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On the Signatures of Torus Knots

Maciej Borodzik, Krzysztof Oleszkiewicz (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

We study properties of the signature function of the torus knot T p , q . First we provide a very elementary proof of the formula for the integral of the signature over the circle. We also obtain a closed formula for the Tristram-Levine signature of a torus knot in terms of Dedekind sums.

On the simple connectivity at infinity of groups

Daniele Ettore Otera (2003)

Bollettino dell'Unione Matematica Italiana

We study the simple connectivity at infinity of groups of finite presentation, and we give a geometric proof of its invariance under quasi-isometry in a special case.

On the structure of closed 3-manifolds

Jan Mycielski (2003)

Fundamenta Mathematicae

We will show that for every irreducible closed 3-manifold M, other than the real projective space P³, there exists a piecewise linear map f: S → M where S is a non-orientable closed 2-manifold of Euler characteristic χ ≡ 2 (mod 3) such that | f - 1 ( x ) | 2 for all x ∈ M, the closure of the set x M : | f - 1 ( x ) | = 2 is a cubic graph G such that S - f - 1 ( G ) consists of 1/3(2-χ) + 2 simply connected regions, M - f(S) consists of two disjoint open 3-cells such that f(S) is the boundary of each of them, and f has some additional interesting properties....

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