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Gram determinants earned traction among knot theorists after Witten's presumption about the existence of a 3-manifold invariant connected to the Jones polynomial. Triggered by the creation of such an invariant by Reshetikhin and Turaev, several mathematicians have explored this line of research ever since. Gram determinants came into play by Lickorish's skein theoretic approach to the invariant. The construction of different bilinear forms is possible through changes in the ambient surface of the Kauffman bracket skein module. Hence, different types of Gram determinants have arisen in knot theory throughout the years; some of these determinants are discussed here. In this paper, we introduce a new version of such a determinant from the Möbius band and prove some important results about its structure. In particular, we explore its connection to the annulus case and factors of its closed formula. [ABSTRACT FROM AUTHOR]

The Gram determinant of type A was introduced by Lickorish in his work on invariants of 3 - manifolds. We generalize the theory of the Gram determinant of type A by evaluating, in the annulus, a bilinear form of non-intersecting connections in the disc. The main result provides a closed formula for this Gram determinant. We conclude the paper by discussing Chen's conjecture about the Gram determinant of type Mb evaluated in the Klein bottle. [ABSTRACT FROM AUTHOR]