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.