  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 four-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 four-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.
