Quotients of the Gordian and H(2)-Gordian graphs

The Gordian graph and H(2)-Gordian graphs of knots are abstract graphs whose vertex sets represent isotopy classes of unoriented knots, and whose edge sets record whether pairs of knots are related by crossing changes or H(2)-moves, respectively. We investigate quotients of these graphs under equivalence relations defined by several knot invariants including the determinant, the span of the Jones polynomial, and an invariant related to tricolorability. We show, in all cases considered, that the quotient graphs are Gromov hyperbolic. We then prove a collection of results about the graph isomorphism type of the quotient graphs. In particular, we find that the H(2)-Gordian graph of links modulo the relation induced by the span of the Jones polynomial is isomorphic with the complete graph on infinitely many vertices.

Hydrogen-bonding in mono-, di- and tetramethylammonium dihydrogenphosphites

The crystal structures of methylammonium and dimethylammonium dihydrogenphosphite (MA⋅H2PO3, I2/a and DMA⋅H2PO3, P 2 1 / c $P{2}_{1}/c$ ) are built of infinite chains of hydrogen bonded H 2 P O 3 − ${\mathrm{H}}_{\mathrm{2}}\mathrm{P}{\mathrm{O}}_{\mathrm{3}}^{-}$ anions. The chains are connected by the ammonium cations via hydrogen bonding to di- (DMA⋅H2PO3) and triperiodic (MA⋅H2PO3) networks. Tetramethylammonium dihydrogenphosphite monohydrate (TMA⋅H2PO3⋅H2O) features temperature dependent dimorphism. The crystal structure of the high-temperature (HT, cubic P213) and low-temperature (LT, orthorhombic P212121) phases were determined at 150 and 100 K, respectively. The hydrogen bonding network in the HT phase is disordered, with H 2 P O 3 − ${\mathrm{H}}_{\mathrm{2}}\mathrm{P}{\mathrm{O}}_{\mathrm{3}}^{-}$ and H2O being located on a threefold axis and is ordered in the LT phase. On cooling, the point symmetry is reduced by an index of 3. The lost symmetry is retained as twin operations, leading to threefold twinning by pseudo-merohedry. The hydrogen-bonding networks of the HT and LT phases can be represented by undirected and directed quotient graphs, respectively.

Symmetric graphs of valency seven and their basic normal quotient graphs

We characterize seven valent symmetric graphs of order 2 p q n 2p{q}^{n} with p < q p\lt q odd primes, extending a few previous results. Moreover, a consequence partially generalizes the result of Conder, Li and Potočnik [On the orders of arc-transitive graphs, J. Algebra 421 (2015), 167–186].

Isotopy classes for 3-periodic net embeddings

Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. Periodic isotopy classifications are obtained for various families of embedded nets with small quotient graphs. The 25 periodic isotopy classes of depth-1 embedded nets with a single-vertex quotient graph are enumerated. Additionally, a classification is given of embeddings of n-fold copies of pcu with all connected components in a parallel orientation and n vertices in a repeat unit, as well as demonstrations of their maximal symmetry periodic isotopes. The methodology of linear graph knots on the flat 3-torus [0,1)3 is introduced. These graph knots, with linear edges, are spatial embeddings of the labelled quotient graphs of an embedded net which are associated with its periodicity bases.

Surface embeddings of the Klein and the Möbius–Kantor graphs

This paper describes an invariant representation for finite graphs embedded on orientable tori of arbitrary genus, with working examples of embeddings of the Möbius–Kantor graph on the torus, the genus-2 bitorus and the genus-3 tritorus, as well as the two-dimensional, 7-valent Klein graph on the tritorus (and its dual: the 3-valent Klein graph). The genus-2 and -3 embeddings describe quotient graphs of 2- and 3-periodic reticulations of hyperbolic surfaces. This invariant is used to identify infinite nets related to the Möbius–Kantor and 7-valent Klein graphs.