Narrow Results

A local move is a pair of tangles with same end points. Habiro defined a system of local moves, C_n-moves, and showed that two knots have the same Vassiliev invariants of order ≤ n - 1 if and only if they are transformed into each other by C_n-moves. We define a local move, β_n-move, which is obtained from a C_n-move by duplicating a single pair of arcs with same end points. Then we immediately have that a C_n+1-move is realized by a β_n-move and that a β_n-move is realized by twice C_n-moves. In this note we study the relation between C_n-move and β_n-move, and in particular, give answers to the following questions: (1) Is a β_n-move realized by a finite sequence of C_n+1-moves? (2) Is C_n-move realized by a finite sequence of β_n-moves?

We define a local move called a CF-move on virtual link diagrams, and show that any virtual knot can be deformed into a trivial knot by using generalized Reidemeister moves and CF-moves. Moreover, we define a new virtual link invariant n(L) for a virtual 2-component link L whose virtual linking number is an integer. Then we give necessary and sufficient conditions for two virtual 2-component links to be deformed into each other by using generalized Reidemeister moves and CF-moves in terms of a virtual linking number and n(L).

Gusarov and Habiro introduced a C_m move, that is strongly related to Vassiliev invariants. In this note, we study a special kind of C_m move, called a non-self C_m move. We show that two links can be transformed into each other by a finite sequence of non-self C_m moves if and only if (1) the two links can be transformed into each other by a finite sequence of C_m moves, and (2) the knot types of corresponding components coincide.

We show that any nontrivial reduced knot projection can be obtained from a trefoil projection by a finite sequence of half-twisted splice operations and their inverses such that the result of each step in the sequence is reduced.

Goussarov, Polyak and Viro defined a finite type invariant and a local move called an n-variation for virtual knots. In this paper, we give the differences of the values of the finite type invariants of degree 2 and 3 between two virtual knots which can be transformed into each other by a 2- and 3-variation, respectively. As a result, we obtain lower bounds of the distance between long virtual knots by 2-variations and the distance between virtual knots by 3-variations by using the values of the finite type invariants of degree 2 and 3, respectively.

We introduce a local move on a link diagram named a region freeze crossing change which is close to a region crossing change, but not the same. We study similarity and difference between region crossing change and region freeze crossing change.

We consider a local move, denoted by $\lambda$, on knot diagrams or virtual knot diagrams.If two (virtual) knots $K_1$ and $K_2$ are transformed into each other by a finite sequence of $\lambda$ moves, the $\lambda$ distance between $K_1$ and $K_2$ is the minimum number of times of $\lambda$ moves needed to transform $K_1$ into $K_2$. By ${\mathrm{\Gamma }}_{\lambda }(K)$, we denote the set of all (virtual) knots which can be transformed into a (virtual) knot $K$ by $\lambda$ moves. A geodesic graph for ${\mathrm{\Gamma }}_{\lambda }(K)$ is the graph which satisfies the following: The vertex set consists of (virtual) knots in ${\mathrm{\Gamma }}_{\lambda }(K)$ and for any two vertices $K_1$ and $K_2$, the distance on the graph from $K_1$ to $K_2$ coincides with the $\lambda$ distance between $K_1$ and $K_2$. When we consider virtual knots and a crossing change as a local move $\lambda$, we show that the $N$-dimensional lattice graph for any given natural number $N$ and any tree are geodesic graphs for ${\mathrm{\Gamma }}_{\lambda }(K)$.

We study the effects of certain local moves on Homflyptand Kauffman polynomials. We show that all Homflypt(or Kauffman) polynomials are equal in a certain nontrivial quotient of the Laurent polynomial ring. As a consequence, we discover some new properties of these invariants.

We extend a notion, an unknotting operation for knots, to a spatial embedding of a graph and study local moves on a diagram of a spatial graph.

The simplest potential counterexample to Y. Nakanishi's, 20 year old, conjecture that the 4-move is an unknotting operation is unknotted using 4-moves (Problem 1.59, 3a in the Kirby Problem List) thus showing it to not be one.