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Fibring the complement of the Fenn-Rolfsen link.

Roger Fenn — 1989

Publicacions Matemàtiques

In this note it is shown that the complement of the singular linked spheres in four dimensions defined by Fenn and Rolfsen can be fibred by tori. Also a symmetry between the two components is revealed which shows that the image provides an example of a Spanier-Whitehead duality. This provides an immediate proof that the α-invariant is non zero.

The equation [B,(A-1)(A,B)] = 0 and virtual knots and links

Stephen BuddenRoger Fenn — 2004

Fundamenta Mathematicae

Let A, B be invertible, non-commuting elements of a ring R. Suppose that A-1 is also invertible and that the equation [B,(A-1)(A,B)] = 0 called the fundamental equation is satisfied. Then this defines a representation of the algebra ℱ = A, B | [B,(A-1)(A,B)] = 0. An invariant R-module can then be defined for any diagram of a (virtual) knot or link. This halves the number of previously known relations and allows us to give a complete solution in the case when R is the quaternions.

Virtual knot theory-unsolved problems

Roger FennLouis H. KauffmanVassily O. Manturov — 2005

Fundamenta Mathematicae

The present paper gives a quick survey of virtual and classical knot theory and presents a list of unsolved problems about virtual knots and links. These are all problems in low-dimensional topology with a special emphasis on virtual knots. In particular, we touch new approaches to knot invariants such as biquandles and Khovanov homology theory. Connections to other geometrical and combinatorial aspects are also discussed.

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