Curve diagrams, laminations, and the geometric complexity of braids

Braids can be represented geometrically as curve diagrams. The geometric complexity of a braid is the minimal complexity of a curve diagram representing it. We introduce and study the corresponding notion of geometric generating function. We compute explicitly the geometric generating function for the group of braids on three strands and prove that it is neither rational nor algebraic, nor even holonomic. This result may appear as counterintuitive. Indeed, the standard complexity (due to the Artin presentation of braid groups) is algorithmically harder to compute than the geometric complexity, yet the associated generating function for the group of braids on three strands is rational.

ON THE INVERSE MAPPING CLASS MONOIDS

Braid groups and mapping class groups have many features in common. Similarly to the notion of inverse braid monoid, inverse mapping class monoid is defined. It concerns surfaces with punctures, but among given n punctures, several can be omitted. This corresponds to braids where the number of strings is not fixed. In the paper we give the analogue of the Dehn–Nilsen–Baer theorem, propose a presentation of the inverse mapping class monoid for a punctured sphere and study the word problem. This shows that certain properties and objects based on mapping class groups may be extended to the inverse mapping class monoids. We also give analogues of Artin presentation with two generators.

TORELLI ACTIONS AND SMOOTH STRUCTURES ON FOUR MANIFOLDS

Artin presentations are discrete equivalents of planar open book decompositions of closed, orientable three manifolds. Artin presentations characterize the fundamental groups of closed, orientable three manifolds. An Artin presentation also determines a smooth, compact, simply conected four manifold that bounds the three dimensional open book. In this way, the study of three and four manifolds may be approached purely group theoretically. In the theory of Artin presentations, elements of the Torelli subgroup act on the topology and smooth structures of the three and four manifolds. We show that the Torelli action can preserve the continuous topological type of a four manifold while changing its smooth structure. This is a new, group theoretic method of altering the smooth structure on a four manifold.

ARTIN PRESENTATIONS I: GAUGE THEORY, 3 + 1 TQFT'S AND THE BRAID GROUPS

We inititate the systematic study of Artin Presentations, (discovered in 1975 by González-Acuña), which characterize the fundamental groups of closed, orientable 3-manifolds, and form a discrete equivalent of the theory of open book decompositions with planar pages of such manifolds. We list and prove the basic properties, state some fundamental problems and describe some of the advantages of the theory: e.g., an Artin Presentation of π1 (M3) does not just determine the closed, orientable 3-manifold M3, but also a canonical, smooth simply-connected cobordism of it, allowing us to tap into 4-dimensional gauge theory (and 3 + 1 TQFT's) in a more direct, purely discrete, functorial manner than others. Thus, in section 4, instead of using PDE's, we show how a canonical action of the commutator subgroup [Pn, Pn] of the pure braid group Pn can be used to study the smooth structures on a closed, smooth-connected 4-manifold with b2 = n, in a systematic way. However, the main purpose of this first paper is to Artin Presentations to set up simple criteria, testable with, say, MAGMA on the computer (where then no knowledge of topology is required) for finding explicit counter-examples to the so-called Weak Poincaré Conjecture: "Every homotopy 3-sphere bounds a smooth, compact, contractible 4-manifold," as well as: "Every irreducible Z-homology 3-sphere Σ, with π1 (Σ) = I (120) is homeomorphic to Σ (2, 3, 5)" and other conjectures implied by Thurston's Geometrization Conjecture. One first philosophical goal is to convince the reader that the truth of these conjectures is at least as unlikely as that of the Andrews-Curtis Conjecture and that ultimately, Artin Presentation Theory is a non-trivial intersection of string/M theory number theory.