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A tridiagonal pair is an ordered pair of diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on the eigenspaces of the other in a block-tridiagonal fashion. We consider a tridiagonal pair (A, A * ) of q-Serre type; for such a pair the maps A and A * satisfy the q-Serre relations. There is a linear map K in the literature that is used to describe how A and A * are related. We investigate a pair of linear maps B = A and B * = tA * + (1 − t)K, where t is any scalar. Our goal is to find a necessary and sufficient condition on t for the pair (B, B * ) to be a tridiagonal pair. We show that (B, B * ) is a tridiagonal pair if and only if t = 0 and P t(q − q −1 ) −2 = 0, where P is a certain polynomial attached to (A, A * ) called the Drinfel'd polynomial.

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Traditionally introduced in terms of advanced topological constructions, many link invariants may also be defined in much simpler terms given their values on a few initial links and a recursive formula on a skein triangle. Then the crucial question to ask is how many initial values are necessary to completely determine such a link invariant. We focus on a specific class of invariants known as nonzero determinant link invariants, defined only for links which do not evaluate to zero on the link determinant. We restate our objective by considering a set S of links subject to the condition that if any three nonzero determinant links belong to a skein triangle, any two of these belonging to S implies that the third also belongs to S. Then we aim to determine a minimal set of initial generators so that S is the set of all links with nonzero determinant. We show that only the unknot is required as a generator if the skein triangle is unoriented. For oriented skein triangles, we show that the unknot and Hopf link orientations form a set of generators.