<p>Starting with a smooth, non-trivial <i>n</i>-dimensional knot <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(K\subset \mathbb {S}^{n+2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, and a beaded <i>n</i>-dimensional necklace subordinated to <i>K</i>, we construct a wild knot with a Cantor set of wild points (<i>i.e</i>,&#xa0;&#xa0;the knot is not locally flat in these points). The construction uses the conformal Schottky group acting on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {S}^{n+2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, generated by inversions on the spheres which are the boundary of the “beads”. We show that if <i>K</i> is a fibered knot, then the wild knot is also fibered. We also study cyclic branched coverings along the wild knots. This work generalizes the result presented in [<CitationRef CitationID="CR8">8</CitationRef>].</p>

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N-dimensional beaded necklaces and higher dimensional wild knots, invariant by a Schottky group

  • Gabriela Hinojosa,
  • Alberto Verjovsky,
  • Juan Pablo Díaz

摘要

Starting with a smooth, non-trivial n-dimensional knot \(K\subset \mathbb {S}^{n+2}\) K S n + 2 , and a beaded n-dimensional necklace subordinated to K, we construct a wild knot with a Cantor set of wild points (i.e,  the knot is not locally flat in these points). The construction uses the conformal Schottky group acting on \(\mathbb {S}^{n+2}\) S n + 2 , generated by inversions on the spheres which are the boundary of the “beads”. We show that if K is a fibered knot, then the wild knot is also fibered. We also study cyclic branched coverings along the wild knots. This work generalizes the result presented in [8].