W. D. Neumann and A. W. Reid conjectured that the figure-eight knot and the two dodecahedral knots are the only hyperbolic knots admitting hidden symmetries. We construct an $n$-component hyperbolic link whose complement admits hidden symmetries for any $n\ge 4$.

In this note, we investigate the tree-depth and tree-width in a heterogeneous random graph obtained by including each edge $e_{ij}$ $(i\neq j)$ of a complete graph $K_{n}$ over $n$ vertices independently with probability $p_{n}(e_{ij})$. When the sequence of edge probabilities satisfies some density assumptions, we show both tree-depth and tree-width are of linear size with high probability. Moreover, we extend the method to random weighted graphs with non-identical edge weights and capture the conditions under which with high probability the weighted tree-depth is bounded by a constant.

We give some new formulae relating an obstruction to the weak approximation on homogeneous spaces to the set of local and global Brauer and R-equivalence classes.