Quantum relatives of the Alexander polynomial. (English. Russian original) Zbl 1149.57024
St. Petersbg. Math. J. 18, No. 3, 391-457 (2007); translation from Algebra Anal. 18, No. 3, 63-157 (2006).
Conway introduced a link invariant that encodes the same information as the enhanced Alexander polynomial (called the Conway function in the paper under review). Later it was shown by Murakami, Rozansky and Saleur, and Reshetikhin that the multivariable Conway function can be defined via representations of quantum groups (see for example [J. Murakami, Pac. J. Math. 157, No. 1, 109–135 (1993; Zbl 0799.57006)] and [L. Rozansky and H. Saleur, Quantum field theory for the multivariable Alexander polynomial, Nuclear Phys. B, 376, 461-509 (1992)]). This allows to bring the study of the Conway function (the Alexander polynomial) into the framework of quantum topology. Note that such an approach was very useful in the case of the Jones polynomial: this link invariant was first generalized to colored links, then to trivalent framed knotted graphs, and its counterparts (Reshetikhin-Turaev invariants) for closed oriented 3-manifolds were discovered along this way. As for the quantum theory of the Alexander polynomial, the first similar steps along this line were made by several authors.
The multivariable Conway function was studied via the methods of quantum topology in two directions: based on the quantum supergroup \(gl(1|1)\) by Reshetikhin, Rozansky, and Saleur and based on the quantum group by Jun Murakami, Deguchi, and Akutsu. In the case of the quantum group \(sl(2)\), a generalization to colored framed trivalent graphs and the face state sum model were obtained, although the results were given in a complicated form and the geometric analysis of them was not considered. In the case of the group \(gl(1|1)\), a generalization to colored framed trivalent graphs was not considered.
In the paper under review, the author develops quantum invariants related to the Alexander polynomial. In the \(gl(1|1)\) direction, the author generalizes the multivariable Conway function to trivalent graphs equipped with colorings and finds a face state sum presentation for this generalization. In the \(sl(2)\) direction, similar results have been obtained by the author, giving simpler formulas than before and a geometric presentation.
Summarized, the relatives of the Alexander polynomial studied by the author, are:
a) the Reshetikhin-Turaev functors \({\mathcal RT}^1\) and \({\mathcal RT}^2\) based on the irreducible representations of quantum groups \(gl(1|1)\) and \(sl(2)\), respectively;
b) a modification \({\mathcal A}^c\) of the functor \({\mathcal RT}^c\) for \(c=1,2\);
c) the invariant \(\Delta^c\) of closed colored framed generic graphs similar to the Conway function \(\nabla\) of links.
The multivariable Conway function was studied via the methods of quantum topology in two directions: based on the quantum supergroup \(gl(1|1)\) by Reshetikhin, Rozansky, and Saleur and based on the quantum group by Jun Murakami, Deguchi, and Akutsu. In the case of the quantum group \(sl(2)\), a generalization to colored framed trivalent graphs and the face state sum model were obtained, although the results were given in a complicated form and the geometric analysis of them was not considered. In the case of the group \(gl(1|1)\), a generalization to colored framed trivalent graphs was not considered.
In the paper under review, the author develops quantum invariants related to the Alexander polynomial. In the \(gl(1|1)\) direction, the author generalizes the multivariable Conway function to trivalent graphs equipped with colorings and finds a face state sum presentation for this generalization. In the \(sl(2)\) direction, similar results have been obtained by the author, giving simpler formulas than before and a geometric presentation.
Summarized, the relatives of the Alexander polynomial studied by the author, are:
a) the Reshetikhin-Turaev functors \({\mathcal RT}^1\) and \({\mathcal RT}^2\) based on the irreducible representations of quantum groups \(gl(1|1)\) and \(sl(2)\), respectively;
b) a modification \({\mathcal A}^c\) of the functor \({\mathcal RT}^c\) for \(c=1,2\);
c) the invariant \(\Delta^c\) of closed colored framed generic graphs similar to the Conway function \(\nabla\) of links.
Reviewer: Leonid Plachta (Gdansk)
MSC:
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
Keywords:
Alexander polynomial; Conway function; quantum topology; quantum supergroup; colored framed trivalent graphs; Reshetikhin-Turaev functorsCitations:
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