  The Birkhoff polytope $\mathcal{B}_d$ consisting of all bistochastic matrices
of order $d$ assists researchers from many areas, including combinatorics,
statistical physics and quantum information. Its subset $\mathcal{U}_d$ of
unistochastic matrices, determined by squared moduli of unitary matrices, is of
a particular importance for quantum theory as classical dynamical systems
described by unistochastic transition matrices can be quantised. In order to
investigate the problem of unistochasticity we introduce the set
$\mathcal{L}_d$ of bracelet matrices that forms a subset of $\mathcal{B}_d$,
but a superset of $\mathcal{U}_d$. We prove that for every dimension $d$ this
set contains the set of factorisable bistochastic matrices $\mathcal{F}_d$ and
is closed under matrix multiplication by elements of $\mathcal{F}_d$. Moreover,
we prove that both $\mathcal{L}_d$ and $\mathcal{F}_d$ are star-shaped with
respect to the flat matrix. We also analyse the set of $d\times d$
unistochastic matrices arising from circulant unitary matrices, and show that
their spectra lie inside $d$-hypocycloids on the complex plane. Finally,
applying our results to small dimensions, we fully characterise the set of
circulant unistochastic matrices of order $d\leq 4$, and prove that such
matrices form a monoid for $d=3$.
