Algebraic and geometric structures inside the Birkhoff polytope

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