The twinplex graph is the graph labeled 
in Fischer and Little (2001). It is a cubic Hamiltonian graph on 12 vertices
with graph crossing number 2 and has maximal
girth among all 12-vertex cubic graphs (Clancy et al. 2020). (There are
two such graphs with maximal girth
5, the other being the twinplex graph of Robertson et al. (2019).)
The twinplex graph has LCF notation ![[-5,-4,4,-4,4,5,-4,5,-4,4,-5,4]](/images/equations/TwinplexGraph/Inline2.svg)
. It is illustrated above in
a number of drawings.
The name appears to refer to its pairing with the cubeplex graph; neither Norine and Thomas (2008) nor Robertson et al. (2019) explain the term.
The twinplex graph is implemented in the Wolfram Language as GraphData["TwinplexGraph"].
Norine and Thomas (2008) use the term "twinplex" to refer to one of the 12-vertex cubic graphs 
and 
from Fischer and Little (2001). Unfortunately, in the
figure supposedly giving drawings for the graphs termed
cubeplex and twinplex in that paper (reproduced
above), both drawings correspond to Fischer and Little's 
, which is the cubeplex
graph.
