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On Alexander polynomials of certain \((2,5)\) torus curves. (English) Zbl 1184.14045

Summary: We compute Alexander polynomials of a torus curve \(C\) of type \((2,5), C:f(x,y)=f_{2}(x,y)^{5}+f_{5}(x,y)^{2}=0\), under the assumption that the origin \(O\) is the unique inner singularity and \(f_{2}=0\) is an irreducible conic. We show that the Alexander polynomial remains the same with that of a generic torus curve as long as \(C\) is irreducible.

MSC:

14H20 Singularities of curves, local rings
14H30 Coverings of curves, fundamental group
14H45 Special algebraic curves and curves of low genus
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