Counting closed 2-manifolds in tubes in hypercubic lattices

Abstract

We consider closed 2-manifolds (2-manifolds without boundary) embedded in tubes in the hypercubic lattice, Z d , ⩾ d 3. For orientable 2-manifolds with fixed genus, ≠ d 4, we prove that the exponential growth rate is independent of the genus and we use this to prove a pattern theorem for manifolds with fixed genus. We prove a similar theorem for the non-orientable case for > d 4. If the genus is not fixed then we prove a pattern theorem and use this to show that 2-manifolds with genus less than any fixed number are exponentially rare so the typical genus increases with the size of the manifold. In four and higher dimensions we prove that orientable manifolds are exponentially rare and are dominated by non-orientable manifolds. In four dimensions, all except exponentially few 2-manifolds, both orientable and non-orientable, contain a local knotted (4, 2)-ball pair.

A random walk with Stu Whittington

“…In addition, similar arguments to those used for 1D knots show that the every good measure of entanglement complexity for p-spheres in a tube in Z p+2 grows at least linearly with the p-volume of the spheres. Similar results can be obtained [17] for closed 2-manifolds in tubes in p ⩾ 3.…”

“…One of the great pleasures of a mathematical life is working with friends on interesting problems. In the list of references at the end of this article, my joint work with Stu and collaborators is listed [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. The following is a list of friends and co-authors I enjoyed while working with Stu Whittington:…”

Models of random knots

“…In another direction, random knot models may extend to knotted 2-spheres or other surfaces in a 4-sphere, and further to randomly embedded manifolds in higher dimensions (Soteros et al 2012;Atapour et al 2015).…”