On Lannes and Polyak-viro type formulas for finite order invariants

Tyurina, S. D.

doi:10.1007/bf02679106

Cited by 5 publications

(11 citation statements)

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“…Finally we get that the cycle d 1 (T T ) shown in (11) is homologous in F 2 \ F 1 to a chain lying in the union of cells of nonmaximal dimensions listed in (7), (8); this homology is provided by the sum of six varieties indicated in the left parts of equalities (12), (13), (14), (15), (16), and (17).…”

Section: The Sum Of Varieties Distinguished By Conditions Of Typesmentioning

confidence: 93%

“…The latter degenerations appear in the lexicographic order: first by the number of colliding pairs of points in R 1 , and then by their positions in R 1 . In (13) first two summands are degenerations of the variety defined by the zigzag when its arrowed endpoint tends to one of boundaries of the corresponding segment, and the third summand belongs to its boundary as the equality of type φ(x) = φ(y) defines a component of the boundary of the set defined by the inequality φ(x) ≥ φ(y).…”

Section: The Sum Of Varieties Distinguished By Conditions Of Typesmentioning

confidence: 99%

“…In other words, the sum of three chains in the left parts of equations (12), (13) and (14) realizes homology between the sum of chains D and E and some chain in the boundary of the cell . We need to find varieties in this cell, whose boundaries include these summands.…”

Section: The Sum Of Varieties Distinguished By Conditions Of Typesmentioning

confidence: 99%

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Combinatorial Formulas for Cohomology of Spaces of Knots

Vassiliev¹

NATO Science Series

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Abstract. We develop homological techniques for finding explicit combinatorial expressions of finite-type cohomology classes of spaces of knots in R n , n ≥ 3, generalizing Polyak-Viro formulas [10] for invariants (i.e. 0-dimensional cohomology classes) of knots in R 3 .As the first applications we give such formulas for the (reduced mod 2) generalized Teiblum-Turchin cocycle of order 3 (which is the simplest cohomology class of long knots R 1 ֒→ R n not reducible to knot invariants or their natural stabilizations), and for all integral cohomology classes of orders 1 and 2 of spaces of compact knots S 1 ֒→ R n . As a corollary, we prove the nontriviality of all these cohomology classes in spaces of knots in R 3 .

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Section: The Sum Of Varieties Distinguished By Conditions Of Typesmentioning

confidence: 93%

Section: The Sum Of Varieties Distinguished By Conditions Of Typesmentioning

confidence: 99%

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Combinatorial Formulas for Cohomology of Spaces of Knots

Vassiliev¹

NATO Science Series

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“…The sum of the third, fourth, and fifth terms in the right-hand side of (15) equals the second summand in (16). Therefore the sum of the right-hand sides of (15)- (17) is equal to the cycle (14), and the sum of chains indicated in the left-hand sides of (15)- (17) is the desired combinatorial formula, i.e., the relative cycle of the space of curves modulo Σ, whose boundary coincides with the cycle generating the degree two Borel-Moore homology group of Σ.…”

Section: Algorithms For Combinatorial Realization Of Knot Invariantsmentioning

confidence: 99%

Combinatorial computation of combinatorial formulas for knot invariants

Vassiliev

2005

Trans. Moscow Math. Soc.

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Abstract. We construct a homology algebraic algorithm for computing combinatorial formulas of all finite degree knot invariants. Its input is an arbitrary weight system, i.e., a virtual principal part of a finite degree invariant, and the output is either a proof of the fact that this weight system actually does not correspond to any knot invariant or an effective description of some invariant with this principal part, i.e., a finite collection of easily described singular chains of full dimension in the space of spatial curves such that the value of this invariant on any generic knot is equal to the sum of multiplicities of these chains in a neighborhood of the knot. (In examples calculated by now, the former possibility never occurred.) This algorithm is formally realized over Z 2 , but its generalization to the case of arbitrary coefficients is just a technical task. The algorithm is based on the study of a complex of chains in the space of smooth curves in the three-dimensional space with a fixed flag of directions, and also in the discriminant variety of this space of curves.

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“…0-dimensional cohomology classes) of knots in R 3 , some combinatorial expressions were obtained by J. Lannes, M. Polyak and O. Viro, P. Cartier, S. Piunikhin, S. Tyurina, a.o., see [38], [45], [53], [54]. It was then proved by M. Goussarov [31] that expressions of Polyak-Viro type exist for any invariants of finite filtration for long knots R 1 → R 3 .…”

Section: 8mentioning

confidence: 99%

Resolutions of discriminants and topology of their complements

Vassiliev

2001

New Developments in Singularity Theory

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Abstract. We study topological invariants of spaces of nonsingular geometrical objects (such as knots, operators, functions, varieties) defined by the linking numbers with appropriate cycles in the complementary discriminant sets of degenerate objects. We describe the main construction of such classes (based on the conical resolutions of discriminants) and list the results for a number of examples.The discriminant subsets of spaces of geometric objects are the sets of all objects with singularities of some chosen type. The important examples are: spaces of polynomials with multiple roots, resultant sets of polynomial systems having common roots, spaces of functions with degenerate singular points, of non-smooth algebraic varieties, of linear operators with zero or multiple eigenvalues, of smooth maps S 1 → M n (n ≥ 3) having singular or self-intersection points, of non-generic plane curves, and many others.The discriminants are usually singular varieties, whose stratifications correspond to the classification of degenerations of the corresponding objects. E.g., the discriminant subset in the space of polynomials x 3 + ax + b is the semicubical parabola (a/3) 3 + (b/2) 2 = 0: its regular points correspond to polynomials with a root of multiplicity exactly 2, and the vertex to the polynomial x 3 . The discriminant in the space of polynomials x 4 +ax 2 +bx+c is the swallowtail, i.e. the surface shown in the right-hand part of Fig. 1: its self-intersection curve consists of polynomials having two double roots, and the semicubical edges correspond to the polynomials with one triple root; the most singular point is the polynomial x 4 with a root of multiplicity 4. Similar stratifications hold for polynomials of all higher degrees: their strata are indexed by the multiplicities and orders in R 1 of all corresponding multiple roots.Usually one is interested in the space of non-singular objects which is the complement of the discriminant Σ, e.g. in the space of polynomials without multiple roots, of smooth varieties, of non-degenerate operators, or of knots, i.e. maps S 1 → R 3 having no self-intersection or singular points.If the total space F of geometric objects is an N -dimensional vector space then the homology groups of these complementary spaces are related by the Alexander duality formula

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On Lannes and Polyak-viro type formulas for finite order invariants

Abstract: Vassiliev's knot invariants can be computed in different ways but many of them as Kontsevich integral are very difficult. We consider more visual diagram formulas of the type Polyak-Viro and give new diagram formula for the two basic Vassiliev invariant of degree 4.

Cited by 5 publications

References 1 publication

Combinatorial Formulas for Cohomology of Spaces of Knots

Combinatorial Formulas for Cohomology of Spaces of Knots

Combinatorial computation of combinatorial formulas for knot invariants

Resolutions of discriminants and topology of their complements

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