Abstract
We present a universal knot polynomials for 2- and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY and Kauffman polynomials at SL and SO/Sp lines on Vogel’s plane, respectively and give their exceptional group’s counterparts on exceptional line. We demonstrate that [m,n]=[n,m] topological invariance, when applicable, take place on the entire Vogel’s plane. We also suggest the universal form of invariant of figure eight knot in adjoint representation, and suggest existence of such universalization for any knot in adjoint and its descendant representations. Properties of universal polynomials and applications of these results are discussed.
Highlights
1.1 Universality in simple Lie algebras and gauge theoriesRepresentation theory is in the basis of our understanding of symmetries of physical theories and plays an increasingly important role with revealing of new hidden symmetries, sometime quite involved, which govern the structure of states and dynamics of string theory and its field theory reductions.Representation theory looks different for different simple Lie algebras, it seems that this is only because representations are classified by weights, not by roots
We present a universal knot polynomials for 2- and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula
We find universal knot polynomials for 2− and 3-strand torus knots, when Rosso-Jones formula [42,43,44,45,46] is available for any representation of any simple Lie algebra, in this case the Rosso-Jones formula itself can be made universal
Summary
Representation theory is in the basis of our understanding of symmetries of physical theories and plays an increasingly important role with revealing of new hidden symmetries, sometime quite involved, which govern the structure of states and dynamics of string theory and its field theory reductions. The sub-sector of representation theory associated with roots shows significant signs of universality: many group-invariant quantities can be represented as values of analytical functions, defined over entire Vogel’s plane (see the definition below), at a special points from Vogel’s table (2.35), (2.36). That idea was based on his study of (his introduced) algebra Λ of three-leg Jacobi diagrams [7], acting on different spaces of diagram and aimed to construction of finite Vassiliev’s invariants of knots These works present first impressive examples of universal quantities, such as e.g. dimensions of adjoint and its descendant representations, and provide a motivation for subsequent developments. Important is problem of existence of a universal form of corresponding duality, which is a basis of our understanding of S-duality in string theories
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