OFFSET
0,1
COMMENTS
Let N be a closed, orientable 3-manifold that admits a triangulation with t tetrahedra. Let F be a Heegaard surface for N. S. Schleimer showed that if g(F) >= 2^{2^{16}t^2}, then the Hempel distance of F (denoted by d(F)) is at most two. In this paper we prove the following generalization:
Let M be an orientable 3-manifold that admits a generalized triangulation with t generalized tetrahedra. Let S be a Heegaard surface for M. If g(S) >= 76t+26, then d(S) <= 2.
LINKS
Tsuyoshi Kobayashi, Yo'av Rieck, A linear bound on the genera of Heegaard splittings with distances greater than two, arXiov:0803.2751 March 20, 2008.
Index entries for linear recurrences with constant coefficients, signature (2,-1).
FORMULA
a(n) = 76*n + 26.
a(n) = 2*a(n-1)-a(n-2). G.f.: 2*(25*x+13)/(x-1)^2. [Colin Barker, Nov 09 2012]
MATHEMATICA
LinearRecurrence[{2, -1}, {26, 102}, 50] (* Harvey P. Dale, Oct 14 2013 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Mar 20 2008
STATUS
approved