A theorem on graph embedding with a relation to hyperbolic volume

Abstract

We prove that a planar cubic cyclically 4-connected graph of odd χ < 0 is the dual of a 1-vertex triangulation of a closed orientable surface. We explain how this result is related to (and applied to prove at a separate place) a theorem about hyperbolic volume of links: the maximal volume of alternating links of given χ < 0 does not depend on the number of their components.

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Correspondence to Alexander Stoimenow.

Additional information

The author was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2011-0027989).

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Stoimenow, A. A theorem on graph embedding with a relation to hyperbolic volume. Combinatorica 36, 557–589 (2016). https://doi.org/10.1007/s00493-014-2840-x

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Mathematics Subject Classiffication (2000)

  • 05C10
  • 57Q15
  • 57M25
  • 57M50