Polyak–viro Formulas for Coefficients of the Conway Polynomial

Chmutov, Sergei; Khoury, Michael Cap; Rossi, Alfred

doi:10.1142/s0218216509007154

Cited by 17 publications

(29 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let G be a Gauss diagram of a string link , see Figure 2(right) for an example. Kravchenko and Polyak [18], consider the following element of a quotient algebra Dias(n) [18, p. 306] of ZA j where the product is defined via tree grafting and relations 6 correspond to Loday's axioms of disassociative 5 We assume a slightly different convention than in [18], see Remark 4.1. 6 essentially encoding the Reidemeister moves algebras [22],…”

Section: Tree Invariantsmentioning

confidence: 99%

Tree invariants and Milnor linking numbers with indeterminacy

Komendarczyk

Michaelides

2020

J. Knot Theory Ramifications

View full text Add to dashboard Cite

The paper concerns the tree invariants of string links, introduced by Kravchenko and Polyak and closely related to the classical Milnor linking numbers also known asμ-invariants. We prove that, analogously as forμ-invariants, certain residue classes of tree invariants yield link homotopy invariants of closed links. The proof is arrow diagramatic and provides a more geometric insight into the indeterminacy through certain tree stacking operations. Further, we show that the indeterminacy of tree invariants is consistent with the original Milnor's indeterminacy. For practical purposes, we also provide a recursive procedure for computing arrow polynomials of tree invariants.

show abstract

Section: Tree Invariantsmentioning

confidence: 99%

Tree invariants and Milnor linking numbers with indeterminacy

Komendarczyk

Michaelides

2020

J. Knot Theory Ramifications

View full text Add to dashboard Cite

show abstract

“…The three loop invariant is an invariant which is given by a Gauss diagram formula [27] with labeled pairs of regions (see below for definition). Much work has been done on finding Gauss diagram formulae for classical and virtual knot invariants [8,7,9,14,4,13]. The aim of this section is to define the three loop invariant, prove that it is an invariant of virtual knots, and provide some examples of its computation.…”

Section: The Three Loop Isotopy Invariantmentioning

confidence: 99%

The three loop isotopy and framed isotopy invariants of virtual knots

Chrisman¹,

Dye²

2014

Topology and its Applications

View full text Add to dashboard Cite

This paper introduces two virtual knot theory "analogues" of a well-known family of invariants for knots in thickened surfaces: the Grishanov-Vassiliev finite-type invariants of order two. The first, called the three loop isotopy invariant, is an invariant of virtual knots while the second, called the three loop framed isotopy invariant, is a regular isotopy invariant of framed virtual knots. The properties of these invariants are investigated at length. In addition, we make precise the informal notion of "analogue". Using this formal definition, it is proved that a generalized three loop invariant is a virtual knot theory analogue of a generalization of the Grishanov-Vassiliev invariants of order two.Theorem 2. Let i, j, k ∈ N ∪ {0} be distinct non-negative integers (i.e. i = j = k = i). Then φ i,j,k : Z[K] → Z is an invariant of virtual knots.Proof. It will be shown that φ i,j,k is invariant under the three Reidemeister moves.

show abstract

“…There are many known extension of the Conway polynomial to virtual knots and virtual long knots. Some of them satisfy a straightforward generalization of the skein relation [4] while some do not [20]. Recently, Chmutov, Khoury, and Rossi showed that there exist two natural extensions of the Conway polynomial ∇ asc and ∇ desc to virtual long knots (see also, [5]) which satisfy a certain skein relation.…”

Section: Property 3: F M -Labelled Conway Polynomialmentioning

confidence: 99%

“…Related work for other knot polynomials has been done by Chmutov and Polyak [5] and Brandenbursky and Polyak [2]. The invariant ∇ asc may be defined as follows [4]. Let D be a Gauss diagram on R and x an arrow of D. We consider R to be identified with the x-axis in R 2 where it bounds the lower half-plane.…”

Section: Property 3: F M -Labelled Conway Polynomialmentioning

confidence: 99%

A lattice of finite-type invariants of virtual knots

Chrisman

2014

Banach Center Publ.

View full text Add to dashboard Cite

Abstract. We construct an infinite commutative lattice of groups whose dual spaces give Kauffman finite-type invariants of long virtual knots. The lattice is based "horizontally" upon the Polyak algebra and extended "vertically" using Manturov's functorial map f . For each n, the n-th vertical line in the lattice contains an infinite dimensional subspace of Kauffman finite-type invariants of degree n. Moreover, the lattice contains infinitely many inequivalent extensions of the Conway polynomial to long virtual knots, all of which satisfy the same skein relation. Bounds for the rank of each group in the lattice are obtained.

show abstract

Polyak–viro Formulas for Coefficients of the Conway Polynomial

Cited by 17 publications

References 3 publications

Tree invariants and Milnor linking numbers with indeterminacy

Tree invariants and Milnor linking numbers with indeterminacy

The three loop isotopy and framed isotopy invariants of virtual knots

A lattice of finite-type invariants of virtual knots

Contact Info

Product

Resources

About