It is known that the Brandt--Lickorish--Millett--Ho polynomial $Q$ contains Casson's knot invariant. Whether there are (essentially) other Vassiliev knot invariants obtainable from $Q$ is an open problem. We show that this is not so up to degree $9$. We also give the (apparently) first examples of knots not distinguished by 2-cable HOMFLY polynomials which are not mutants. Our calculations provide evidence of a negative answer to the question whether Vassiliev knot invariants of degree $d \le 10$ are determined by the HOMFLY and Kauffman polynomials and their 2-cables, and for the existence of algebras of such Vassiliev invariants not isomorphic to the algebras of their weight systems.

MSC Classifications:

57M25, 57M27, 20F36, 57M50 show english descriptions Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Invariants of knots and 3-manifolds
Braid groups; Artin groups
Geometric structures on low-dimensional manifolds 57M25 - Knots and links in $S^3$ {For higher dimensions, see 57Q45}
57M27 - Invariants of knots and 3-manifolds
20F36 - Braid groups; Artin groups
57M50 - Geometric structures on low-dimensional manifolds

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