Distinguishing Mutant knots

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