On realizabilty of Gauss diagrams and constructions of meanders

The problem concerning which Gauss diagrams can be realized by knots is an old one and has been solved in several ways. In this paper, we present a direct approach to this problem. We show that the needed conditions for realizability of a Gauss diagram can be interpreted as follows “the number of exits = the number of entrances” and the sufficient condition is based on the Jordan curve theorem. Further, using matrices, we redefine conditions for realizability of Gauss diagrams and then we give an algorithm to construct meanders.

A note on coverings of virtual knots

For a virtual knot [Formula: see text] and an integer [Formula: see text], the [Formula: see text]-covering [Formula: see text] is defined by using the indices of chords on a Gauss diagram of [Formula: see text]. In this paper, we prove that for any finite set of virtual knots [Formula: see text], there is a virtual knot [Formula: see text] such that [Formula: see text], [Formula: see text], and otherwise [Formula: see text].

Virtual Seifert surfaces

A virtual knot that has a homologically trivial representative [Formula: see text] in a thickened surface [Formula: see text] is said to be an almost classical (AC) knot. [Formula: see text] then bounds a Seifert surface [Formula: see text]. Seifert surfaces of AC knots are useful for computing concordance invariants and slice obstructions. However, Seifert surfaces in [Formula: see text] are difficult to construct. Here, we introduce virtual Seifert surfaces of AC knots. These are planar figures representing [Formula: see text]. An algorithm for constructing a virtual Seifert surface from a Gauss diagram is given. This is applied to computing signatures and Alexander polynomials of AC knots. A canonical genus of AC knots is also studied. It is shown to be distinct from the virtual canonical genus of Stoimenow–Tchernov–Vdovina.

Parity conditions for realizability of Gauss diagrams

We consider a problem of realizability of Gauss diagrams by closed plane curves where the plane curves have only double points of transversal self-intersection. We formulate the necessary and sufficient conditions for realizability. These conditions are based only on the parity of double and triple intersections of the chords in the Gauss diagram.

Enumeration of ribbon 2-knots presented by virtual arcs with up to four crossings

We enumerate ribbon 2-knots presented by virtual arc diagrams with up to four classical crossings. We use a linear Gauss diagram for a virtual arc diagram.

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.

On Gauss diagrams of periodic virtual knots

In this paper, we study the Gauss diagrams for periodic virtual knots (Theorem 3.1) and show that the virtual knot corresponding to a periodic Gauss diagram is equivalent to the periodic virtual knot whose factor is the virtual knot corresponding to the factor Gauss diagram (Theorem 3.2). We give formulae for the writhe polynomial and the affine index polynomial of periodic virtual knots by using those of factor knots (Corollary 4.2, Corollary 4.6).

PARITY AND PROJECTION FROM VIRTUAL KNOTS TO CLASSICAL KNOTS

We construct various functorial maps (projections) from virtual knots to classical knots. These maps are defined on diagrams of virtual knots; in terms of Gauss diagram each of them can be represented as a deletion of some chords. The construction relies upon the notion of parity. As corollaries, we prove that the minimal classical crossing number for classical knots. Such projections can be useful for lifting invariants from classical knots to virtual knots. Different maps satisfy different properties.

VASSILIEV INVARIANTS FROM PARITY MAPPINGS

Parity mappings (weights) from the chords of a Gauss diagram to the integers are defined. The parity of the chords is used to construct families of invariants of Gauss diagrams and consequently, virtual knots. Each family forms a set of degree n Vassiliev invariants for n ≥ 1.

INVARIANTS OF CLOSED BRAIDS VIA COUNTING SURFACES

A Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. In this paper we present simple formulas for an infinite family of invariants in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram associated with a closed braid. We then identify the resulting invariants with partial derivatives of the HOMFLY-PT polynomial.