We define polynomial tangle invariants $\nabla_T^s$ via Kauffman states and Alexander codes and investigate some of their properties. In particular, we prove symmetry relations for $\nabla_T^s$ of 4-ended tangles and deduce that the multivariable Alexander polynomial is invariant under Conway mutation. The invariants $\nabla_T^s$ can be interpreted naturally via Heegaard diagrams for tangles. This leads to a categorified version of $\nabla_T^s$: a Heegaard Floer homology $\widehat{ \operatorname{ HFT}}$ for tangles, which we define as a bordered sutured invariant. We discuss a bigrading on $\widehat{ \operatorname{ HFT}}$ and prove symmetry relations for $\widehat{ \operatorname{ HFT}}$ of 4-ended tangles that echo those for $\nabla_T^s$.
Categories: math.GT