Diagrammatic formulae of Viro-Polyak type for knot invariants of finite order

Tyurina, S. D.

doi:10.1070/rm1999v054n03abeh000172

Cited by 6 publications

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“…The arrow diagram formulas presented by Polyak and Viro give an easy way to compute some link invariants of low degree, and have been used by A. Stoimenow to obtain bounds for these invariants and some polynomial link invariants [7]. Arrow diagram formulas for other knot invariants have been obtained by S. D. Tyurina [8]. Homological methods for finding arrow diagram formulas have recently been developed by V. I. Vassiliev [9].…”

Section: Arrow Diagram Formulasmentioning

confidence: 99%

A diagrammatic approach to link invariants of finite degree

Östlund¹

2004

MATH. SCAND.

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In [5] M. Polyak and O. Viro developed a graphical calculus of diagrammatic formulas for Vassiliev link invariants, and presented several explicit formulas for low degree invariants. M. Goussarov [2] proved that this arrow diagram calculus provides formulas for all Vassiliev knot invariants. The original note [5] contained no proofs, and it also contained some minor inaccuracies. This paper fills the gap in literature by presenting the material of [5] with all proofs and details, in a selfcontained form. Furthermore, a compatible coalgebra structure, related to the connected sum of knots, is introduced on the algebra of based arrow diagrams with one circle.

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Section: Arrow Diagram Formulasmentioning

confidence: 99%

A diagrammatic approach to link invariants of finite degree

Östlund¹

2004

MATH. SCAND.

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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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Diagram Invariants of Knots and the Kontsevich Integral

Tyurina¹

2006

J Math Sci

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Diagrammatic formulae of Viro-Polyak type for knot invariants of finite order

Abstract: The possible mechanisms for anomalous decay Z+Z + l~y are discussed.

Cited by 6 publications

References 1 publication

A diagrammatic approach to link invariants of finite degree

A diagrammatic approach to link invariants of finite degree

Resolutions of discriminants and topology of their complements

Diagram Invariants of Knots and the Kontsevich Integral

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