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On calculations of the twisted Alexander ideals for spatial graphs, handlebody-knots and surface-links. (English) Zbl 1394.57001

The twisted Alexander ideal of a knot is a generalization of the Alexander ideal, and is derived from the fundamental group of the knot exterior, an abelianization and a group representation. In this paper, the authors adopt Wada’s version of the twisted Alexander ideal, extended for a finitely presentable group with an abelianization and a group representation; see [M. Wada, Topology 33, No. 2, 241–256 (1994; Zbl 0822.57006)]. The authors calculate the twisted Alexander ideals for spatial graphs, handlebody-knots and surface-links. A spatial graph is a finite and labeled graph embedded in \(S^3\). For spatial graphs, they calculate and show the nontriviality of the twisted Alexander ideal for Suzuki’s theta-\(n\) curve for an integer \(n \geq 3\) such that \(n \equiv 1,5 \pmod{6}\), although the Alexander ideal is trivial for this case. A handlebody-knot is a handlebody embedded in \(S^3\). They calculate the twisted Alexander ideals for handlebody-knots of genus two with up to six crossings in the table due to A. Ishii et al. [J. Knot Theory Ramifications 21, No. 4, 1250035, 9 p. (2012; Zbl 1236.57015)]. A surface-link is a closed surface locally flatly embedded in \(\mathbb{R}^4\). They calculate the twisted Alexander ideals for surface-links in Yoshikawa’s table, correcting three calculations in the table due to K. Yoshikawa [Osaka J. Math. 31, No. 3, 497–522 (1994; Zbl 0861.57033)].

MSC:

57M05 Fundamental group, presentations, free differential calculus
57M15 Relations of low-dimensional topology with graph theory
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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References:

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