We show a close connection between generalized generalized spin models (four-weight spin models) and exactly solvable models. The defining relations of four-weight spin models are discussed from the viewpoint of the star-triangle relations. For an illustration we derive symmetric spin models from the self-dual $Z_N$ model, and then construct various four-weight spin models through gauge transformations. We also show that the gauge transformations do not change the values of the link invariants derived from the four-weight spin models.

We study antipodal distance-regular covers of complete graphs. The first part of the paper gives an introduction to the basic theory and the main constructions. In the second part, we turn to linear covers, which can be described geometrically.

We consider the permutation action of $E \simeq {}^2E_6 (2)$ on the cosets of its maximal subgroup $F \simeq Fi_{22}$. We calculate the intersection matrices and character table of the centralizer algebra corresponding to this action. There are three reasons for the interest in this particular representation. Firstly, it is a sporadic multiplicity-free action of a simple group of exceptional Lie type. Secondly, $E$ and $F$ are $Y$-groups $Y_{333}$ and $Y_{332}$, respectively, factorized over their centers. We believe that the intersection matrices we have calculated might be useful for a computer-free identification of $Y_{333}$ with $2^3 \cdot {}^2E_6 (2)$. Thirdly, the permutation group considered is the one induced by the involution centralizer on the set of points fixed by an involution in the action of the Baby Monster $F_2$ on the cosets of the Fischer group $Fi_{23}$. The latter action has the largest rank (namely 23) among the primitive multiplicity-free actions of the sporadic simple groups and the calculation of its character table is an open problem.

There have been a number of striking new results concerning translation planes of characteristic 2, obtained using orthogonal and symplectic spreads. The impetus for this came from coding theory. This paper surveys the geometric advances, while providing a hint of their coding-theoretic connections.

A connected graph of diameter $d$ is said to be almost bipartite if it contains no cycle of length $2\ell + 1$ for all $\ell < d$. An almost bipartite distance-regular graph $\Gamma = (X, E)$ is 2-homogeneous if and only if there are constants $\gamma_1, \ldots, \gamma_d$ such that $|\Gamma_{i-1}(u) \cap \Gamma_1(x) \cap \Gamma_1(y)| = \gamma_i$ holds for all $u \in X$ and for all $x, y \in \Gamma_i(u)$ with $\partial(x, y) = 2$ $\ (i=1, \ldots, d)$.

In this paper, almost bipartite 2-homogeneous distance-regular graphs are classified. This determines triangle-free connected graphs affording spin models (for link invariants) with certain weights.

We define a quantum matroid to be any finite nonempty poset $P$ satisfying the conditions R, SL, M, AU below.

R: $P$ is ranked.

SL: $P$ is a (meet) semilattice.

M: For all $x \in P$, the interval $[0, x]$ is a modular atomic lattice.

AU: For all $x, y \in P$ satisfying $\operatorname{rank}(x) < \operatorname{rank}(y)$, there exists an atom $a \in P$ such that $a \le y$, $a \not\le x$, and such that $x \lor a$ exists in $P$.

Condition AU is the augmentation axiom.

We develop a theory of quantum matroids. Although we deal at length with the general case, our emphasis is on quantum matroids $P$ with the following extra structure: We say $P$ is nontrivial if $P$ has rank $D \ge 2$, and $P$ is not a modular atomic lattice. In what follows suppose $P$ is nontrivial. We say $P$ is $q$-line regular whenever each rank 2 element in $P$ covers exactly $q + 1$ elements of $P$. We say $P$ is $\beta$-dual-line regular whenever each element in $P$ with rank $D - 1$ is covered by exactly $\beta + 1$ elements of $P$. We say $P$ is $\alpha$-zig-zag regular whenever for all pairs $x, y \in P$ such that $\operatorname{rank}(x)=D-1$, $\operatorname{rank}(y)=D$, and such that $x$ covers $x \wedge y$, there exists exactly $\alpha + 1$ pairs $x', y' \in P$ such that $y'$ covers $x$, $y'$ covers $x'$, and such that $y$ covers $x'$. We say $P$ is regular whenever $P$ is line regular, dual-line regular, and zig-zag regular. We prove the following theorem.

Theorem. Let $D$ denote an integer at least 4. Then a poset $P$ is a nontrivial regular quantum matroid of rank $D$ if and only if $P$ is isomorphic to one of the following:

(i) A truncated Boolean algebra $B(D, N)$, $(D < N)$.

(ii) A Hamming matroid $H(D, N)$, $(2 \le N)$.

(iii) A truncated projective geometry $L_q(D, N)$, $(D < N)$.

(iv) An attenuated space $A_q(D, N)$, $(D < N)$.

(v) A classical polar space of rank $D$.