The chromatic polynomial of fatgraphs and its categorification

Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered as the Euler characteristic. For plane graphs, we show that our chromatic homology can be recovered from the Khovanov homology of an associated link. We apply this connection with Khovanov homology to show that the torsion-free part of our chromatic homology is independent of the choice of planar embedding of a graph. We extend our construction and categorify the Bollobás-Riordan polynomial (a generalization of the Tutte polynomial to embedded graphs). We prove that both our chromatic homology and the Khovanov homology of an associated link can be recovered from this categorification.     

Determination of histamine in tomatoes by capillary electrophoresis

An improved capillary electrophoresis assay for histamine in crude extracts was developed and used to determine histamine levels in a selection of tomato fruits and pastes. Performance in terms of reproducibility and sensitivity was optimized by use of a high sensitivity detector flow cell, sodium hydroxide rinses, and a voltage gradient. The method was linear down to 0.2 microgram/mL (signal-to-noise ratio = 4:1), which was below the endogenous level in all samples.     

High frequency optogalvanic signals and CO2 laser stabilisation

The normal optogalvanic effect used in the stabilisation of CO₂ lasers decreases in amplitude with increase in frequency and disappears at 2–3 kHz. We show that there is also a high frequency optogalvanic effect in CO₂-N₂-He-Xe laser gas mixtures up to ～ 100 kHz, explain its origin and use it to frequency stabilise a laser with long term (hours) frequency drift of < 50 kHz and frequency re-settability of < 100 kHz.     

Bipartite partial duals and circuits in medial graphs

It is well known that a plane graph is Eulerian if and only if its geometric dual is bipartite. We extend this result to partial duals of plane graphs. We then characterize all bipartite partial duals of a plane graph in terms of oriented circuits in its medial graph.     

A reinvestigation of asymmetric induction in diels-alder reactions to chiral acrylates.

The chiral induction in the Diels-Alder addition $\text{1}$ → $\text{2}$, assessed reliably by ¹⁹F-NMR-spectroscopy of the endo-esters $\text{4}$, varied between 47 - 93% in favor of the 2-(R)-adducts $\text{2}$ depending on the auxiliary chiral group and the Lewis-acid catalyst.     

Excluded Minors and the Ribbon Graphs of Knots

In this article we consider minors of ribbon graphs (or, equivalently, cellularly embedded graphs). The theory of minors of ribbon graphs differs from that of graphs in that contracting loops is necessary and doing this can create additional vertices and components. Thus, the ribbon graph minor relation is incompatible with the graph minor relation. We discuss excluded minor characterizations of minor closed families of ribbon graphs. Our main result is an excluded minor characterization of the family of ribbon graphs that represent knot and link diagrams.     

On the Potts Model Partition Function in an External Field

We study the partition function of the Potts model in an external (magnetic) field, and its connections with the zero-field Potts model partition function. Using a deletion-contraction formulation for the partition function Z for this model, we show that it can be expanded in terms of the zero-field partition function. We also show that Z can be written as a sum over the spanning trees, and the spanning forests, of a graph G. Our results extend to Z the well-known spanning tree expansion for the zero-field partition function that arises though its connections with the Tutte polynomial.