Closed 4-braids and the Jones unknot conjecture

The Jones problem is a question whether there is a non-trivial knot with the trivial Jones polynomial in one variable q . The answer to this fundamental question is still unknown despite numerous attempts to explore it. In braid presentation, the case of 4-strand braids is already open. S. Bigelow showed in 2000 that if the Burau representation for 4-strand braids is unfaithful, then there is an infinite number of non-trivial knots with the trivial two-variable HOMFLY-PT polynomial and hence, with the trivial Jones polynomial, since it is obtained from the HOMFLY-PT polynomial by the specialisation of one of the variables A=q^{2} .In this paper, we study 4-strand braids and ask whether there are non-trivial knots with the trivial Jones polynomial but a non-trivial HOMFLY-PT polynomial. We have discovered that there is a whole 1-parameter family, parameterised by the writhe number, of 2-variable polynomials that can be HOMFLY-PT polynomials of some knots. We explore various properties of the obtained hypothetical HOMFLY-PT polynomials and suggest several checks to test these formulas. A generalisation is also proposed for the case of a large number of strands.