×

Cyclic branched coverings of 2-bridge knots. (English) Zbl 0952.57001

A cyclic presentation \(G_n(w)\) of a group is a presentation with \(n\) (ordered) generators whose relations are obtained by cyclically permuting the generators in a word \(w\) in these generators. The polynomial associated to a presentation \(G_n(w)\) has as its coefficients the exponent sums of the generators in the word \(w\). The question arises when such presentations \(G_n(w)\) are geometric, that is, correspond to spines of closed 3-manifolds \(M_n(w)\), and in particular whether such 3-manifolds arise as cyclic branched coverings of knots in the 3-sphere; also, does in this case the above polynomial coincide with the Alexander polynomials of the knot? In the present paper, the cyclic branched coverings of the 2-bridge knots are studied. Using the representation of 3-manifolds by railroad systems (and also by pairwise identification of the boundary faces of a triangulated 3-ball) it is shown that the cyclic branched coverings of the 2-bridge knots admit cyclic presentations which are geometric (arising from Heegaard diagrams with rotational symmetries), and that the above polynomials are in fact the Alexander polynomials of the corresponding knots.

MSC:

57M12 Low-dimensional topology of special (e.g., branched) coverings
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
57R65 Surgery and handlebodies
PDFBibTeX XMLCite
Full Text: EuDML