Notices of the International Consortium of Chinese Mathematicians

Volume 4 (2016)

Number 2

Strong embeddings and $2$-isomorphism

Pages: 5 – 13

DOI: https://dx.doi.org/10.4310/ICCM.2016.v4.n2.a3

Authors

Rani Hod (Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts, U.S.A.)

An Huang (Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts, U.S.A.)

Mark Kempton (Department of Mathematics, Brandeis University, Waltham, Massachusetts, U.S.A.)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Abstract

We present two conjectures related to strong embeddings of a graph into a surface. The first conjecture relates equivalence of integer quadratic forms given by the Laplacians of graphs, $2$-isomorphism of $2$-connected graphs, and strong embeddings of graphs. We prove various special cases of this conjecture, and give evidence for it. The second conjecture, motivated by ideas from physics and number theory, gives a lower bound on the number of strong embeddings of a graph. If true, this conjecture would imply the well-known Strong Embedding Conjecture.

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Published 7 April 2017