Kyoto Journal of Mathematics

Lattice multipolygons

Akihiro Higashitani and Mikiya Masuda

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We discuss generalizations of some results on lattice polygons to certain piecewise linear loops which may have a self-intersection but have vertices in the lattice Z2. We first prove a formula on the rotation number of a unimodular sequence in Z2. This formula implies the generalized twelve-point theorem of Poonen and Rodriguez-Villegas. We then introduce the notion of lattice multipolygons, which is a generalization of lattice polygons, state the generalized Pick’s formula, and discuss the classification of Ehrhart polynomials of lattice multipolygons and also of several natural subfamilies of lattice multipolygons.

Article information

Kyoto J. Math., Volume 57, Number 4 (2017), 807-828.

Received: 28 April 2014
Revised: 1 July 2016
Accepted: 6 July 2016
First available in Project Euclid: 22 June 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05A99: None of the above, but in this section
Secondary: 51E12: Generalized quadrangles, generalized polygons 57R91: Equivariant algebraic topology of manifolds

lattice polygon twelve-point theorem Pick’s formula Ehrhart polynomial toric topology


Higashitani, Akihiro; Masuda, Mikiya. Lattice multipolygons. Kyoto J. Math. 57 (2017), no. 4, 807--828. doi:10.1215/21562261-2017-0016.

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  • [1] M. Beck and S. Robins,Computing the Continuous Discretely, Undergrad. Texts in Math., Springer, New York, 2007.
  • [2] W. Castryck,Moving out the edges of a lattice polygon, Discrete Comput. Geom.47(2012), 496–518.
  • [3] R. Diaz and S. Robins,Pick’s formula via the Weierstrass $\wp$-function, Amer. Math. Monthly102(1995), 431–437.
  • [4] W. Fulton,An Introduction to Toric Varieties, Ann. of Math. Stud.113, Princeton Univ. Press, Princeton, 1993.
  • [5] B. Grünbaum and G. C. Shephard,Pick’s theorem, Amer. Math. Monthly100(1993), 150–161.
  • [6] A. Hattori and M. Masuda,Theory of multifans, Osaka J. Math.40(2003), 1–68.
  • [7] A. M. Kasprzyk and B. Nill,Reflexive polytopes of higher index and the number $12$, Electron. J. Combin.19(2012), no. 9.
  • [8] M. Masuda,Unitary toric manifolds, multi-fans and equivariant index, Tohoku Math. J. (2)51(1999), 237–265.
  • [9] T. Oda,Convex Bodies and Algebraic Geometry, Ergeb. Math. Grenzgeb. (3), Springer, Berlin, 1988.
  • [10] G. Pick,Geometrisches zur Zahlentheorie, Sitzenber. Lotos (Prague)19(1899), 311–319.
  • [11] B. Poonen and F. Rodriguez-Villegas,Lattice polygons and the number $12$, Amer. Math. Monthly107(2000), 238–250.
  • [12] D. Repovsh, M. Skopenkov, and M. Tsentsel$^{\prime}$,An elementary proof of the twelve lattice point theorem(Russian), Mat. Zametki77(2005), 117–120; English translation in Math. Notes77(2005), 108–111.
  • [13] P. R. Scott,On convex lattice polygons, Bull. Austral. Math. Soc.15(1976), 395–399.
  • [14] R. T. Živaljević,Rotation number of a unimodular cycle: An elementary approach, Discrete Math.313(2013), 2253–2261.