Intersections of loops and the Andersen-Mattes-Reshetikhin algebra (Q2840166)

Let \(\alpha_1,\alpha_2\in\hat{\pi}\) be two free homotopy classes of loops on an oriented 2-dimensional surface, \(F\). A Lie bracket was defined on \({\boldsymbol{Z}}[\hat{\pi}]\) by \textit{W. M. Goldman} [Invent. Math. 85, 263--302 (1986; Zbl 0619.58021)]. The minimal (unsigned) intersection number \(\#(\alpha_1,\alpha_2)\) amongst smooth representatives of \(\alpha_1\), \(\alpha_2\) is bounded below by the number of terms in a reduced expression of \([\alpha_1,\alpha_2]\in{\boldsymbol{Z}}[\hat{\pi}]\). \textit{M. Chas} [Geom. Dedicata 144, 25--60 (2010; Zbl 1186.57015)] showed that equality holds if one of the loops contains a power of a simple loop and further that equality holds for a particular \(\alpha_1\) and for \textit{all} \(\alpha_2\) \textit{only if} \(\alpha_1\) contains a power of a simple loop. Furthermore, \textit{M. Chas} [Topology 43, No. 3, 543--568 (2004; Zbl 1050.57014)] gave an explicit example on a 3-holed sphere where equality does not hold.NEWLINENEWLINEThe current paper shows that if the Goldman (Lie) bracket is replaced in a suitable way by the Andersen-Mattes-Reshetikhin Poisson bracket, then equality is obtained.NEWLINENEWLINEBy a \textsl{geometric chord diagram on} \(F\) is meant a smooth map of a chord diagram (circles and arcs) into \(F\) in which the arcs are collapsed to points. The Andersen-Mattes-Reshetikhin chord algebra [\textit{J. E. Andersen} et al., Topology 35, No. 4, 1069--1083 (1996; Zbl 0857.58009)] is defined as the quotient of the vector space generated by generic geometric chord diagrams on \(F\) up to homotopy, by the ideal generated by 4T relations. The Poisson structure \(\{\cdot,\cdot\}\) on this algebra is defined on chord diagrams as a signed sum over intersection points of geometric chord diagram representatives, of the new chord diagram obtained from the union by adding a chord to correspond to that intersection point. A free homotopy class of loops on \(F\) can be viewed as a chord diagram on \(F\) with one circle and no chords.NEWLINENEWLINEThe paper presents two new results: (1) for distinct \(\alpha_1\), \(\alpha_2\), \(\#(\alpha_1,\alpha_2)\) is exactly the number of terms in a reduced expression of \(\{\alpha_1,\alpha_2\}\); (2) for a non-trivial free homotopy class \(\alpha\) and arbitrary non-zero distinct \(p,q\in\boldsymbol{Z}\), the number of terms in a reduced expression of \(\{\alpha^p,\alpha^q\}\) is \(2|pq|(\#(\alpha)-n+1)\) where \(\#(\alpha)\) denotes the minimal self-intersection number of \(\alpha\) and \(n\in{\boldsymbol{N}}\) is maximal for which there exists \(\beta\in\pi_1(F)\) with \(\alpha=\beta^n\).NEWLINENEWLINEResults similar to the second theorem, but connecting self-intersection numbers with Goldman brackets were obtained in [\textit{M. Chas} and \textit{F. Krongold}, J. Topol. Anal. 2, No. 3, 395--417 (2010; Zbl 1245.57021)] and \textit{M. Chas} and \textit{S. Gadgil} [The Goldman bracket determines intersection numbers for surfaces and orbifolds, \url{arXiv:1209.0634v2}].NEWLINENEWLINEThe proof of theorem 1 deals separately with cases of sphere, annulus, torus, other compact surface with or without boundary and non-compact \(F\), the main work being in the penultimate case. There a pants decomposition is used to place a Riemannian metric of constant negative curvature on the surface in such a way that the boundary is geodesic and uses the two fundamental facts that every free homotopy class has a unique representative as a geodesic loop and that all non-trivial subgroups of \(\pi_1(F)\) are infinite cyclic. The cases where \(\alpha_1\), \(\alpha_2\) do and do not contain powers of the same loop are also dealt with separately, the former being the most intricate.NEWLINENEWLINEThe paper ends with a detailed analysis of the example of Chas [Zbl 1050.57014] (Example 5.6) on a 3-holed sphere \(F\), with classes \(\alpha_1\), \(\alpha_2\) for which the Goldman bracket vanishes but the minimal intersection number is 2. They evaluate the Andersen-Mattes-Reshetikhin Poisson bracket \(\{\alpha_1,\alpha_2\}\) demonstrating how to prove that its reduced form has two terms and explain the general algorithm by which this Poisson bracket can be used to calculate minimal intersection numbers of free homotopy classes of loops.