamma1 0 1 1 0 Gamma1 1 0 0 # gives the Alexander polynomial for 9 2 59 . There is one pair of ....

C. Adams, M. Hildebrand, and J. Weeks. Hyperbolic invariants of knots and links. Trans. Amer. Math. Soc. 326(1991), 1--56.

....It is well known that the complement in S 3 of the Borromean rings B has the hyperbolic structure [13] Therefore by the hyperbolic Dehn surgery theorem [13] almost all knot complements of our family B i j are hyperbolic. Moreover all of them are hyperbolic for i 1, j 1, i j 5 (see [1]) By Mostov s rigidity theorem both symmetries and oe are also isometries of the corresponding hyperbolic structure on the knot complement of B i j in S 3 . Since for any hyperbolic knot there exist such n that for all k n the orbifolds with this knot as its singular set and k as its ....

....now such hyperbolic orbifolds B i j (k) corresponding to the knots of our family. In this way, for example, holds, that B 1 1 (k) is hyperbolic for k 4 and that the other B i j (k) i 1, j 1, i j 5, are hyperbolic for k 3 as one may verify using the direct calculations of SnapPea [1]. Obviously these hyperbolic orbifolds B i j (k) have still the same hyperbolic orientationpreserving isometries and oe of order two. Factorizing the hyperbolic orbifold B i j (k) by the hyperbolic isometry we obtain a hyperbolic orbifold still with underlying space S 3 and the singular ....

[Article contains additional citation context not shown here]

Adams, C., Hildebrand, M., Weeks, J. , Hyperbolic invariants of knots and links, Trans. AMS, 1991, v. 326, 1, 1-56.

....let S 3 n L i be a non compact manifold which is a complement to L i in the sphere S 3 and let L i (2) be an orbifold with the underlying space S 3 and the link L i as the singular set. Hyperbolic volumes of these manifolds and orbifolds can be founded using SnapPea program of J. Weeks [1], 12] 29] In last column there are given notations of links L i according to [5] and [24] vol(M i ) vol(L i (2) vol(S 3 n L i ) L i M 1 = W (5; Gamma2; 5; Gamma1) 0; 9427 : 0; 4713 : 9; 4270 : 9 49 M 2 = W (1; 1; 5; Gamma1) 0; 9813 : 0; 4906 : 5; 6387 : 10 ....

Adams C., Hildebrand M., Weeks J., Hyperbolic invariants of knots and links, Trans. Amer. Math. Soc., 326 (1991), 1--56.

....Riemann zeta value. It follows from [2] that ZK is reducible, with unspecified 1 rational coefficients, to Bloch Wigner dilogarithms fD(z k ) j z k 2 Kg. The volume of a hyperbolic manifold, for which the single complex place field K is the invariant trace field [7] is systematically [8] reducible to such dilogarithms and, moreover, is expected to be some unspecified rational multiple of the very specific construct (1) Accordingly, we seek coprime integers a and b such that a b vol(M) ZK (2) for a manifold M with a single complex place invariant trace field K. We call a=b ....

C. Adams, M.V. Hildebrand and J.R Weeks, Hyperbolic invariants of knots and links, Trans. AMS, 326 (1991) 1--56.

....hence is an invariant of the knot itself. In fact, by work of Gromov, Jorgensen, and Thurston, the set of volumes of hyperbolic 3 manifolds is well ordered. Hyperbolic volume is an e ective invariant for distinguishing knots. It distinguishes nearly all hyperbolic knots with up to 10 crossings ( AHW91] A noncompact nite volume hyperbolic 3 manifold (e.g. a knot complement) can be canonically decomposed into a nite number of ideal hyperbolic polyhedra ( EP88, Wee93] Ideal tetrahedra form natural building blocks from which to construct hyperbolic 3 manifolds. In [CHW] it is shown that ....

....and have a correspondingly small volume, yet have a very large number of crossings. We will consider various examples of these phenomena in Section 6. There are many interesting aspects of knot theory and hyperbolic structures, for a more thorough discussion see the survey [CR] See also [AHW91] for a table of hyperbolic invariants for the knots and links in the 10 crossing tables. 1.3. Surgery and Complexity. In the study of Dehn surgery, the number of ideal tetrahedra is a far better measure of the complexity of a knot than is the crossing number. Dehn surgery is the process of ....

C. Adams, M. Hildebrand, and J. Weeks, Hyperbolic invariants of knots and links, Trans. Amer. Math. Soc. 326 (1991), no. 1, 1-56.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC