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A new point of view in the theory of knot and link invariants. (English) Zbl 1002.57026
From the text: “Recent progress in string theory has led to a reformulation of quantum group polynomial invariants for knots and links into new polynomial invariants whose coefficients can be understood in topological terms. We describe in detail how to construct the new polynomials and we conjecture their general structure. This leads to new conjectures on the algebraic structure of the quantum group polynomial invariants. We also describe the geometrical meaning of the coefficients in terms of the enumerative geometry of Riemann surfaces with boundaries in a certain Calabi-Yau threefold.”
Let \(\mathcal L\) be a knot and \(\xi\in B_N\) be a braid, whose closure is \(L\). For any representations of the quantum group \(U_q(sl(n))\), \(\xi\) induces by means of the universal \(R\)-matrix an endomorphism of the representation \(V^{\otimes N}\). The quantum trace of this endomorphism is an invariant for \(\mathcal L\), called quantum group polynomial invariant. One can generalize this construction to links of \(L\) components, \(L\geq 1\), where each set of \(L\) representations defines a polynomial invariant.
The reformulated quantum polynomial invariants are constructed as follows. Let \(\mathcal L\) be a link of \(L\) components. For each set of \(L\) highest weight irreducible representations of \(U_q(sl(n))\), \(R_1,R_2,\dots, R_L\), let \(W(R_1,R_2,\dots,R_L)\) be the corresponding invariant. Using these polynomial invariants and other constants from representation theory of the symmetric groups, one forms a power series called the generating functional of quantum group polynomial invariants of \(\mathcal L\). The logarithm of this functional when expanded in a series gives us the reformulated quantum group polynomial invariants.
The authors conjecture that these new polynomial invariants can be described in a specific way by integer invariants of \(\mathcal L\), which have some topological content.

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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