Related papers: Every Reidemeister move is needed for each knot ty…

Polyak proved that all oriented versions of Reidemeister moves for knot and link diagrams can be generated by a set of just four oriented Reidemeister moves, and that no fewer than four oriented Reidemeister moves generate them all. We…

The Reidemeister theorem states that any link in $3$-space can be encoded by a diagram (a suitably decorated projection) on a plane, and provides a finite set of combinatorial moves relating two diagrams of the same link up to isotopy. In…

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,…

For each diagram $D$ of a $2$-knot, we provide a way to construct a new diagram $D'$ of the same knot such that any sequence of Roseman moves between $D$ and $D'$ necessarily involves branch points. The proof is done by developing the…

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…

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…

Any knot diagram can be transformed into the unknot by a series of unknotting operations. This paper introduces the diagonal move, a novel unknotting operation that generalizes and unifies several existing moves. We prove that the diagonal…

Let K and K' be 2-knots. Suppose that K and K' are ribbon-move equivalent. Then the Farber-Levine pairing for K is equivalent to that for K' and the (Z-)torsion part of the first Alexander module of $K$ is isomorphic to that of K' as Z[Z]…

Both classical and virtual knots arise as formal Gauss diagrams modulo some abstract moves corresponding to Reidemeister moves. If we forget about both over/under crossings structure and writhe numbers of knots modulo the same Reidemeister…

Every classical or virtual knot is equivalent to the unknot via a sequence of extended Reidemeister moves and the so-called forbidden moves. The minimum number of forbidden moves necessary to unknot a given knot is an invariant we call the…

We propose some natural generalizations of Reidemeister moves that do not increase the number of crossings in the generated diagrams. Experimentations make us conjecture that this class of monotonic moves is complete for computing canonical…

Two knots in three-space are S-equivalent if they are indistinguishable by Seifert matrices. We show that S-equivalence is generated by the doubled-delta move on knot diagrams. It follows as a corollary that a knot has trivial Alexander…

In this paper, we study knot diagrams for which the underlying graph has treewidth two. We give a linear time algorithm for the following problem: given a knot diagram of treewidth two, does it represent the unknot? We also show that for a…

In this paper, we prove that given two cubical links of dimension two in ${\mathbb R}^4$, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the…

We consider knot theories possessing a {\em parity}: each crossing is decreed {\em odd} or {\em even} according to some universal rule. If this rule satisfies some simple axioms concerning the behaviour under Reidemeister moves, this leads…

We give a generating set of the generalized Reidemeister moves for oriented singular links. We use it to introduce an algebraic structure arising from the study of oriented singular knots. We give some examples, including some…

We study the number of Reidemeister type III moves using Fox n-colorings of knot diagrams.

The Knight Move Conjecture claims that the Khovanov homology of any knot decomposes as direct sums of some "knight move" pairs and a single "pawn move" pair. This is true for instance whenever the Lee spectral sequence from Khovanov…

We define a local move for knots and links called the {\em one-two-way pass-move}, abbreviated briefly as the {\em $1$-$2$-move}. The $1$-$2$-move is motivated from the pass-move and the $\#$-move, and it is a hybrid of them. We show that…

Polyak proved that the set $\{\Omega1a,\Omega1b,\Omega2a,\Omega3a\}$ is a minimal generating set of oriented Reidemeister moves. One may distinguish between forward and backward moves, obtaining $32$ different types of moves, which we call…