Distinguishing Mutant knots

Abstract 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 S U ( 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.