Counting the spanning trees of the 3-cube using edge slides

We give a direct combinatorial proof of the known fact that the 3-cube has 384 spanning trees, using an "edge slide" operation on spanning trees. This gives an answer in the case n=3 to a question implicitly raised by Stanley. Our argument also gives a bijective proof of the n=3 case of a weighted count of the spanning trees of the n-cube due to Martin and Reiner.Comment: 17 pages, 9 figures. v2: Final version as published in the Australasian Journal of Combinatorics. Section 5 shortened and restructured; references added; one figure added; some typos corrected; additional minor changes in response to the referees' comment

Finite subset spaces of closed surfaces

The kth finite subset space of a topological space X is the space exp_k X of non-empty finite subsets of X of size at most k, topologised as a quotient of X^k. The construction is a homotopy functor and may be regarded as a union of configuration spaces of distinct unordered points in X. We show that the finite subset spaces of a connected 2-complex admit "lexicographic cell structures" based on the lexicographic order on I^2 and use these to study the finite subset spaces of closed surfaces. We completely calculate the rational homology of the finite subset spaces of the two-sphere, and determine the top integral homology groups of exp_k Sigma for each k and closed surface Sigma. In addition, we use Mayer-Vietoris arguments and the ring structure of H^*(Sym^k Sigma) to calculate the integer cohomology groups of the third finite subset space of Sigma closed and orientable.Comment: 40 pages, 5 .eps figure

Generalised knot groups distinguish the square and granny knots (with an appendix by David Savitt)

Given a knot K we may construct a group G_n(K) from the fundamental group of K by adjoining an nth root of the meridian that commutes with the corresponding longitude. These "generalised knot groups" were introduced independently by Wada and Kelly, and contain the fundamental group as a subgroup. The square knot SK and the granny knot GK are a well known example of a pair of distinct knots with isomorphic fundamental groups. We show that G_n(SK) and G_n(GK) are non-isomorphic for all n>1. This confirms a conjecture of Lin and Nelson, and shows that the isomorphism type of G_n(K), n>1, carries more information about K than the isomorphism type of the fundamental group. An appendix by David Savitt contains some results on representations of the trefoil group in PSL(2,p) that are needed for the proof.Comment: 25 pages, 5 figures, to appear in JKTR. v3: example of the target groups added; slight correction to the construction of the target groups; references updated; some changes to notation. v2: section 4.2 expanded to give overview of proo

Level structures on the Weierstrass family of cubics

Let W -> A^2 be the universal Weierstrass family of cubic curves over C. For each N >= 2, we construct surfaces parametrizing the three standard kinds of level N structures on the smooth fibers of W. We then complete these surfaces to finite covers of A^2. Since W -> A^2 is the versal deformation space of a cusp singularity, these surfaces convey information about the level structure on any family of curves of genus g degenerating to a cuspidal curve. Our goal in this note is to determine for which values of N these surfaces are smooth over (0,0). From a topological perspective, the results determine the homeomorphism type of certain branched covers of S^3 with monodromy in SL_2(Z/N).Comment: LaTeX, 12 pages; added section giving a topological interpretation of the result

Some Ramsey-type results on intrinsic linking of n-complexes

Define the complete n-complex on N vertices to be the n-skeleton of an (N-1)-simplex. We show that embeddings of sufficiently large complete n-complexes in R^{2n+1} necessarily exhibit complicated linking behaviour, thereby extending known results on embeddings of large complete graphs in R^3 (the case n=1) to higher dimensions. In particular, we prove the existence of links of the following types: r-component links, with the linking pattern of a chain, necklace or keyring; 2-component links with linking number at least lambda in absolute value; and 2-component links with linking number a non-zero multiple of a given integer q. For fixed n the number of vertices required for each of our results grows at most polynomially with respect to the parameter r, lambda or q.Comment: 26 pages, 4 figures. v3: references added, some typos corrected, order of Thms 1.4 and 1.5 reversed, other minor changes in response to referee's comments. v2: added reference to arXiv:0705.2026 and updated abstract and introduction in view of that paper; improved bound in Thm 1.4 from O(p^4) to O(p^2); some additional discussion of results; typos correcte

Monadic transductions and definable classes of matroids

A transduction provides us with a way of using the monadic second-order language of a structure to make statements about a derived structure. Any transduction induces a relation on the set of these structures. This article presents a self-contained presentation of the theory of transductions for the monadic second-order language of matroids. This includes a proof of the matroid version of the Backwards Translation Theorem, which lifts any formula applied to the images of the transduction into a formula which we can apply to the pre-images. Applications include proofs that the class of lattice-path matroids and the class of spike-minors can be defined by sentences in monadic second-order logic

Finite subset spaces of graphs and punctured surfaces

The kth finite subset space of a topological space X is the space exp_k X of non-empty finite subsets of X of size at most k, topologised as a quotient of X^k. The construction is a homotopy functor and may be regarded as a union of configuration spaces of distinct unordered points in X. We calculate the homology of the finite subset spaces of a connected graph Gamma, and study the maps (exp_k phi)_* induced by a map phi:Gamma -> Gamma' between two such graphs. By homotopy functoriality the results apply to punctured surfaces also. The braid group B_n may be regarded as the mapping class group of an n-punctured disc D_n, and as such it acts on H_*(exp_k D_n). We prove a structure theorem for this action, showing that the image of the pure braid group is nilpotent of class at most floor((n-1)/2).Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-29.abs.htm