TY - JOUR

T1 - On chirality of toroidal embeddings of polyhedral graphs

AU - Barthel, Senja

PY - 2017/7/1

Y1 - 2017/7/1

N2 - We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [2, 3]. Building on this and using the chirality of torus knots and links [9, 10], we prove that the nontrivial embeddings of simple 3-connected planar graphs in the standard torus are chiral. For the case that the spatial graph contains a nontrivial knot, the statement was shown by Castle et al. [5]. We give an alternative proof using minors instead of the Euler characteristic. To prove the case in which the graph embedding contains a nonsplit link, we show the chirality of Hopf ladders with at least three rungs, thus generalizing a theorem of Simon [12].

AB - We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [2, 3]. Building on this and using the chirality of torus knots and links [9, 10], we prove that the nontrivial embeddings of simple 3-connected planar graphs in the standard torus are chiral. For the case that the spatial graph contains a nontrivial knot, the statement was shown by Castle et al. [5]. We give an alternative proof using minors instead of the Euler characteristic. To prove the case in which the graph embedding contains a nonsplit link, we show the chirality of Hopf ladders with at least three rungs, thus generalizing a theorem of Simon [12].

KW - chirality

KW - knots and links

KW - templating on a toroidal substrate

KW - Topological graphs

KW - topology and chemistry

UR - http://www.scopus.com/inward/record.url?scp=85021720064&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85021720064&partnerID=8YFLogxK

U2 - 10.1142/S021821651750050X

DO - 10.1142/S021821651750050X

M3 - Article

AN - SCOPUS:85021720064

VL - 26

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

SN - 0218-2165

IS - 8

M1 - 1750050

ER -