## Cusp structures of alternating links.(English)Zbl 0810.57010

Given a finite, connected graph $$\Gamma$$ in a 2-sphere in $$S^ 3$$ there is naturally associated an alternating link $${\mathcal L}_ \Gamma$$ whose projection to the 2-sphere meets each complementary region of $$\Gamma$$ in its inscribed polygon. There is an associated decomposition of the complement $$S^ 3 - {\mathcal L}_ \Gamma$$ into ideal polyhedra as introduced by Thurston in his notes on “The Geometry and Topology of 3- manifolds”. This paper is a study of these constructions as tools in the understanding of the 3-manifolds $$S^ 3 - {\mathcal L}_ \Gamma$$ and their Dehn fillings relative to questions such as the existence of incompressible surfaces, the existence of (possibly singular) metrics of negative curvature, determination by their fundamental group, and so on. Numerous examples of these phenomena are provided.
Reviewer: J.Hempel (Houston)

### MSC:

 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M50 General geometric structures on low-dimensional manifolds 57M25 Knots and links in the $$3$$-sphere (MSC2010)
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### References:

 [1] [AR1] Aitchison, I.R., Rubinstein, J.H.: An introduction to polyhedral metrics of non-positive curvature on 3-manifolds. In: Geometry of Low-Dimensional Manifolds, vol. II: Symplectic Manifolds and Jones-Witten Theory pp. 127-161. Cambridge: Cambridge University Press, 1990 · Zbl 0735.57005 [2] [AR2] Aitchison, I.R., Rubinstein, J.H.: Combinatorial cubings, cusps, and the dodecahedral knots. In: Proceedings of the Research Semester in Low Dimensional Topology at Ohio State University. Topology90 (to appear) · Zbl 0773.57010 [3] [AR3] Aitchison, I.R., Rubinstein, J.H.: Canonical surgery on alternating link diagrams, In: Proceedings of the International Conference on Knots, Osaka 1990 (to appear) · Zbl 0765.57005 [4] [AR4] Aitchison, I.R., Rubinstein, J.H.: Polyhedral metrics of non-positive curvature on 3-manifolds with cusps. (In preparation) [5] [BGS] Ballman, W., Gromov, M., Schroeder, V.: Manifolds of nonpositive curvature. Boston: Birkh?user 1985 · Zbl 0591.53001 [6] [BH] Bleiler, S., Hodgson, C.: Spherical space forms and Dehn surgery. Proceedings of the International Conference on Knots, Osaka 1990 (to appear) [7] [Co1] Coxeter, H.S.M.: Regular Polytopes. London: Methuen & Co. 1948 · Zbl 0031.06502 [8] [Co2] Coxeter, H.S.M.: Regular honeycombs in hyperbolic space. In: Proc. I.C.M., 1954. Amsterdam: North-Holland 1956 · Zbl 0056.38603 [9] [Gr] Gromov, M.: Hyperbolic manifolds groups and actions. Riemann surfaces and related topics. In: Kra, I., Maskit, B. (eds.) Stonybrook Conference Proceedings. (Ann. Math. Stud., vol. 97, pp. 183-214) Princeton: Princeton University Press 1981 [10] [GT] Gromov, M., Thurston, W.P.: Pinching constants for hyperbolic manifolds. Invent. Math.89, 1-12 (1987) · Zbl 0646.53037 [11] [HKW] de la Harpe, P., Kervaire, M., Weber, C.: On the Jones polynomial. Enseign. Math.32, 271-335 (1986) · Zbl 0622.57002 [12] [HS] Hass, J., Scott, P.: Homotopy equivalence and homeomorphism of 3-manifolds. (Preprint MSRI July 1989) [13] [HRS] Hass, J., Rubinstein, H., Scott, P.: Covering spaces of 3-manifolds. Bull. Am. Math. Soc.16, 117-119 (1987) · Zbl 0624.57016 [14] [Ha] Hatcher, A.: Hyperbolic structures of arithmetic type on some link complements. J. Lond. Math. Soc.27, 345-355 (1983) · Zbl 0516.57001 [15] [Ho1] Hodgson, C.: Notes on the orbifold thorem. (In preparation) [16] [Ho2] Hodgson, C.: Private communication. (Melbourne 1989) [17] [La] Lawson, T.C.: Representing link complements by identified polyhedra. (Preprint) [18] [Me1] Menasco, W.W.: Polyhedra representation of link complements. (Contemp Math., vol. 20, pp. 305-325) Providence, RI: Am. Math. Soc. 1983 · Zbl 0524.57005 [19] [Me2] Menasco, W.W.: Closed incompressible surfaces in alternating knot and link complements. Topology23, 37-44 (1984) · Zbl 0525.57003 [20] [Re] Reid, A.W.: Totally geodesic surfaces in hyperbolic 3-manifolds. (Preprint); Proc Edinb. Math. Soc. (to appear) [21] [Ro] Rolfsen, D.: Knots and Links. Berkeley: Publish or Perish 1976 · Zbl 0339.55004 [22] [Ta] Takahashi, M.: On the concrete construction of hyperbolic structures of 3-manifolds. (Preprint) [23] [Th] Thurston, W.P.: The geometry and topology of 3-manifolds. Princeton University Lecture Notes 1978 [24] [Wa] Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math.87, 56-88 (1968) · Zbl 0157.30603 [25] [We1] Weeks, J.R.: Hyperbolic structures on three-manifolds. PhD dissertation, Princeton 1985 [26] [We2] Weeks, J.R.: Programs for hyperbolic structures on three-manifolds. Macintosh II version 1990
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