The Gram determinant of the type B Temperley-Lieb algebra

In this paper, we solve a problem posed by Rodica Simion regarding type B Gram determinants. We present this in a fashion influenced by the work of W.B.R.Lickorish on Witten-Reshetikhin-Turaev invariants of 3-manifolds. The roots of the determinant were predicted by Dabkowski and Przytycki, and the complete factorization was conjectured by Gefry Barad. We will give a detailed history of this problem in a sequel paper in which we also plan to address other related questions by Simion, and connect the problem to Frenkel-Khovanov's work.