Invariants of curves and fronts via Gauss diagrams

Polyak, Michael

doi:10.1016/s0040-9383(97)00013-x

Cited by 40 publications

(57 citation statements)

References 2 publications

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“…There are close connections between the finite type invariants of such knots and those of the underlying curve (Polyak 1998). For example, the expected value of the Casson invariant c 2 is one eighth the defect, a first-order invariant of the curve.…”

Section: Random Planar Curvesmentioning

confidence: 95%

Models of random knots

Even-Zohar

2017

J Appl. and Comput. Topology

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The study of knots and links from a probabilistic viewpoint provides insight into the behavior of ''typical'' knots, and opens avenues for new constructions of knots and other topological objects with interesting properties. The knotting of random curves arises also in applications to the natural sciences, such as in the context of the structure of polymers. We present here several known and new randomized models of knots and links. We review the main known results on the knot distribution in each model. We discuss the nature of these models and the properties of the knots they produce. Of particular interest to us are finite type invariants of random knots, and the recently studied Petaluma model. We report on rigorous results and numerical experiments concerning the asymptotic distribution of such knot invariants. Our approach raises questions of universality and classification of the various random knot models.

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Section: Random Planar Curvesmentioning

confidence: 95%

Models of random knots

Even-Zohar

2017

J Appl. and Comput. Topology

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“…mutual orientations of these components at these points) and weights (which are polynomial functions of the indices of these points with respect to different components of the curve), see [63]. Strong generalizations of these index-type invariants were found in [40], see also [46]. Similar expressions were found in [50], [51] for the simplest strangeness invariant in problem (si) of [8], [9] and some of its generalizations; combinatorial expressions for the invariants J + , J − were constructed in [76].…”

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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“….,~j+ ~ ~J+ cusp births safe self-ta, ngencies There are several combinatorial formulas for calculating the values of J+ on curves without cusps (see a review in [8]) and Polyak's formula [18] for curves with cusps.…”

Section: Index Maslov Index and Perestroikas To Each Componentmentioning

confidence: 99%

Kauffman bracket of plane curves

Chmutov

Goryunov

1996

Commun.Math. Phys.

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Abstract:We lower the Kauffman bracket for links in a solid torus (see [16]) to generic plane fronts. It turns out that the bracket can be entirely defined in terms of a front itself without using the Legendrian lifting. We show that all the coefficients of the lowered bracket are in fact Vassilev type invariants of Arnold's J+-theory [3,4]. We calculate their weight systems. As a corollary we obtain that the first coefficient is essentially the quantum deformation of the Bennequin invariant introduced recently by M. Polyak [19].There exists a straightforward way to get an invariant of an immersed cooriented hypersurface C in a smooth manifold N. We lift C to the manifold M of cooriented contact elements of N. This gives us an embedded submanifold Lc. Now we take the value of a known invariant of embeddings on Lc ~-~ M as the invariant of our initial immersion C ~-~ N.The manifold M of cooriented contact elements is the spherisation of the cotangent bundle of N: M = ST*N. It has a natural contact structure. Our lifting Lc is a Legendrian submanifold with respect to this structure. The hypersurface C is called the front of Lc. The above procedure defines an invariant not only on immersed C q-~ N but also on submanifolds with some "admissible" singularities which may appear as singularities of fronts of smooth Legendrian submanifolds generically embedded into M. The simplest situation is N = R E. The "admissible" singularities in this case are cusps. Thus we can induce an invariant on collections of closed oriented and cooriented plane curves which may have only double points and cusps as singularities. The manifold M of contact elements of the plane is the solid torus M = R 2 x S 1. So the lifted submanifolds are Legendrian links in it. This general approach was used in [12] to define an invariant of an immersed plane curve. There a Kontsevich type integral [11] was taken as a known invariant of knots in a solid torus.

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Invariants of curves and fronts via Gauss diagrams

Cited by 40 publications

References 2 publications

Models of random knots

Models of random knots

Resolutions of discriminants and topology of their complements

Kauffman bracket of plane curves

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