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A174684 The number of (tame) knots and links with Matveev complexity n for closed manifolds, given by Heard in the form of a surgery description when the underlying space is not S^3. 0
3, 4, 13, 32 (list; graph; refs; listen; history; text; internal format)
Matveev complexity is additive under connected sum, hence to know the complexity of any manifold one only needs to know that of its connected summands; and, with a few easy exceptions, the complexity of a prime manifold equals the minimal number of tetrahedra required to triangulate it. Heard determines all (0,1,2)-irreducible pairs up to complexity 5, allowing disconnected graphs but forbidding components without vertices in complexity 5. The result is a list of 129 pairs, of which 123 are hyperbolic with parabolic meridians. Petronio (p.31) notes that Matveev showed with a (complicated) theoretical argument that no closed manifold of complexity smaller than 9 can be hyperbolic.
Martelli shows that there are precisely 4 closed orientable hyperbolic 3-manifolds of complexity 9, and they coincide with those of smallest known volume. Heard gives pictures of the hyperbolic graphs up to complexity 4, as figure Figure 16, Complexity 1 (1 example), p.28; Figure 17, Complexity 2 (4 examples), p.28; Figure 18, Complexity 3 (8 examples), p.29; Figures 19-22. Complexity 4 (32 examples), pp.30-33. See pp.27-33 in Heard pictures of the hyperbolic graphs up to complexity 4, given in the form of a surgery description when the underlying space is not S^3.
For each graph, we give the name and the volume of the hyperbolic structure with parabolic meridians. The figures were produced using Orb and the census of knotted graphs. The theoretical framework underlying the papers is twofold, being based on Matveev's theory of spines and on Thurston's idea (later developed by several authors) of constructing hyperbolic structures via triangulations. Many of these results were obtained (or suggested) by computer investigations.
B. Martelli and C. Petronio, 3-manifolds having complexity at most 9, Experiment. Math. 10 (2001), 207-237.
S. V. Matveev, Complexity theory of three-dimensional manifolds, Acta Appl. Math. 19 (1990), 101-130.
Damian Heard, Craig Hodgson, Bruno Martelli, Carlo Petronio, Hyperbolic graphs of small complexity, April 30, 2008.
*** I am not sure that I properly understood all definitions end enumerations of the two arXiv papers for a(3), which might not be 32 if I misunderstood ***
a(0) = 3 because of the unknot in S^3, the core of Heegaard torus in P^3, and the core of Heegaard torus in L(3; 1). a(1) = 4 because of the trefoil in S^3, the Hopf link in S^3, the core of Heegaard torus in L(4; 1), and core of Heegaard torus in L(5; 2). See table 10, p.26 of Heard. a(2) = 13, which are described in Table 10, p.26 of Heard. In complexity 3 only knots and links in the 3-sphere are described in that table; those not in the 3-sphere are shown in later tables.
Sequence in context: A111954 A192872 A036672 * A286917 A084315 A194649
Jonathan Vos Post, Mar 26 2010

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