Arc shift move and region arc shift move for twisted knots

In this paper, we study the unknotting operation for twisted knots, called arc shift move. First, we find a family of twisted knots with arc shift number $n$ for any given $n \in \mathbb{N}$. Then we define a new unknotting operation, called the region arc shift move for twisted knots and find family of twisted knots whose region arc shift number is less than or equal to $n$ for any given $n \in \mathbb{N}$. Later, we explore bounds for region arc shift number and forbidden number.

💡 Research Summary

This paper investigates unknotting operations for twisted knots, focusing on the arc shift move and its regional extension, the region arc shift move. Twisted knots are generalizations of virtual knots that include classical crossings, virtual crossings, and bars on arcs. The authors first recall the extended Reidemeister moves (R₁‑R₃, V₁‑V₄, T₁‑T₃) that define diagrammatic equivalence, and they introduce the forbidden moves (F₁‑F₄, T₄) which, when allowed, trivialize any Gauss diagram.

The arc shift move is defined on an “arc” that connects exactly two crossings (classical or virtual). Depending on whether a bar lies on the arc, there are two types of moves (Type 1 without a bar, Type 2 with a bar). Applying an arc shift cuts the arc near its endpoints, swaps the loose ends, and inserts a virtual crossing; the order and signs of the two original crossings are altered. The arc shift number A(K) of a twisted knot K is the minimal number of such moves (together with the extended Reidemeister moves) needed to transform any diagram of K into a trivial twisted knot (either the unknot without a bar or with a single bar). A(K) is shown to be a knot invariant, and the odd writhe J(K) provides a lower bound: J(K) ≤ A(K).

To demonstrate that any natural number n can occur as an arc shift number, the authors construct a family {K₁,…,Kₙ}. Each knot Kₙ is built from n identical blocks Bᵢ, each containing three classical crossings (cᵢ, dᵢ, d′ᵢ) and a bar. By analyzing the Gauss diagram, they compute the index of each crossing: ind(dᵢ)=−2n+4i−3, ind(d′ᵢ)=−2n+4i−2, ind(cᵢ)=1. Consequently, all dᵢ and cᵢ are odd crossings, giving J(Kₙ)=2n. Lemma 2.7 then yields A(Kₙ)≥n. A concrete unknotting sequence is exhibited: one arc shift applied to each block reduces the block to two parallel strands; after n such moves, Kₙ becomes the trivial knot without a bar. Hence A(Kₙ)=n.

The paper also presents two distinct unknotting sequences for the same range of n. In the first sequence the final trivial knot has no bar; in the second it retains a single bar. Both sequences consist of knots Kₙ and K′ₙ whose arc shift numbers equal n. The distinction between the two families is verified using the polynomial invariant Q(s,t), defined via over‑ and under‑affine indices and the parity of bars on smoothed diagrams. Explicit calculations show that Q distinguishes Kₙ from K′ₙ, confirming that the families are genuinely different despite sharing the same arc shift number.

Next, the authors generalize the region arc shift (R.A.S.) move from virtual knots to twisted knots. A region is a connected component of the 4‑valent graph obtained by placing a vertex at each classical and virtual crossing. The R.A.S. move consists of performing an arc shift on every arc that bounds a chosen region simultaneously. Proposition 4.1 proves that applying the same R.A.S. move twice restores the original diagram, mirroring the virtual‑knot case. Crucially, the authors demonstrate that each forbidden move (F₁/F₂, F₃/F₄, and T₄) can be realized by a single R.A.S. move on an appropriately chosen region, using a combination of Type 1 and Type 2 arc shifts. Figures 21‑23 illustrate these realizations. Consequently, the R.A.S. move is an unknotting operation: any sequence of forbidden moves can be replaced by a sequence of region arc shifts.

Building on this, the paper introduces two new numerical invariants. The forbidden number F(K) is the minimal number of forbidden moves required to unknot K, while the region arc shift number R(K) is the minimal number of region arc shifts needed. By examining how Q(s,t) changes under each forbidden move, the authors obtain lower bounds for F(K). They also relate R(K) to the previously known invariants: since each region arc shift can be simulated by a bounded number of ordinary arc shifts, R(K) ≤ c·A(K) for some constant c, and because each forbidden move can be realized by a region arc shift, F(K) ≤ R(K). These inequalities provide a hierarchy of unknotting complexities for twisted knots.

Overall, the paper makes several significant contributions: (1) it constructs explicit families of twisted knots realizing any prescribed arc shift number, (2) it defines and validates the region arc shift move as a comprehensive unknotting operation for twisted knots, (3) it introduces and bounds new invariants (forbidden number and region arc shift number) using the polynomial Q(s,t) and existing invariants such as odd writhe, and (4) it demonstrates that these new invariants refine our understanding of the unknotting complexity beyond what is captured by the arc shift number alone. The results enrich the toolkit for studying twisted knots and suggest further avenues, such as exploring algorithmic computation of R(K) and F(K), or extending these concepts to higher‑dimensional knotted objects.