Mathematical researches

A knot is a circle tied in the three dimensional space which can be deformed continuously. In order to distinguish knots, a bundle of invariants have been developed. Among them, the Jones polynomial (1985) is one of the most famous and studied. Surprisingly, its hability to detect the unknot is not decided yet. Recently, a paper from Eliahou and Fromentin (2017) shows a new point of view for this problem. By considering the Jones polynomial modulo an integer m, the existence of non-trivial knots seen as the unknot by this polynomial invariant is more approachable. Their work also presents an answer for the cases m=2^r. In this memoir, we show that it is possible to construct knots to answer cases m^r from one having a trivial Jones polynomial modulo m. That allows to give an other answer for cases m=2^r, but also for cases m=3^r. We try to give a generalized construction of the Eliahou and Fromentin one too, but the lack of examples make this only theoretical so far.

The Kauffman bracket (1987) is a tool to construct the Jones polynomial from a knot diagram. Its connections with generators of the Temperley-Lieb algebra, which also span the diagram monoid, and the braid theory have already been studied, notably by Kauffman himself (1990). We continue to investigate these different connections so that we can learn more on the Jones polynomial, and on these connected theories too. We bring as example a new matrix representation of braids, which provide a new method to compute the Jones polynomial associated to these braids closed in various way. We also study the diagram monoid thank to an embedding inside permutations, and develop as such a new product for permutations similar to the classical composition.