In this paper, we introduce a new nontrivial filtration, called F-order, for classical and virtual knot invariants; this filtration produces filtered knot invariants, which are called finite type invariants similar to Vassiliev knot invariants. Finite type invariants introduced by Goussarov, Polyak, and Viro are well-known, and we call them finite type invariants of GPV-order. We show that for any positive integer $n$ and for any classical knot $K$, there exist infinitely many of nontrivial classical knots, all of whose finite type invariants of GPV-order $< n$, coincide with those of $K$ (Theorem 1). Further, we show that for any positive integer n, there exists a nontrivial virtual knot whose finite type invariants of our F-order $< n$ coincide with those of the trivial knot (Theorem 2). In order to prove Theorem 1 (Theorem 2, resp.), we define an n-triviality via a certain unknotting operation, called virtualization (forbidden moves, resp.), and for any positive integer n, find an n-trivial classical knot (virtual knot, resp.).
The (ordinary) unknotting-number of 1-dimensional knots, which is defined by using the crossing-change, is a very basic and important invariant. It is very natural to consider the `unknotting-number' associated with other local-moves on n-dimensional knots, where n is a natural number. In this paper we prove the following facts. For the ribbon-move on 2-knots, which is a kind of local-move on knots, we have the following: There is a ribbon-move-unknotting-number-two 2-knot. The ribbon-move-unknotting-number of 2-knots is unbounded. For the pass-move on 1-knots, which is a kind of local-move on…
In this paper we introduce the notion of an unknotting index for virtual knots. We give some examples of computation by using writhe invariants, and discuss a relationship between the unknotting index and the virtual knot module. In particular, we show that for any non-negative integer $n$ there exists a virtual knot whose unknotting index is $(1,n)$.
We study a local twist move on welded knots that is an analog of the virtualization move on virtual knots. Since this move is an unknotting operation we define an invariant, unknotting twist number, for welded knots. We relate the unknotting twist number with warping degree and welded unknotting number, and establish a lower bound on the twist number using Alexander quandle coloring. We also study the Gordian distance between welded knots by twist move and define the corresponding Gordian complex.
The virtual unknotting number of a virtual knot is the minimal number of crossing changes that makes the virtual knot to be the unknot, which is defined only for virtual knots virtually homotopic to the unknot. We focus on the virtual knot obtained from the standard (p,q)-torus knot diagram by replacing all crossings on one overstrand into virtual crossings and prove that its virtual unknotting number is equal to the unknotting number of the $(p,q)$-torus knot, i.e. it is (p-1)(q-1)/2.
A knot in the 3-sphere is said to have zero negative unknotting number if it can be transformed into the unknot by performing only positive crossing changes. In this paper, we provide an obstruction for a knot to having zero negative unknotting number, and discuss its application to two classes of knots.
By Thurston's hyperbolization theorem, irreducible handlebody-knots are classified into three classes: hyperbolic, toroidal, and atoroidal cylindrical. It is known that a non-trivial handlebody-knot of genus two has a finite symmetry group if and only if it is atoroidal. The paper investigates the topology of cylindrical handlebody-knots of genus two that admit an unknotting annulus; we show that the symmetry group is trivial if the unknotting annulus is unique and of type $2$.
We study relations between unknotting number and crossing number of a spatial embedding of a handcuff-graph and a theta curve. It is well known that for any non-trivial knot $K$ twice the unknotting number of $K$ is less than or equal to the crossing number of $K$ minus one. We show that this is extended to handlebody-knots. We also characterize the handlebody-knots which satisfy the equality.
Dalian University of Technology（大连理工大学）；Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences（俄罗斯科学院西伯利亚分院索博列夫数学研究所）
The unknotting number of knots is a difficult quantity to compute, and even its behavior under basic satelliting operations is not understood. We establish a lower bound on the unknotting number of cable knots and iterated cable knots purely in terms of the winding number of the pattern. The proof uses Alishahi-Eftekhary's bounds on unknotting number from knot Floer homology together with Hanselman-Watson's computation of the knot Floer homology of cables in terms of immersed curves in the punctured torus.
Vassiliev introduced filtered invariants of knots using an unknotting operation, called crossing changes. Goussarov, Polyak, and Viro introduced other filtered invariants of virtual knots, which order is called GPV-order, using an unknotting operation, called virtualization. We defined other filtered invariants, which order is called $F$-order, of virtual knots using an unknotting operation, called forbidden moves. In this paper, we show that the set of virtual knot invariants of $F$-order $\le n+1$ is strictly stronger than that of $F$-order $\le n$ and that of GPV-order $\le 2n+1$. To obtain…
Given a knot K in S^3, let u^-(K) (respectively, u^+(K)) denote the minimum number of negative (respectively, positive) crossing changes among all unknotting sequences for K. We use knot Floer homology to construct the invariants l^-(K), l^+(K) and l(K), which give lower bounds on u^-(K), u^+(K) and the unknotting number u(K), respectively. The invariant l(K) only vanishes for the unknot, and is greater than or equal to the ν^-(K). Moreover, the difference l(K)-ν^-(K) can be arbitrarily large. We also present several applications towards bounding the unknotting number, the alteration number an…
Ito-Takimura recently defined a splice-unknotting number $u^-(D)$ for knot diagrams. They proved that this number provides an upper bound for the crosscap number of any prime knot, asking whether equality holds in the alternating case. We answer their question in the affirmative. (Ito has independently proven the same result.) As an application, we compute the crosscap numbers of all prime alternating knots through at least 13 crossings, using Gauss codes.
Let $D$ be a diagram of an alternating knot with unknotting number one. The branched double cover of $S^3$ branched over $D$ is an L-space obtained by half integral surgery on a knot $K_D$. We denote the set of all such knots $K_D$ by $\mathcal D$. We characterize when $K_D\in \mathcal D$ is a torus knot, a satellite knot or a hyperbolic knot. In a different direction, we show that for a given $n>0$, there are only finitely many L-space knots in $\mathcal D$ with genus less than $n$.
We show that there is a knot satisfying the property that for each minimal crossing number diagram of the knot and each single crossing of the diagram, changing the crossing results in a diagram for a knot whose unknotting number is at least that of the original knot, thus giving a counterexample to the Bernhard-Jablan Conjecture.
We introduce a new knot diagram invariant called the Self-Crossing Index (SCI). Using SCI, we provide bounds for unknotting two families of framed unknots. For one of these families, unknotting using framed Reidemeister moves is significantly harder than unknotting using regular Reidemeister moves. We also investigate the relation between SCI and Arnold's curve invariant St, as well as the relation with Hass and Nowik's invariant, which generalizes cowrithe. In particular, the change of SCI under Ω3 moves depends only on the forward/backward character of the move, similar to how the change of …
Twisted knot theory, introduced by M.O.Bourgoin, is a generalization of virtual knot theory. It is easily shown that any virtual knot can be deformed into a trivial knot by a finite sequence of generalized Reidemeister moves and two "forbidden moves" $F_{1}$ and $F_{2}$. Similarly, we show that any twisted knot also can be deformed into a trivial knot or a trivial knot with a bar by a finite sequence of extended Reidemeister moves and three "forbidden moves" $T_{4}$, $F_{1}$ (or $F_{2}$) and $F_{3}$ (or $F_{4}$) .
In this paper, we show that the Gluck twist of certain satellite $2$-knots in a $4$-manifold do not change the diffeomorphism type in three different ways: one is directly from the definition of the satellite $2$-knot, and the other two are by finding an equivalent description of the satellite $2$-knot. Furthermore, using the new description, we gave infinite number of new examples of $2$-knots which are unknotted by connected summing a single standard real projective plane.
New lower bounds on the unknotting number of a knot are constructed from the classical knot signature function. These bounds can be twice as strong as previously known signature bounds. They can also be stronger than known bounds arising from Heegaard Floer and Khovanov homology. Results include new bounds on the Gordian distance between knots and information about four-dimensional knot invariants. By considering a related non-balanced signature function, bounds on the unknotting number of slice knots are constructed; these are related to the property of double-sliceness.
Humboldt-Universität zu Berlin（柏林洪堡大学）；Universität Heidelberg（海德堡大学）；ETH Zürich（苏黎世联邦理工学院）；Universität Regensburg（雷根斯堡大学）；Universidad de Sevilla（塞维利亚大学）
这是Gordon-Reid [Tangle decompositions of tunnel number one knots and links，J.纽结理论及其分支，4（1995）389-409]和Thompson [Thin position and bridge number for knots in the 3-sphere，Topology，36（1997）505-507]的定理的结果，如果将一个隧道号结置于薄的位置，也会在桥上。我们表明，在这样一个薄的介绍，隧道可以使水平，使它位于一个水平的领域。这解决了Morimoto提出的一个问题[A note on unknotting tunnels for 2-bridge knots，Bulletin of Faculty of Engineering Takushoku University，3（1992）219-225]，他表明，如果已知任何unknotting tunnel都可以被制成水平的，那么用于2-bridge knots的unknotting tunnel的（现在已知的）分类将很快出现。
