Explicit R and Racah matrices are given for the symmetric representation of SO(5) to compute Kauffman polynomials via a generalized Reshetikhin-Turaev construction.
Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid
9 Pith papers cite this work. Polarity classification is still indexing.
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abstract
Character expansion is introduced and explicitly constructed for the (non-colored) HOMFLY polynomials of the simplest knots. Expansion coefficients are not the knot invariants and can depend on the choice of the braid realization. However, the method provides the simplest systematic way to construct HOMFLY polynomials directly in terms of the variable A=q^N: a much better way than the standard approach making use of the skein relations. Moreover, representation theory of the simplest quantum group SU_q(2) is sufficient to get the answers for all braids with m<5 strands. Most important we reveal a hidden hierarchical structure of expansion coefficients, what allows one to express all of them through extremely simple elementary constituents. Generalizations to arbitrary knots and arbitrary representations is straightforward.
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Khovanov-Rozansky invariants are recast as a bicomplex of local operators D and conjugations χ^(±), with nilpotency on closed diagrams allowing reductions that simplify the hypercube construction.
Authors propose shaded A-polynomials A_a(ℓ_b, m_c) for SU(N) via CG chords from huge representations of U_q(su_N) in the classical limit, with examples for knots 3_1, 4_1, 5_1 in su_3.
A modified Goeritz matrix is defined for bipartite link diagrams that reduces HOMFLY-PT computation for any N to matrix algebra.
Alexander polynomials appear in two opposite roles in two Kashaev phases of Chern-Simons theory due to co-existing branches in the quasiclassical limit with non-trivial versus vanishing classical actions.
In the Kashaev limit the non-homogeneous quantum A-polynomial splits into phases tied to zero action and deformed hyperbolic volume, with a byproduct expectation that the classical A-polynomial at L=1 is proportional to the Alexander polynomial.
The paper outlines the generalization of cabling-based entangling gates to SU(N) anyons and identifies differences and new problems that arise.
citing papers explorer
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Racah matrices for the symmetric representation of the SO(5) group
Explicit R and Racah matrices are given for the symmetric representation of SO(5) to compute Kauffman polynomials via a generalized Reshetikhin-Turaev construction.
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Reductions in Khovanov-Rozansky operator formalism
Khovanov-Rozansky invariants are recast as a bicomplex of local operators D and conjugations χ^(±), with nilpotency on closed diagrams allowing reductions that simplify the hypercube construction.
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Shading A-polynomials via huge representations of $U_q(\mathfrak{su}_N)$
Authors propose shaded A-polynomials A_a(ℓ_b, m_c) for SU(N) via CG chords from huge representations of U_q(su_N) in the classical limit, with examples for knots 3_1, 4_1, 5_1 in su_3.
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Analogue of Goeritz matrices for computation of bipartite HOMFLY-PT polynomials
A modified Goeritz matrix is defined for bipartite link diagrams that reduces HOMFLY-PT computation for any N to matrix algebra.
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Two roles of Alexander in two Kashaev phases
Alexander polynomials appear in two opposite roles in two Kashaev phases of Chern-Simons theory due to co-existing branches in the quasiclassical limit with non-trivial versus vanishing classical actions.
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More on Kashaev limits of the quantum $A$-polynomials
In the Kashaev limit the non-homogeneous quantum A-polynomial splits into phases tied to zero action and deformed hyperbolic volume, with a byproduct expectation that the classical A-polynomial at L=1 is proportional to the Alexander polynomial.
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Entangling gates for the SU(N) anyons
The paper outlines the generalization of cabling-based entangling gates to SU(N) anyons and identifies differences and new problems that arise.
- A relation between the HOMFLY-PT and Kauffman polynomials via characters
- Braid twists and HZ factorisation via character expansion