We introduce two sequences of two-variable polynomials {L n K (t, )} ∞ n=1 and {F n K (t, )} ∞ n=1 , expressed in terms of index value of a crossing and n-dwrithe value of a virtual knot K, where t and are variables. Basing on the fact that n-dwrithe is a flat virtual knot invariant we prove that L n K and F n K are virtual knot invariants containing Kauffman affine index polynomial as a particular case. Using L n K we give sufficient conditions when virtual knot does not admit cosmetic crossing change.

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in [Formula: see text]. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using [Formula: see text]-move and forbidden moves. In this paper, we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high-dimensional simplex in both the Gordian complexes, i.e. by region crossing change and by the arc shift move. For any given knot (respectively, virtual knot) diagram we construct an infinite family of knots (respectively, virtual knots) such that any two distinct members of the family have distance one by region crossing change (respectively, arc shift move). We show that the constructed virtual knots have the same affine index polynomial.

Abstract. We discuss polynomial representations for 2-bridge knots and determine the minimal degree sequence for all such knots. We apply the connection between rational tangles and 2-bridge knots.1. Introduction. It is known that, up to ambient isotopy, every noncompact knot is equivalent to some polynomial knot [10]. In fact, one can say that the set of all non-compact knot types is the same as the set of all polynomial knot types because the isotopy between two equivalent polynomial knots can be given by a one-parameter family of polynomial embeddings. Thus, for a given knot type K (always non-compact for us) there exist three real polynomials f (t), g(t) and h(t) such that the map t → (f (t), g(t), h(t)) represents K. If deg f (t) = l, deg g(t) = m and deg h(t) = n, then we say that the triple (l, m, n) is a degree sequence of K. We define (l, m, n) to be the minimal degree sequence for the knot type K if it is minimal amongst all degree sequences of K with respect to the lexicographic ordering in N 3 . A knot can be well understood if its minimal degree sequence is known. In fact a polynomial knot with the minimal degree sequence has a diagram that cannot be reduced further. Thus, to determine the minimal degree sequence of a given knot type is an interesting problem.In our earlier papers we have found a degree sequence for all torus knots ([6], [7]) and also the minimal degree sequence for torus knots of type (2, 2n + 1) ([7]). The minimal degree sequence for a general torus knot of type (p, q) is still not known, except for the special case when q = 2p − 1 ([5]). For convenience we state some known partial results:

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F-polynomials for virtual knots were defined by Kaur, Prabhakar and Vesnin in 2018 using flat virtual knot invariants. These polynomials naturally generalize Kauffman’s affine index polynomial and use smoothing in the classical crossing of a virtual knot diagram. In this paper, we introduce weight functions for ordered orientable virtual and flat virtual links. A flat virtual link is an equivalence class of virtual links with respect to a local symmetry changing a type of classical crossing in a diagram. By considering three types of smoothing in classical crossings of a virtual link diagram and suitable weight functions, there is provided a recurrent construction for new invariants. It is demonstrated by explicit examples that newly defined polynomial invariants are stronger than F-polynomials.