Related papers: A polynomial upper bound on Reidemeister moves

We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in $S^2$ and in $\R^2$.

We provide an explicit upper bound on the number of Reidemeister moves required to pass between two diagrams of the same link. This leads to a conceptually simple solution to the equivalence problem for links.

We provide an upper bound on the number of ordered Reidemeister moves required to pass between two diagrams of the same link. This bound is in terms of the number of unordered Reidemeister moves required.

In this paper, a link diagram is said to be minimal if no Reidemeister move I or II can be applied to it to reduce the number of crossings. We show that for an arbitrary diagram D of a link without a trivial split component, a minimal…

Using unknotting number, we introduce a link diagram invariant of Hass and Nowik type, which changes at most by 2 under a Reidemeister move. As an application, we show that a certain infinite sequence of diagrams of the trivial…

For each link type $K$ in the 3-sphere, we show that there is a polynomial $p_K$ such that any two diagrams of $K$ with $c_1$ and $c_2$ crossings differ by at most $p_K(c_1) + p_K(c_2)$ Reidemeister moves. As a consequence, the problem of…

We present three "hard" diagrams of the unknot. They require (at least) three extra crossings before they can be simplified to the trivial unknot diagram via Reidemeister moves in $\mathbb{S}^2$. Both examples are constructed by applying…

If a rectangular diagram represents the trivial knot, then it can be deformed into the trivial rectangular diagram with only four edges by a finite sequence of merge operations and exchange operations, without increasing the number of…

There is a positive constant $c_1$ such that for any diagram $D$ representing the unknot, there is a sequence of at most $2^{c_1 n}$ Reidemeister moves that will convert it to a trivial knot diagram, $n$ is the number of crossings in $D$. A…

Consider a robot that remembers only the starting position and walks along a knot once on a knot diagram, switching every undercrossing it meets until it returns to the starting position. We observe that the robot produces an ascending…

We first prove that, infinitely many pairs of trivial knot diagrams that are transformed into each other by applying Reidemeister moves I and III are NOT transformed into each other by a sequence of the Reidemeister moves I that increase…

Deformations of knots and links in ambient space can be studied combinatorially on their diagrams via local modifications called Reidemeister moves. While it is well-known that, in order to move between equivalent diagrams with Reidemeister…

Let $D$ be a knot diagram, and let ${\mathcal D}$ denote the set of diagrams that can be obtained from $D$ by crossing exchanges. If $D$ has $n$ crossings, then ${\mathcal D}$ consists of $2^n$ diagrams. A folklore argument shows that at…

In this study of the Reidemeister moves within the classical knot theory, we focus on hard diagrams of knots and links, categorizing them as either rigid or shaky based on their adaptability to certain moves. We establish that every link…

We prove that deciding if a diagram of the unknot can be untangled using at most $k$ Riedemeister moves (where $k$ is part of the input) is NP-hard. We also prove that several natural questions regarding links in the $3$-sphere are NP-hard,…

We introduce an up-down coloring of a virtual-link diagram. The colorabilities give a lower bound of the minimum number of Reidemeister moves of type II which are needed between two 2-component virtual-link diagrams. By using the notion of…

We define a set of restricted Reidemeister moves and show that if $K$ is obtained from $K_0\,\#\,K_1$ using those moves, then the crossing number of $K$ is at least $c(K_0)+c(K_1)$. We also explore topological interpretations of this…

In the previous paper, we considered a link diagram invariant of Hass and Nowik type using regular smoothing and unknotting number, to estimate the number of Reidemeister moves needed for unlinking. In this paper, we introduce a new link…

This note has an experimental nature and contains no new theorems. We introduce certain moves for classical knot diagrams that for all the very many examples we have tested them on give a monotonic complete simplification. A complete…

We introduce the non-self OU sequence and the OU number for link diagrams. Using these, we give a lower bound for the number of necessary Reidemeister moves of type III between two diagrams of the same link.