Links In R3 Research Articles

The quantum UV-IR map for line defects in mathfrak{gl} (3)-type class S theories

We consider the quantum UV-IR map for line defects in class S theories of mathfrak{gl} (3)-type. This map computes the protected spin character which counts framed BPS states with spin for the bulk-defect system. We give a geometric method of computing this map motivated by the physics of five-dimensional mathcal{N} = 2 supersymmetric Yang-Mills theory, and compute it explicitly in various examples. As a spin-off we propose a new way of computing a certain specialization of the HOMFLY polynomial for links in ℝ3, as a sum over BPS webs attached to the link.

Invariants in Quantum Geometry

In quantum geometry, we consider a set of loops, a compact orientable surface and a solid compact spatial region, all inside ℝ × ℝ3 = ℝ4, which forms a triple. We want to define an ambient isotopic equivalence relation on such triples, so that we can obtain equivalence invariants. These invariants describe how these submanifolds are causally related to or ‘linked’ with each other, and they are closely associated with the linking number between links in ℝ3. Because we distinguish the time-axis from spatial subspace in ℝ4, we see that these equivalence relations will also imply causality.

On the group of ring motions of an H-trivial link

In this paper we compute a presentation for the group of ring motions of the split union of a Hopf link with Euclidean components and a Euclidean circle. A key part of this work is the study of a short exact sequence of groups of ring motions of general ring links in R3. This sequence allowed us to build the main result from the previously known case of the ring group with one component, which a particular case of the ring groups studied by Brendle and Hatcher. This work is a first step towards the computation of a presentation for groups of motions of H-trivial links with an arbitrary number of components.

On Jones' subgroup of R. Thompson group F

Recently Vaughan Jones showed that the R. Thompson group F encodes in a natural way all knots and links in R3, and a certain subgroup F→ of F encodes all oriented knots and links. We answer several questions of Jones about F→. In particular we prove that the subgroup F→ is generated by x0x1, x1x2, x2x3 (where xi, i∈N are the standard generators of F) and is isomorphic to F3, the analog of F where all slopes are powers of 3 and break points are 3-adic rationals. We also show that F→ coincides with its commensurator. Hence the linearization of the permutational representation of F on F/F→ is irreducible. We show how to replace 3 in the above results by an arbitrary n, and to construct a series of irreducible representations of F defined in a similar way. Finally we analyze Jones' construction and deduce that the Thompson index of a link is linearly bounded in terms of the number of crossings in a link diagram.

A reduced set of moves on one-vertex ribbon graphs coming from links

Every link in R 3 \mathbb {R}^3 can be represented by a one-vertex ribbon graph. We prove a Markov type theorem on this subset of link diagrams.

SIGNATURE, NULLITY AND DETERMINANT OF CHECKERBOARD COLORABLE VIRTUAL LINKS

In this paper, we present the Goeritz matrix for checkerboard colorable virtual links or, equivalently, checkerboard colorable links in thickened surfaces Sg × [0, 1], which is an extension of the Goeritz matrix for classical knots and links in ℝ3. Using this, we show that the signature, nullity and determinant of classical oriented knots and links extend to those of checkerboard colorable oriented virtual links.

Criteria of prime links in terms of pseudo-characters

Pseudo-characters of groups have recently found applications in the theory of classical knots and links in ℝ3. More precisely, there is a connection between pseudo-characters of Artin’s braid groups and properties of links represented by braids. In the present work, this connection is investigated and the notion of kernel pseudo-characters of braid groups is introduced. It is proved that a kernel pseudo-character ϕ and a braid β satisfy Ιϕ(β)І > C ϕ, where C ϕ is the defect of ϕ, then β represents a prime link (i.e., a link that is noncomposite, nonsplit, and nontrivial). Furthermore, the space of braid group pseudo-characters is studied and a way to obtain nontrivial kernel pseudo-characters from an arbitrary braid group pseudo-character that is not a homomorphisrn is described. This allows one to use an arbitrary nontrivial braid group pseudo-character for recognition of prime knots and links. Bibliography: 17 titles.

INVARIANTS OF SURFACE LINKS IN ℝ4 VIA SKEIN RELATION

Using Yoshikawa's surface diagram, we constructed new invariants of ambient isotopy classes of smoothly embedded closed surfaces in ℝ4 via a state-sum model similar to the Kauffman's state-sum model for the Jones polynomial for classical knots and links in ℝ3. It is shown that the invariants can also be defined by skein relation and thus they are calculated from Yoshikawa's surface diagrams recurrently. Some of the properties of the invariants are given and explicit computations for several surfaces are included.

Links in an open book decomposition and in the standard contact structure

We study a relationship between arc presentations of links in R 3 \mathbb {R}^3 and Legendrian links in R 3 \mathbb {R}^3 with the standard tight contact structure. We determine the arc indices of torus knots.

Polynomial invariants of links in the projective space

This thesis consists of two papers about knots and links in the real projective space RP3. We present an algorithm to unknot diagrammatically knots and links in RP3. We extend two classical polynomial invariants of links in R3 to links in RP3.In the first paper, the notion of descending diagram is extended from links in R3 to nonoriented links in RP3. With the help of this extended notion, any nonoriented diagram of a projective link can be unknotted with some crossing changes. It is also shown that the notion of descending diagram cannot be extended to oriented projective links.In the second paper, Homfly and Kauffman skein modules of the real projective space are computed. Both modules are free and generated by an infinite set of links, described in the paper. This explicit computation gives an extension of the Homfly and Kauffman polynomial invariants of links in R3 to links in RP3. It is also shown that the extended Homfly and Kauffman polynomials allow to measure how far a projective link is from being affine.

Diagrammatic Unknotting of Knots and Links in the Projective Space

In the classical knot theory there is a well-known notion of descending diagram. From an arbitrary diagram one can easily obtain, by some crossing changes, a descending diagram which is a diagram of the unknot or unlink. In this paper the notion of descending diagram for knots and links in ℝ3 is extended to the case of nonoriented knots and links in the projective space. It is also shown that this notion cannot be extended to oriented links.

Links on T -Polyhedra: Examples of Gusarov's Cubic Spaces

Any link in ℝ3 is isotopic to a link lying on the union T of three half-planes with common boundary line. A nontrivial theory of knots and links on T was developed by the same author in an earlier paper. In the present paper, the results obtained are interpreted in the context of M. Gusarov's theory of invariants of finite degree (cubic space theory). Bibliography: 6 titles.

On Conway Algebras and the Homflypt Polynomial

In 1984 J. H. Przytycki and P. Traczyk introduced the notion of Conway algebras. They had proved that for every Conway algebra there exists an invariant of links in ℝ3, which takes values in that algebra. In this paper we show that any link invariant obtained in this way is not better than the Homflypt (Jones-Conway) polynomial.

Polygon Indices and Superbridge Indices of Torus Knots and Links

The minimal number of straight line segments required to form a given knot or link in ℝ3 is determined for a family of torus knots and links.

QUADRISECANTS OF KNOTS AND LINKS

We show that every non-trivial tame knot or link in ℝ3 has a quadrisecant, i.e. four collinear points. The quadrisecant must be topologically non-trivial in a precise sense. As an application, we show that a nonsingular, algebraic surface in ℝ3 which is a knotted torus must have degree at least eight.