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Twisted Alexander polynomial and matrix-weighted zeta function. (English) Zbl 1453.57009

Summary: The twisted Alexander polynomial is an invariant of the pair of a knot and its group representation. Herein, we introduce a digraph obtained from an oriented knot diagram, which is used to study the twisted Alexander polynomial of knots. In this context, we show that the inverse of the twisted Alexander polynomial of a knot may be regarded as the matrix-weighted zeta function that is a generalization of the Ihara-Selberg zeta function of a directed weighted graph.

MSC:

57K14 Knot polynomials
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
11S40 Zeta functions and \(L\)-functions
15A15 Determinants, permanents, traces, other special matrix functions
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[1] Y. Ihara. On discrete subgroups of the two by two projective linear group over p-adic fields. J. Math. Soc. Japan 18(3) (1966), 219-235. · Zbl 0158.27702
[2] A. Ishii and K. Oshiro. Augmented Alexander matrices and generalization of twisted Alexander invariants and quandle cocycle invariants. Preprint.
[3] B. Jiang and S. Wang. Twisted topological invariants associated with representation. Topics in Knot Theory (NATO Advanced Science Institute. Series C. Mathematical and Physical Sciences, 399). Kluwer Academic, Dordrecht, 1993, pp. 211-227. · Zbl 0815.55001
[4] T. Kitano. Twisted Alexander polynomial and Reidemeister torsion. Pacific J. Math. 174 (1996), 431-442. · Zbl 0863.57001
[5] X.-S. Lin. Representation of knot groups and twisted Alexander polynomials. Acta Math. Sin. 17 (2001), 361-380. · Zbl 0986.57003
[6] X.-S. Lin and Z. Wang. Random walk on knot diagram, colored Jones polynomial and Ihara-Selberg zeta function. Knots, Braids, and Mapping Class Groups - Papers Dedicated to Joan S. Birman (New York, 1998) (Studies in Advanced Mathematics, 24). American Mathematical Society, Providence, RI, 2001, pp. 107-121. · Zbl 0992.57001
[7] T. Morifuji. Representations of knot groups into SL(2, ℂ) and twisted Alexander polynomials. Handbook of Group Actions. Vol. I (Advanced Lectures in Mathematics, 31). International Press, Boston, and Higher Education Press of China, 2015, pp. 527-576.
[8] I. Sato, H. Mitsuhashi and H. Morita. A matrix-weighted zeta function of a graph. Linear Multilinear Algebra 62(1) (2014), 114-125. · Zbl 1286.05100
[9] J.-P. Serre. Trees. Springer, New York, 1980. · Zbl 0548.20018
[10] T. Sunada. L-functions in geometry and some applications. Curvature and Topology of Riemannian Manifolds (Lecture Notes in Mathematics, 1201). Springer, Berlin, 1986, pp. 266-284. · Zbl 0605.58046
[11] T. Sunada. Fundamental groups and Laplacians (in Japanese). Kinokuniya, Tokyo, 1988. · Zbl 0646.58027
[12] M. Wada. Twisted Alexander polynomial for finitely presentable groups. Topology 33(2) (1994), 241-256. · Zbl 0822.57006
[13] Y.
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