Modularity of Nahm Sums Dual to Zagier's Rank-Three Examples

In 2007, Zagier identified twelve sets of rank-three modular Nahm sums and proved the modularity of three of them. The modularity of the remaining examples was confirmed by Wang. In this paper, we investigate the Nahm sums dual to Zagier's rank-three examples, numbered according to their order in Zagier's list. Combining our results with earlier work, the modularity of all these duals is established except those corresponding to the ninth and twelfth examples. For the ninth example, we prove three of the four sets of identities required to establish the proposed product representations and leave the remaining set as a conjecture, thereby obtaining a conditional modularity result. For the twelfth example, we formulate conjectural product identities that would imply the modularity of its dual. Our proofs rely on Rogers--Ramanujan type identities that express the relevant Nahm sums as finite combinations of infinite products. Along the way, we prove four rank-four tadpole Nahm sum identities previously conjectured by Cao and Wang and by the present authors. We also discover and prove the modularity of several new rank-three Nahm sums.