Инварианты и когомологии первого порядка для пространств вложений самопересекающихся кривых в $\mathbb R^n$

The present monograph is devoted to low-dimensional topology in the context of two thriving theories: parity theory and theory of graph-links, the latter being an important generalization of virtual knot theory constructed by means of intersection graphs. Parity theory discovered by the second-named author leads to a new perspective in virtual knot theory, the theory of cobordisms in two-dimensional surfaces, and other new domains of topology. Theory of graph-links highlights a new combinatorial approach to knot theory.

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The van Kampen obstruction and its relatives

Melikhov

2009

Proc. Steklov Inst. Math.

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Abstract. We review a cochain-free treatment of the classical van Kampen obstruction ϑ to embeddability of an n-polyhedron into R 2n and consider several analogues and generalizations of ϑ, including an extraordinary lift of ϑ which in the manifold case has been studied by J.-P. Dax. The following results are obtained.• The mod2 reduction of ϑ is incomplete, which answers a question of Sarkaria.• An odd-dimensional analogue of ϑ is a complete obstruction to linkless embeddability (="intrinsic unlinking") of the given n-polyhedron in R 2n+1 .• A "blown up" 1-parameter version of ϑ is a universal type 1 invariant of singular knots, i.e. knots in R 3 with a finite number of rigid transverse double points. We use it to decide in simple homological terms when a given integer-valued type 1 invariant of singular knots admits an integral arrow diagram (= Polyak-Viro) formula.• Settling a problem of Yashchenko in the metastable range, we obtain that every PL manifold N, non-embeddable in a given R m , m ≥, contains a subset X such that no map N → R m sends X and N \ X to disjoint sets.• We elaborate on McCrory's analysis of the Zeeman spectral sequence to geometrically characterize "k-co-connected and locally k-co-connected" polyhedra, which we embed in R 2n−k for k < n−3 2 extending the Penrose-Whitehead-Zeeman theorem.

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Инварианты и когомологии первого порядка для пространств вложений самопересекающихся кривых в $\mathbb R^n$

Cited by 3 publications

References 17 publications

Explicit formulas for Vasil'ev invariants of the fourth order for knots

Explicit formulas for Vasil'ev invariants of the fourth order for knots

Parity in knot theory and graph-links

The van Kampen obstruction and its relatives

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