The Non-Commutative $A_1$ $T$-system and its positive Laurent property

We define a non-commutative version of the $A_1$ T-system, which underlies frieze patterns of the integer plane. This system has discrete conserved quantities and has a particular reduction to the known non-commutative Q-system for $A_1$. We solve the system by generalizing the flat $GL_2$ connection method used in the commuting case to a 2$\times$2 flat matrix connection with non-commutative entries. This allows to prove the non-commutative positive Laurent phenomenon for the solutions when expressed in terms of admissible initial data. These are rephrased as partition functions of paths with non-commutative weights on networks, and alternatively of dimer configurations with non-commutative weights on ladder graphs made of chains of squares and hexagons.