Polynomial Invariants Of Knots Research Articles

Yamada polynomial and associated link of θ-curves

The discovery of polynomial invariants of knots and links, ignited by V. F. R. Jones, leads to the formulation of polynomial invariants of spatial graphs. The Yamada polynomial, one of such invariants, is frequently utilized for practical distinguishment of spatial graphs. Especially for θ-curves, the polynomial is an ambient isotopy invariant after a normalization. On the other hand, to each θ-curve, a 3-component link can be associated as an ambient isotopy invariant. The benefit of associated links is that invariants of links can be utilized as invariants of θ-curves.In this paper we investigate the relation between the normalized Yamada polynomial of θ-curves and the Jones polynomial of their associated links, and show that the two polynomials are equivalent for brunnian θ-curves as a corollary. For our purpose the Jaeger polynomial of spatial graphs is observed, a specialization of which is equivalent to the Yamada polynomial.

Link polynomials derived from magnetic graphs

We introduce a graph diagram which can be regarded as a generalized link diagram. By using it, we construct two polynomial invariants for knots and links which are distinct from both the HOMFLY and the Kauffman polynomials. We show some features of the polynomials including relationships with the HOMFLY and the Kauffman polynomials.

Polynomial Invariants of Knots and Links

The class of knots corresponding to d-diagrams with m vertical and n horizontal chords is considered and the set of their invariant polynomials is calculated. Bibliography: 2 titles.

A New Point of View in the Theory of Knot and Link Invariants

Recent progress in string theory has led to a reformulation of quantum-group polynomial invariants for knots and links into new polynomial invariants whose coefficients can be understood in topological terms. We describe in detail how to construct the new polynomials and we conjecture their general structure. This leads to new conjectures on the algebraic structure of the quantum-group polynomial invariants. We also describe the geometrical meaning of the coefficients in terms of the enumerative geometry of Riemann surfaces with boundaries in a certain Calabi-Yau threefold.

On link invariants and topological string amplitudes

We explicitly show that the new polynomial invariants for knots, upto nine crossings, agree with the Ooguri-Vafa conjecture relating Chern-Simons gauge theory to topological string theory on the resolution of the conifold.

A path-integral approach to polynomial invariants of links

A self-contained, genuinely three-dimensional, monodromy-matrix based, nonperturbative, covariant path-integral approach to polynomial invariants of knots and links in the framework of (topological) quantum Chern–Simons field theory is proposed. The idea of ‘‘physical’’ observables represented by an auxiliary topological quantum-mechanical model in an external gauge field is introduced as a replacement for the limited notion of the Wilson loop. The possibility of using various generalizations of the Chern–Simons action (also higher-dimensional ones) as well as a purely functional language thereby arises. The theory is quantized in the framework of the antibracket-antifield formalism of Batalin and Vilkovisky, which is well adapted to this case. Using Stokes’s theorem and formal translational invariance of the path-integral measure we derive a monodromy matrix corresponding to an arbitrary pair of irreducible representations of an arbitrary semisimple Lie group.

Index of subfactors and statistics of quantum fields

The endomorphism semigroup End(M) of an infinite factor M is endowed with a natural conjugation (modulo inner automorphisms) {Mathematical expression}, where I³ is the canonical endomorphism of M into I(M). In Quantum Field Theory conjugate endomorphisms are shown to correspond to conjugate superselection sectors in the description of Doplicher, Haag and Roberts. On the other hand one easily sees that conjugate endomorphisms correspond to conjugate correspondences in the setting of A. Connes. In particular we identify the canonical tower associated with the inclusion[Figure not available: see fulltext.] relative to a sector I. As a corollary, making use of our previously established index-statistics correspondence, we completely describe, in low dimensional theories, the statistics of a selfconjugate superselection sector I with 3 or less channels, in particular of sectors with statistical dimension d(I)<2, by obtaining the braid group representations of V. Jones and Birman, Wenzl and Murakami. The statistics is thus described in these cases by the polynomial invariants for knots and links of Jones and Kauffman. Selconjugate sectors are subdivided into real and pseudoreal ones and the effect of this distinction on the statistics is analyzed. The FYHLMO polynomial describes arbitrary 2-channels sectors. © 1990 Springer-Verlag.

Polynomial invariants of knots in lens spaces

We characterize the polynomials that can arise as Alexander polynomial of (homotopically nontrivial) knots in classical lens spaces. Our main geometric argument contains an algorithm to produce knots realizing these polynomials.

Jones polynomial invariants for knots and satellites

AbstractResults of Kirillov and Reshetikhin on constructing invariants of framed links from the quantum group SU(2)q are adapted to give a simple formula relating the invariants for a satellite link to those of the companion and pattern links used in its construction. The special case of parallel links is treated first. It is shown as a consequence that any SU(2)q-invariant of a link L is a linear combination of Jones polynomials of parallels of L, where the combination is determined explicitly from the representation ring of SU(2). As a simple illustration Yamada's relation between the Jones polynomial of the 2-parallel of L and an evaluation of Kauffman's polynomial for sublinks of L is deduced.

Invariants of plane algebraic curves via representations of the braid groups

Recently new remarkable polynomial invariants of knots in 3-sphere were introduced using some representations of the braid groups which were shown to be effective in distinguishing knots ([-J], [-FYHLMO]). The purpose of this note is to introduce, using representation of the braid groups, an invariant of continuous equisingular families of plane algebraic curves. A pleasant feature is that this invariant is defined for any representation of the braid group (no coherence of representations for different number of strings is required), though for many representations this invariant is trivial. In the case of the reduced Burau representation, this invariant essentially coincides with the Alexander polynomial of plane curves introduced in [-L 1]. This fact is the counterpart of the well-known relation between the Alexander polynomial of closed braids and the reduced Burau representation. The definition of this invariant depends on braid monodromy associated with the curve ([M]). In particular we obtain a direct (not involving calculations of the fundamental group, though computationally cumbersome) method of computation of the Alexander polynomial via braid monodromy. In the next section we shall recall the background on continuous equisingular families of plane algebraic curves and their braid monodromies. In the Sect. 3 we introduce the invariant and in the last section we consider the case of reduced Burau representations.

Examples on polynomial invariants of knots and links. II

Examples on polynomial invariants of knots and links. II

The New Polynomial Invariants of Knots and Links

(1988). The New Polynomial Invariants of Knots and Links. Mathematics Magazine: Vol. 61, No. 1, pp. 3-23.

The New Polynomial Invariants of Knots and Links

The New Polynomial Invariants of Knots and Links

Examples on polynomial invariants of knots and links

Examples on polynomial invariants of knots and links

Polynomial Invariants of Knots of Codimension Two

Polynomial Invariants of Knots of Codimension Two