Volume 18, issue 2 (2014)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 20
Issue 4, 1807–2438
Issue 3, 1257–1806
Issue 2, 629–1255
Issue 1, 1–627

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Subscriptions
Author Index
To Appear
Contacts
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Ehrhart theory of polytopes and Seiberg–Witten invariants of plumbed $3$–manifolds

Tamás László and András Némethi

Geometry & Topology 18 (2014) 717–778
Abstract

Let M be a rational homology sphere plumbed 3–manifold associated with a connected negative-definite plumbing graph. We show that its Seiberg–Witten invariants equal certain coefficients of an equivariant multivariable Ehrhart polynomial. For this, we construct the corresponding polytope from the plumbing graph together with an action of H1(M, ) and we develop Ehrhart theory for them. At an intermediate level we define the ‘periodic constant’ of multivariable series and establish their properties. In this way, one identifies the Seiberg–Witten invariant of a plumbed 3–manifold, the periodic constant of its ‘combinatorial zeta function’ and a coefficient of the associated Ehrhart polynomial. We make detailed presentations for graphs with at most two nodes. The two node case has surprising connections with the theory of affine monoids of rank two.

Keywords
$3$–manifolds, $\mathbb{Q}$–homology spheres, plumbed $3$–manifolds, Seiberg–Witten invariant, Ehrhart theory, equivariant Ehrhart polynomials, affine monoids, polytopes, periodic constant, surface singularities
Mathematical Subject Classification 2010
Primary: 14E15, 57M27
Secondary: 52B20, 06F05, 57R57
References
Publication
Received: 22 November 2012
Revised: 10 June 2013
Accepted: 19 July 2013
Published: 20 March 2014
Proposed: Ronald Stern
Seconded: Ronald Fintushel, Simon Donaldson
Authors
Tamás László
Central European University
Nador u. 9
1051 Budapest
Hungary
http://www.renyi.hu/~ltamas
András Némethi
MTA Rényi Institute of Mathematics
Reáltanoda u. 13-15
1053 Budapest
Hungary
http://www.renyi.hu/~nemethi