On unknotting operations of rotation type

2015 ◽  
Vol 24 (10) ◽  
pp. 1540009
Author(s):  
Yongju Bae ◽  
Byeorhi Kim
Keyword(s):  
Finite Sequence ◽  
Reidemeister Moves

An unknotting operation is a local move on a knot diagram such that any knot diagram can be transformed into a diagram of the unknot by a finite sequence of the operations and Reidemeister moves. In this paper, we introduce a new local move H(T) on a knot diagram which is obtained by the rotation of a tangle diagram T and study their properties. As an application, we prove that the H(T)-move is an unknotting operation for any descending tangle diagram T.

scholarly journals (1, 2) AND WEAK (1, 3) HOMOTOPIES ON KNOT PROJECTIONS

2013 ◽  
Vol 22 (14) ◽  
pp. 1350085 ◽  
Author(s):  
NOBORU ITO ◽  
YUSUKE TAKIMURA
Keyword(s):  
Equivalence Relation ◽  
Finite Sequence ◽  
Reidemeister Move ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Homotopy Classes ◽  
The Third ◽  
Reidemeister Moves ◽  
Necessary And Sufficient

In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 2.2). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. 3). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary 4.1).

FORBIDDEN MOVES UNKNOT A VIRTUAL KNOT

2001 ◽  
Vol 10 (01) ◽  
pp. 89-96 ◽  
Author(s):  
TAIZO KANENOBU
Keyword(s):  
Finite Sequence ◽  
Virtual Knot ◽  
Reidemeister Moves

We show that any virtual knot can be deformed into a trivial knot by a finite sequence of generalized Reidemeister moves and "forbidden moves". We also show that a Δ-unknotting operation is realized by a finite sequence of generalized Reidemeister moves and "forbidden moves".

scholarly journals CUBULATED MOVES AND DISCRETE KNOTS

2013 ◽  
Vol 22 (14) ◽  
pp. 1350079 ◽  
Author(s):  
GABRIELA HINOJOSA ◽  
ALBERTO VERJOVSKY ◽  
CYNTHIA VERJOVSKY MARCOTTE
Keyword(s):  
Cyclic Permutation ◽  
Finite Sequence ◽  
The Other ◽  
Reidemeister Moves

In this paper, we prove that two cubic knots K1, K2 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 Reidemeister moves for classical tame knots. We use this fact to describe a cubic knot in a discrete way, as a cyclic permutation of contiguous vertices of the ℤ3-lattice (with some restrictions); moreover, we describe a regular diagram of a cubic knot in terms of such cyclic permutations.

scholarly journals Realizing exterior Cromwell moves on rectangular diagrams by Reidemeister moves

2014 ◽  
Vol 23 (05) ◽  
pp. 1450023
Author(s):  
Tatsuo Ando ◽  
Chuichiro Hayashi ◽  
Yuki Nishikawa
Keyword(s):  
Upper Bound ◽  
Knot Theory ◽  
Finite Sequence ◽  
Reidemeister Moves ◽  
Recognition Algorithms ◽  
Russian Math

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 edges, which was shown by Dynnikov in [Arc-presentations of links: Monotone simplification, Fund. Math. 190 (2006) 29–76; Recognition algorithms in knot theory, Uspekhi Mat. Nauk 58 (2003) 45–92. Translation in Russian Math. Surveys 58 (2003) 1093–1139]. Using this, Henrich and Kauffman gave in [Unknotting unknots, preprint (2011), arXiv:1006.4176v4 [math.GT]] an upper bound for the number of Reidemeister moves needed for unknotting a knot diagram of the trivial knot. However, exchange or merge moves on the top and bottom pairs of edges of rectangular diagrams are not considered in [Unknotting unknots, preprint (2011), arXiv:1006.4176v4 [math.GT]]. In this paper, we show that there is a rectangular diagram of the trivial knot which needs such an exchange move for being unknotted, and study upper bound of the number of Reidemeister moves needed for realizing such an exchange or merge move.

Theory of models with generalized atomic formulas

1960 ◽  
Vol 25 (1) ◽  
pp. 1-26 ◽  
Author(s):  
H. Jerome Keisler
Keyword(s):  
Natural Number ◽  
Finite Sequence ◽  
Sufficient Condition ◽  
Elementary Extension ◽  
Necessary And Sufficient Condition ◽  
Elementary Class ◽  
Relational Systems ◽  
Necessary And Sufficient

IntroductionWe shall prove the following theorem, which gives a necessary and sufficient condition for an elementary class to be characterized by a set of sentences having a prescribed number of alternations of quantifiers. A finite sequence of relational systems is said to be a sandwich of order n if each is an elementary extension of (i ≦ n—2), and each is an extension of (i ≦ n—2). If K is an elementary class, then the statements (i) and (ii) are equivalent for each fixed natural number n.

A note on Spitzer's lemma and its application to the maximum of partial sums of dependent random variables

1976 ◽  
Vol 13 (2) ◽  
pp. 361-364 ◽  
Author(s):  
M. E. Solari ◽  
J. E. A. Dunnage
Keyword(s):  
Infinite Sequence ◽  
Random Variables ◽  
Finite Sequence ◽  
Partial Sums ◽  
Dependent Random Variables ◽  
Finite Segment ◽  
Exchangeable Random Variables ◽  
Partial Sum

We give an expression for the expectation of max (0, S1, …, Sn) where Sk is the kth partial sum of a finite sequence of exchangeable random variables X1, …, Xn. When the Xk are also independent, the formula we give has already been obtained by Spitzer; and when the sequence is a finite segment of an infinite sequence of exchangeable random variables, it is a consequence of a theorem of Hewitt.

SAFE ETERNAL 1-SECURE SETS IN GRAPHS

2021 ◽  
Vol 10 (5) ◽  
pp. 2537-2548
Author(s):  
K.R. Kumar ◽  
E.N. Satheesh
Keyword(s):  
Directed Graphs ◽  
Finite Sequence ◽  
Dominating Sets ◽  
Secure Set

An eternal $1$-secure set, in a graph $G = (V, E)$ is a set $D \subset V$ having the property that for any finite sequence of vertices $r_1, r_2, \ldots, r_k$ there exists a sequence of vertices $v_1, v_2, \ldots, v_k$ and a sequence $ D = D_0, D_1, D_2, \ldots, D_k$ of dominating sets of $G$, such that for each $i$, $1 \leq i \leq k$, $D_{i} = (D_{i-1} - \{v_i\}) \cup \{r_i\}$, where $v_i \in D_{i-1}$ and $r_i \in N[v_i]$. Here $r_i = v_i$ is possible. The cardinality of the smallest eternal $1$-secure set in a graph $G$ is called the eternal $1$-security number of $G$. In this paper we study a variations of eternal $1$-secure sets named safe eternal $1$-secure sets. A vertex $v$ is safe with respect to an eternal $1$-secure set $S$ if $N[v] \bigcap S =1$. An eternal 1 secure set $S$ is a safe eternal 1 secure set if at least one vertex in $G$ is safe with respect to the set $S$. We characterize the class of graphs having safe eternal $1$-secure sets for which all vertices - excluding those in the safe $1$-secure sets - are safe. Also we introduce a new kind of directed graphs which represent the transformation from one safe 1 - secure set to another safe 1-secure set of a given graph and study its properties.

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