The Non-Commutative A ₁ T-System and its Positive Laurent Property

We define a non-commutative version of the A₁ 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 A₁. We solve the system by generalizing the flat GL₂ connection method used in the commuting case to a 2 × 2 flat matrix connection with non-commutative entries. This allows us 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.