  Knot theory is actively studied both by physicists and mathematicians as it
provides a connecting centerpiece for many physical and mathematical theories.
One of the challenging problems in knot theory is distinguishing mutant knots.
Mutant knots are not distinguished by colored HOMFLY-PT polynomials for knots
colored by either symmetric and or antisymmetric representations of $SU(N)$.
Some of the mutant knots can be distinguished by the simplest non-symmetric
representation $[2,1]$. However there is a class of mutant knots which require
more complex representations like $[4,2]$. In this paper we calculate
polynomials and differences for the mutant knot polynomials in representations
$[3,1]$ and $[4,2]$ and study their properties.
