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We show that there exist non-trivial piecewise linear (PL) knots with isolated singularities $S^{n - 2} \subset S ⁿ$ , n ≥ 5, whose complements have the homotopy type of a circle. This is in contrast to the case of smooth, PL locally flat, and topological locally flat knots, for which it is known that if the complement has the homotopy type of a circle, then the knot is trivial.

Spacefilling knots.

Schmitt, Peter (1997)

Beiträge zur Algebra und Geometrie

Special positions for surfaces bounded by closed braids.

Lee Rudolph (1985)

Revista Matemática Iberoamericana

Stably hyperbolic ...-hermitian forms and doubly sliced knots.

Eva Bayer-Fluckiger, Neal W. Stoltzfus (1986)

Journal für die reine und angewandte Mathematik

State-sum invariants of knotted curves and surfaces from quandle cohomology.

Carter, J.Scott, Jelsovsky, Daniel, Kamada, Seiichi, Langford, Laurel, Saito, Masahico (1999)

Electronic Research Announcements of the American Mathematical Society [electronic only]

Submanifolds of codimension two and homology equivalent manifolds

Sylvain E. Cappell, Julius L. Shaneson (1973)

Annales de l'institut Fourier

In this paper new methods of studying codimension two embeddings of manifolds are outlined. Results are stated on geometric periodicity of knot cobordism. The group of local knots of a manifold in a 2-plane bundle is introduced and computed, and applied to $C^{o}$ -close embeddings. General codimension two splitting theorems are discussed, with applications to equivariant knots and knot cobordism. A general existence theorem for P.L. (non-locally flat) embeddings is also given.The methods involve some...

Sur le rôle de la monodromie entière dans la topologie des singularités

Françoise Michel, Claude Weber (1986)

Annales de l'institut Fourier

Nous considérons l’action de la monodromie sur l’homologie de la fibre de Milnor d’une singularité complexe. Cette action est plus compliquée que prévu : en effet nous montrons que, sur $Z$ , elle n’est, en général, pas somme directe de modules cycliques. Nous donnons également des exemples prouvant que la monodromie rationnelle ne détermine pas la monodromie entière et que la monodromie entière ne détermine pas la topologie.

Surfaces in 3-space that do not lift to embeddings in 4-space

J. Carter, Masahico Saito (1998)

Banach Center Publications

A necessary and sufficient condition for an immersed surface in 3-space to be lifted to an embedding in 4-space is given in terms of colorings of the preimage of the double point set. Giller's example and two new examples of non-liftable generic surfaces in 3-space are presented. One of these examples has branch points. The other is based on a construction similar to the construction of Giller's example in which the orientation double cover of a surface with odd Euler characteristic is immersed...

Surfaces in the complex projective plane and their mapping class groups.

Hirose, Susumu (2005)

Algebraic & Geometric Topology

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