Narrow Results

In this paper, we give a definition of $\mathbb{Z}$-valued functions from the ambient isotopy classes of spherical/plane curves derived from chord diagrams, denoted by ${\sum }_i{\alpha }_i{x}_i$. Then, we introduce certain elements of the free $\mathbb{Z}$-module generated by the chord diagrams with at most $l$ chords, called relators of Type (I) ((SII), (WII), (SIII), or (WIII), respectively), and introduce another function ${\sum }_i{\alpha }_i{\tilde{x}}_i$ derived from ${\sum }_i{\alpha }_i{x}_i$. The main result (Theorem 1) shows that if ${\sum }_i{\alpha }_i{\tilde{x}}_i$ vanishes for the relators of Type (I) ((SII), (WII), (SIII), or (WIII), respectively), then ${\sum }_i{\alpha }_i{x}_i$ is invariant under the Reidemeister move of type RI (strong RII, weak RII, strong RIII, or weak RIII, respectively) that is defined in [N. Ito and Y. Takimura, $(1,2)$ and weak $(1,3)$ homotopies on knot projections, J. Knot Theory Ramifications22 (2013) 1350085 14 pp].

We show that any two bridge positions of a handlebody-knot are stably equivalent.

In this paper a classification of Reidemeister moves, which is the most refined, is introduced. In particular, this classification distinguishes some Ω₃-moves that only differ in how the three strands that are involved in the move are ordered on the knot.

To transform knot diagrams of isotopic knots into each other one must in general use Ω₃-moves of at least two different classes. To show this, knot diagram invariants that jump only under Ω₃-moves are introduced.

Knot diagrams of isotopic knots can be connected by a sequence of Reidemeister moves of only six, out of the total of 24, classes. This result can be applied in knot theory to simplify proofs of invariance of diagrammatical knot invariants. In particular, a criterion for a function on Gauss diagrams to define a knot invariant is presented.

How many Reidemeister moves do we need for unknotting a given diagram of the trivial knot? The absolute value of the writhe gives a lower bound of the number of Reidemeister I moves. That of a complexity of knot diagram "cowrithe" works for Reidemeister II, III moves.

In Appendix A, we give an example of an infinite sequence of diagrams D_n of the trivial knot with an O(n) number of crossings such that the author expects the number of Reidemeister moves needed for unknotting it to be O(n²). However, writhe and cowrithe do not prove this.

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten–Reshetikhin–Turaev invariant of three manifolds.

A knot diagram has an associated looped interlacement graph, obtained from the intersection graph of the Gauss diagram by attaching loops to the vertices that correspond to negative crossings. This construction suggests an extension of the Kauffman bracket to an invariant of looped graphs, and an extension of Reidemeister equivalence to an equivalence relation on looped graphs. The graph bracket polynomial can be defined recursively using the same pivot and local complementation operations used to define the interlace polynomial, and it gives rise to a graph Jones polynomial V_G(t) that is invariant under the graph Reidemeister moves.

It is well known that there is a one-to-one correspondence between signed plane graphs and link diagrams via the medial construction. The component number of the corresponding link diagram is however independent of the signs of the plane graph. Determining this number may be one of the first problems in studying links by using graphs. Some works in this aspect have been done. In this paper, we investigate the component number of links corresponding to lattices. Firstly we provide some general results on component number of links. Then, via these results, we proceed to determine the component number of links corresponding to lattices with free or periodic boundary conditions and periodic lattices with one cap (i.e. spiderweb graphs) or two caps.

Let D be an oriented classical or virtual link diagram with directed universe . Let C denote a set of directed Euler circuits, one in each connected component of U. There is then an associated looped interlacement graph whose construction involves very little geometric information about the way D is drawn in the plane; consequently is different from other combinatorial structures associated with classical link diagrams, like the checkerboard graph, which can be difficult to extend to arbitrary virtual links. is determined by three things: the structure of as a 2-in, 2-out digraph, the distinction between crossings that make a positive contribution to the writhe and those that make a negative contribution, and the relationship between C and the directed circuits in arising from the link components; this relationship is indicated by marking the vertices where C does not follow the incident link component(s). We introduce a bracket polynomial for arbitrary marked graphs, defined using either a formula involving matrix nullities or a recursion involving the local complement and pivot operations; the marked-graph bracket of is the same as the Kauffman bracket of D. This provides a unified combinatorial description of the Jones polynomial that applies seamlessly to both classical and non-classical virtual links.

In earlier work we introduced the graph bracket polynomial of graphs with marked vertices, motivated by the fact that the Kauffman bracket of a link diagram D is determined by a looped, marked version of the interlacement graph associated to a directed Euler system of the universe graph of D. Here we extend the graph bracket to graphs whose vertices may carry different kinds of marks, and we show how multiply marked graphs encode interlacement with respect to arbitrary (undirected) Euler systems. The extended machinery brings together the earlier version and the graph-links of Ilyutko and Manturov [J. Knot Theory Ramifications18 (2009) 791–823]. The greater flexibility of the extended bracket also allows for a recursive description much simpler than that of the earlier version.

In the present paper we construct an equivalence between the set of graph-knots and the set of homotopy classes of looped graphs. Moreover, the graph-knot and the homotopy class constructed from a given knot are related by this equivalence. This equivalence is given by a simple formula.

Arnold introduced invariants J⁺, J^- and St for generic planar curves. It is known that both J⁺/2 + St and J^-/2 + St are invariants for generic spherical curves. Applying these invariants to underlying curves of knot diagrams, we can obtain lower bounds for the number of Reidemeister moves required for unknotting. J^- /2 + St works well to count the minimum number of unmatched RII moves. However, it works only up to a factor of two for RI moves. Let w denote the writhe for a knot diagram. We show that J^-/2 + St ± w/2 also gives sharp counts for the number of required RI moves, and demonstrate that it gives a precise estimate for a certain family of diagrams of the unknot with the underlying curve r = 2 + cos(nθ/(n + 1)), (0 ≤ θ ≤ 2(n + 1)π).

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.

A ribbon chord diagram, or simply a chord diagram, of a ribbon surface-link in the 4-space is introduced. Links, virtual links and welded virtual links can be described naturally by chord diagrams with the corresponding moves, respectively. Some moves on chord diagrams are introduced by overseeing these special moves. Then the faithful equivalence on ribbon surface-links is stated in terms of the moves on chord diagrams. This answers questions by Nakanishi and Marumoto affirmatively. The faithful TOP-equivalence on ribbon surface-links derives the same result. By combining a previous result on TOP-triviality of a surface-knot, a ribbon surface-knot is DIFF-trivial if and only if the fundamental group is an infinite cyclic group. This corrects an erroneous proof in Yanagawa's old paper.

Recently, the author discovered an interesting class of knot-like objects called free knots. These purely combinatorial objects are equivalence classes of Gauss diagrams modulo Reidemeister moves (the same notion in the language of words was introduced by Turaev [Topology of words, Proc. Lond. Math. Soc.95(3) (2007) 360–412], who thought all free knots to be trivial). As it turned out, these new objects are highly nontrivial, see [V. O. Manturov, Parity in knot theory, Mat. Sb.201(5) (2010) 65–110], and even admit nontrivial cobordism classes [V. O. Manturov, Parity and cobordisms of free knots, Mat. Sb.203(2) (2012) 45–76]. An important issue is the existence of invariants where a diagram evaluates to itself which makes such objects "similar" to free groups: An element has its minimal representative which "lives inside" any representative equivalent to it. In this paper, we consider generalizations of free knots by means of (finitely presented) groups. These new objects have lots of nontrivial properties coming from both knot theory and group theory. This connection allows one not only to apply group theory to various problems in knot theory but also to apply Reidemeister moves to the study of (finitely presented) groups. Groups appear naturally in this setting when graphs are embedded in surfaces.

In 2006 C. Hayashi gave a lower bound for the number of Reidemeister moves in deformation of two equivalent knot diagrams by using writhe and cowrithe. It can be naturally extended for two virtually isotopic virtual knot diagrams. We introduce a polynomial q_K(t) of a virtual knot diagram K and give lower bounds for the number of Reidemeister moves in deformation of two virtually isotopic knots by using q_K(t). We give an example which shows that the polynomial q_K(t) is useful to map out a sequence of Reidemeister moves to deform a virtual knot diagram to another virtually isotopic one.

We introduce an up–down coloring of a virtual-link (or classical-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 (or classical-link) diagrams. By using the notion of a quandle cocycle invariant, we give a method to detect the necessity of Reidemeister moves of type II between two given virtual-knot (or classical-knot) diagrams. As an application, we show that for any virtual-knot diagram $D$, there exists a diagram $D^{\prime }$ representing the same virtual-knot such that any sequence of generalized Reidemeister moves between them includes at least one Reidemeister move of type II.

A string figure is topologically a trivial knot lying on an imaginary plane orthogonal to the fingers with some crossings. The fingers prevent cancellation of these crossings. As a mathematical model of string figure, we consider a knot diagram on the $xy$-plane in $xyz$-space missing some straight lines parallel to the $z$-axis. These straight lines correspond to fingers. We study minimal number of crossings of these knot diagrams under Reidemeister moves missing these lines.

We give a definition of an integer-valued function ${\sum }_i{\alpha }_i{x}_i^{\ast }$ derived from arrow diagrams for the ambient isotopy classes of oriented spherical curves. Then, we introduce certain elements of the free $\mathbb{Z}$-module generated by the arrow diagrams with at most $l$ arrows, called relators of Type ($\check{\mathrm{\text{I}}}$) (($\check{\mathrm{\text{SII}}}$), ($\check{\mathrm{\text{WII}}}$), ($\check{\mathrm{\text{SIII}}}$) or ($\check{\mathrm{\text{WIII}}}$), respectively), and introduce another function ${\sum }_i{\alpha }_i{\tilde{x}}_i^{\ast }$ to obtain ${\sum }_i{\alpha }_i{x}_i^{\ast }$. One of the main results shows that if ${\sum }_i{\alpha }_i{\tilde{x}}_i^{\ast }$ vanishes on finitely many relators of Type ($\check{\mathrm{\text{I}}}$) (($\check{\mathrm{\text{SII}}}$), ($\check{\mathrm{\text{WII}}}$), ($\check{\mathrm{\text{SIII}}}$) or ($\check{\mathrm{\text{WIII}}}$), respectively), then ${\sum }_i{\alpha }_i{x}_i^{\ast }$ is invariant under the deformation of type RI (strong RII, weak RII, strong RIII or weak RIII, respectively). The other main result is that we obtain new functions of arrow diagrams with up to six arrows explicitly. This computation is done with the aid of computers.

This paper mainly studies the minimum number of colorings for all non-trivially 19-colored diagrams of any 19-colorable knot K. By using some special Reidemeister move, we successfully eliminated 13 colors from 19 colors. It can be seen that for any 19-colorable knot K, at least six colors are enough to color K, that is, the minimum number of 19-colorable knot is six.

We introduce a ribbon presentation to describe a ribbon knot, and equivalences between them. Superspun knots of classical knots are well known to be ribbon knots, and we give a construction of ribbon presentations of the knots by means of classical knot diagrams. We can then extend the classical Reidemeister moves into higher dimensions, and we prove that the presentations of a superspun knot are stably equivalent by using these moves.