Vassiliev Invariants Of Knots Research Articles

Finite type invariants for knotoids

We extend the theory of Vassiliev (or finite type) invariants for knots to knotoids using two different approaches. Firstly, we take closures on knotoids to obtain knots and we use the Vassiliev invariants for knots, proving that these are knotoid isotopy invariant. Secondly, we define finite type invariants directly on knotoids, by extending knotoid invariants to singular knotoid invariants via the Vassiliev skein relation. Then, for spherical knotoids we show that there are non-trivial type-1 invariants, in contrast with classical knot theory where type-1 invariants vanish. We give a complete theory of type-1 invariants for spherical knotoids, by classifying linear chord diagrams of order one, and we present examples arising from the affine index polynomial and the extended bracket polynomial.

On the skein theory of dichromatic links and invariants of finite type

In [Dichromatic link invariants, Trans. Amer. Math. Soc. 321(1) (1990) 197–229], Hoste and Kidwell investigated the skein theory of oriented dichromatic links in [Formula: see text]. They introduced a multi-variable polynomial invariant [Formula: see text]. We use special substitutions for some of the parameters of the invariant [Formula: see text] to show how to deduce invariants of finite type from [Formula: see text] using partial derivatives. Then we consider the 2-component 1-trivial dichromatic links. We study the Vassiliev invariants of the 2-component in the complement of the 1-component, which is equivalent to studying Vassiliev invariants for knots in [Formula: see text] We give combinatorial formulas for the type-zero and type-one invariants and we connect these invariants to existing invariants such as Aicardi's invariant. This provides us with a topological meaning of the first partial derivative, which is also shown to be universal as a type-one invariant.

Type-two invariants for knots in the solid torus

We introduce a natural filtration in the space of knots and singular knots in the solid torus, and start the study of the type-two Vassiliev invariants with respect to this filtration. The main result of the work states that any such invariant within the second term of this filtration in the space of knots with zero winding number is a linear combination of seven explicitly described Gauss diagram invariants. This introduces a basis (and a universal invariant) for the type-two Vassiliev invariants for knots with zero winding number. Then we formalize the problem of exploring the set of all type-two invariants for knots with zero winding number.

Chern–Simons theory, Vassiliev invariants, loop quantum gravity and functional integration without integration

This paper is an exposition of the relationship between Witten’s Chern–Simons functional integral and the theory of Vassiliev invariants of knots and links in three-dimensional space. We conceptualize the functional integral in terms of equivalence classes of functionals of gauge fields and we do not use measure theory. This approach makes it possible to discuss the mathematics intrinsic to the functional integral rigorously and without functional integration. Applications to loop quantum gravity are discussed.

Connection matrices and Lie algebra weight systems for multiloop chord diagrams

We give necessary and sufficient conditions for a weight system on multiloop chord diagrams to be obtainable from a metrized Lie algebra representation, in terms of a bound on the ranks of associated connection matrices. Here a multiloop chord diagram is a graph with directed and undirected edges so that at each vertex, precisely one directed edge is entering and precisely one directed edge is leaving, and each vertex is incident with precisely one undirected edge. Weight systems on multiloop chord diagrams yield the Vassiliev invariants for knots and links. The k-th connection matrix of a function f on the collection of multiloop chord diagrams is the matrix with rows and columns indexed by k-labeled chord tangles and with entries equal to the f value on the join of the tangles.

Arrow-diagram formulas for fourth-order invariants of knots

In this paper, we study explicit arrow-diagram formulas for fourth-order Vassiliev invariants of knots announced by several authors. We show that, in fact, these formulas do not determine any knot invariants.

FINITE TYPE INVARIANTS OF LINKS WITH A FIXED LINKING MATRIX

In this paper we introduce two theories of finite type invariants for framed links with a fixed linking matrix. We show that these theories are different from, but related to, the theory of Vassiliev invariants of knots and links. We will take special note of the case of zero linking matrix. i.e., zero-framed algebraically split links. We also study the corresponding spaces of "chord diagrams".

On knot adjacency

A knot K is called n-adjacent to the unknot if it admits a projection that contains n disjoint single crossings such that changing any 0<m⩽n of these crossings, yields a projection of the unknot. Using a result of Gabai [D. Gabai, J. Differential Geom. 26 (1987) 445–503] we characterize knots that are n-adjacent to the unknot as these obtained from the unknot by n “finger moves” determined by a certain kind of trivalent graphs (Brunnian Suzuki n-graphs). Using this characterization we derive vanishing results about abelian invariants as well as Vassiliev invariants of knots that are n-adjacent to the unknot. Finally, we partially settle a conjecture of [Kalfagianni, X.-S. Lin, Preprint, 1999].

Vassiliev invariants and the cubical knot complex

We construct a cubical CW-complex CK( M) whose rational cohomology algebra contains Vassiliev invariants of knots in the 3-manifold M. We construct CK( R 3) by attaching cells to CK( R 3) for every degenerate 1-singular and 2-singular knot, and we show that π 1( CK( R 3))=1 and π 2( CK( R 3))= Z . We give conditions for Vassiliev invariants to be nontrivial in cohomology. In particular, for R 3 we show that v 2 uniquely generates H 2( CK, D), where D is the subcomplex of degenerate singular knots. More generally, we show that any Vassiliev invariant coming from the Conway polynomial is nontrivial in cohomology. The cup product in H ∗(CK) provides a new graded commutative algebra of Vassiliev invariants evaluated on ordered singular knots. We show how the cup product arises naturally from a cocommutative differential graded Hopf algebra of ordered chord diagrams.

Vassiliev invariants of knots in a spatial graph

Dedicated to Professor Kazuaki Kobayashi for his 60th birthday. We show that the Vassiliev invariants of the knots contained in an embedding of a graph G into R 3 satisify certain equations that are independent of the choice of the embedding of G. By a similar observation we define certain edge-homotopy invariants andvertex-homotopy invariants of spatial graphs basedon the Vassiliev invariants of the knots containedin a spatial graph. A graph G is calledad aptable if, given a knot type for each cycle of G, there is an embedding of G into R 3 that realizes all of these knot types. As an application we show that a certain planar graph is not adaptable.

On a Hopf Algebra in Graph Theory

We introduce and start the study of a bialgebra of graphs, which we call the 4- bialgebra , and of the dual bialgebra of 4- invariants . The 4-bialgebra is similar to the ring of graphs introduced by W. T. Tutte in 1946, but its structure is more complicated. The roots of the definition are in low dimensional topology, namely, in the recent theory of Vassiliev knot invariants. In particular, 4-invariants of graphs determine Vassiliev invariants of knots. The relation between the two notions is discussed.

Chromatic weight systems and the corresponding knot invariants

We prove that chromatic weight systems, introduced by Chmutov, Duzhin and Lando, can be expressed in terms of weight systems associated with direct sums of the Lie algebras \({\mathfrak{gl}}_n\) and \({\mathfrak{so}}_n\). As a consequence the Vassiliev invariants of knots corresponding to the chromatic weight systems distinguish exactly the same knots as a one-variable specialization \(\Upsilon\) of the Homfly and Kauffman polynomial.

On the number of chord diagrams

We present some combinatorial results in counting various kinds of s.c. chord diagrams. Latter are basic objects in the theory of Vassiliev invariants for knots.

THE INTERSECTION GRAPH CONJECTURE FOR LOOP DIAGRAMS

The study of Vassiliev invariants for knots can be reduced to the study of the algebra of chord diagrams modulo certain relations (as done by Bar-Natan). Chmutov, Duzhin and Lando defined the idea of the intersection graph of a chord diagram, and conjectured that these graphs determine the equivalence class of the chord diagrams. They proved this conjecture in the case when the intersection graph is a tree. This paper extends their proof to the case when the graph contains a single loop, and determines the size of the subalgebra generated by the associated "loop diagrams." While the conjecture is known to be false in general, the extent to which it fails is still unclear, and this result helps to answer that question.

ON VASSILIEV INVARIANTS NOT COMING FROM SEMISIMPLE LIE ALGEBRAS

We prove a refinement of Vogel's statement that the Vassiliev invariants of knots coming from semisimple Lie algebras do not generate all Vassiliev invariants. This refinement takes into account the second grading on the Vassiliev invariants induced by cabling of knots. As an application we get an amelioration of the actually known lower bounds for the dimensions of the space of Vassiliev invariants.

Vassiliev invariants: A new framework for quantum gravity

We show that Vassiliev invariants of knots, appropriately generalized to the spin network context, are loop differentiable in spite of being diffeomorphism invariant. This opens the possibility of defining rigorously the constraints of quantum gravity as geometrical operators acting on the space of Vassiliev invariants of spin nets. We show how to explicitly realize the diffeomorphism constraint on this space and present proposals for the construction of Hamiltonian constraints.

ENUMERATION OF CHORD DIAGRAMS AND AN UPPER BOUND FOR VASSILIEV INVARIANTS

We treat an enumeration problem of chord diagrams, which is shown to yield an upper bound for the dimension of the space of Vassiliev invariants for knots. We give an asymptotical estimate for this bound. As an aside, we present a trivial proof for the bound D!.