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A Graphical Construction of the sl(3) Invariant for Virtual Knots

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arxiv 1207.0719 v3 pith:FO4QVYVV submitted 2012-07-03 math.GT

classification math.GT
keywords invariantknotsvirtualbracketconstructconstructionhomflyptkuperberg
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By generalizing the Kuperberg sl(3) bracket, we construct a graph-valued analogue of the Homflypt sl(3) invariant for virtual knots. The restriction of this invariant for classical knots coincides with the usual Homflypt sl(3) invariant, and for virtual knots and graphs it provides new information that allows one to prove minimality theorems and to construct new invariants for free knots. We formulate this new invariant for virtual braids as well, and show that it leads to the construction of a trace function on the virtual Hecke algebra. Finally, we show that the Penrose coloring bracket is a special case of the Kuperberg bracket, and we raise new questions about the extension of the present work.

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  1. On geometric bases for A-polynomials II: $\mathfrak{su}_3$ and Kuberberg bracket

    hep-th 2025-05 conditional novelty 6.0 of 10

    A new arcade-based planarization technique plus the Kuperberg bracket yields a closed system of classical relations toward su3 A-polynomials, demonstrated on the trefoil.

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