Homfly polynomial via an invariant of colored plane graphs
Summary: After the first discovery of the quantum invariant associated with SU(2) by \textit{V. F. R. Jones} [Bull. Am. Math. Soc., New Ser. 12, 103-111 (1985; Zbl 0564.57006)], the invariants associated with \(\text{SU}(n)\) were found by several authors [\textit{P. Freyd}, \textit{D. Yetter}, \textit{J. Hoste}, \textit{W. B. R. Lickorish}, \textit{K. Millett}, and \textit{A. Ocneanu}, ibid., 239-246 (1985; Zbl 0572.57002)]. It was first proved by \textit{V. G. Turaev} [Invent. Math. 92, No. 3, 527-553 (1988; Zbl 0648.57003)] that all these come from so-called ``quantum groups'', especially from their \(R\)-matrices corresponding to the vector representations. There also exist various quantum invariants corresponding to other representations [see for example \textit{A. N. Kirillov} and \textit{N. Yu. Reshetikhin}, Adv. Ser. Math. Phys. 7, 285-337 (1989; Zbl 0742.17018); \textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1-26 (1990; Zbl 0768.57003); \textit{H. R. Morton}, in `Topics in knot theory', NATO ASI Ser., Ser. C, Math. Phys. Sci. 399, 107-155 (1993; Zbl 0816.57003)]. The aim of this paper is to give a graphical way to define \(\text{SU}(n)\) quantum invariants for links. To do this we first construct an invariant of colored, oriented, trivalent, plane graphs for each \(n(\geq 2)\). Then we show that the \(\text{SU}(n)\) polynomial invariant corresponding to the vector representation (HOMFLY polynomial) can be defined by using our graph invariant. We can also show that our invariant defines the \(\text{SU}(n)\) polynomial invariant corresponding to the anti-symmetric tensors of the vector representation. We note that our graph invariant for \(\text{SU}(3)\) was first introduced by \textit{G. Kuperberg} in [Int. J. Math. 5, No. 1, 61-85 (1994; Zbl 0797.57008)]. The second and the third authors used it in [J. Knot Theory Ramifications 6, No. 3, 373-404 (1997; Zbl 0949.57011)] to construct magic elements and defined the quantum SU(3) invariants for 3-manifolds. Now, the just cited and the present paper together give an elementary and self-contained proof of the existence of magic elements for SU(3) and so that of the quantum SU(3) invariants of 3-manifolds just like \textit{W. B. R. Lickorish} did for SU(2) in [J. Knot Theory Ramifications 2, No. 2, 171-194 (1993; Zbl 0793.57003)] using the Kauffmann bracket [\textit{L. H. Kauffman}, Topology 26, 395-407 (1987; Zbl 0622.57004)]. See [\textit{Y. Yokota}, Math. Ann. 307, No. 1, 109-138 (1997; Zbl 0953.57009)] for a similar approach to \(\text{SU}(n)\) invariants of 3-manifolds. We also note that our graph invariant may be obtained (not checked yet) by direct computations of the universal \(R\)-matrix. But the advantage of our definition is that it does not require any knowledge of quantum groups nor representation theory. On the contrary we can recover the \(R\)-matrix of the quantum group \(U_q({\mathfrak sl}(n, \mathbb{C}))\) corresponding at least to the vector representation.
scientific article; zbMATH DE number 1558729