Search

This paper presents evidence supporting the surprising conjecture that in thetopological category the slice genus of a satellite knot $P(K)$ is boundedabove by the sum of the slice genera of $K$ and $P(U)$. Our main resultestablishes this conjecture for a variant of the topological slice genus, the$\mathbb{Z}$-slice genus. As an application, we show that the $(n,1)$-cable ofany 3-genus 1 knot (e.g. the figure 8 or trefoil knot) has topological slicegenus at most 1. Further, we show that the lower bounds on the slice genuscoming from the Tristram-Levine and Casson-Gordon signatures cannot be used todisprove the conjecture. Notably, the conjectured upper bound does not involvethe algebraic winding number of the pattern $P$. This stands in stark contrastwith the smooth category, where for example there are many genus 1 knots whose$(n,1)$-cables have arbitrarily large smooth 4-genera.

In this paper, we compute explicitly the Khovanov polynomials (using the computer program from katlas.org) for the two simplest families of the satellite knots, which are the twisted Whitehead doubles and the two-strand cables. We find that a quantum group decomposition for the HOMFLY polynomial of a satellite knot can be extended to the Khovanov polynomial, whose quantum group properties are not manifest. Namely, the Khovanov polynomial of a twisted Whitehead double or two-strand cable (the two simplest satellite families) can be presented as a naively deformed linear combination of the pattern and companion invariants. For a given companion, the satellite polynomial “smoothly” depends on the pattern but for the “jump” at one critical point defined by the [Formula: see text]-invariant of the companion knot. A similar phenomenon is known for the knot Floer homology and [Formula: see text]-invariant for the same kind of satellites.

The ropelength of a knot is the minimum contour length of a tube of unit radius that traces out the knot in three dimensional space without self-overlap, colloquially the minimum amount of rope needed to tie a given knot. Theoretical upper and lower bounds have been established for the asymptotic relationship between crossing number and ropelength and stronger bounds have been conjectured, but numerical bounds have only been calculated exhaustively for knots and links with up to 11 crossings, which are not sufficiently complex to test these conjectures. Existing ropelength measurements also have not established the complexity required to reach asymptotic scaling with crossing number. Here, we investigate the ropelength of knots and links beyond the range of tested crossing numbers, past both the 11-crossing limit as well as the 16-crossing limit of the standard knot catalog. We investigate torus knots up to 1023 crossings, establishing a stronger upper bound for T(p,2) knots and links and demonstrating power-law scaling in T(p,p+1) below the proven limit. We investigate satellite knots up to 42 crossings to determine the effect of a systematic crossing-increasing operation on ropelength, finding that a satellite knot typically has thrice the ropelength of its companion. We find that ropelength is well described by a model of repeated Hopf links in which each component adopts a minimal convex hull around the cross-section of the others, and derive formulae that heuristically predict the crossing-ropelength relationship of knots and links without free parameters., Comment: 14 pages, 8 figures

A well-known conjecture in knot theory says that the proportion of hyperbolic knots among all of the prime knots of $n$ or fewer crossings approaches $1$ as $n$ approaches infinity. In this article, it is proved that this conjecture contradicts several other plausible conjectures, including the 120-year-old conjecture on additivity of the crossing number of knots under connected sum and the conjecture that the crossing number of a satellite knot is not less than that of its companion.

We show an infinite family of satellite knots that can be unknotted by a single band move, but such that there is no band unknotting the knots which is disjoint from the satellite torus., Comment: 10 pages, 5 figures

Let $\Sigma_0,\Sigma_1$ be closed oriented surfaces. Two oriented knots $K_0 \subset \Sigma_0 \times [0,1]$ and $K_1 \subset \Sigma_1 \times [0,1]$ are said to be (virtually) concordant if there is a compact oriented $3$-manifold $W$ and a smoothly and properly embedded annulus $A$ in $W \times [0,1]$ such that $\partial W=\Sigma_1 \sqcup -\Sigma_0$ and $\partial A=K_1 \sqcup -K_0$. This notion of concordance, due to Turaev, is equivalent to concordance of virtual knots, due to Kauffman. A prime virtual knot, in the sense of Matveev, is one for which no thickened surface representative $K \subset \Sigma \times [0,1]$ admits a nontrivial decomposition along a separating vertical annulus that intersects $K$ in two points. Here we prove that every knot $K \subset \Sigma \times [0,1]$ is concordant to a prime satellite knot and a prime hyperbolic knot. For homologically trivial knots in $\Sigma \times [0,1]$, we prove this can be done so that the Alexander polynomial is preserved. This generalizes the corresponding results for classical knot concordance, due to Bleiler, Kirby-Lickorish, Livingston, Myers, Nakanishi, and Soma. The new challenge for virtual knots lies in proving primeness. Contrary to the classical case, not every hyperbolic knot in $\Sigma \times [0,1]$ is prime and not every composite knot is a satellite. Our results are obtained using a generalization of tangles in $3$-balls we call complementary tangles. Properties of complementary tangles are studied in detail., Comment: 32 pages, 25 figures; v2--typos corrected, some proofs streamlined

Let K be a nontrivial knot in S 3 and t ( K ) its tunnel number. For any ( p ≥ 2 , q ) -slope in the torus boundary of a closed regular neighborhood of K in S 3 , denoted by K ⋆ , it is a nontrivial cable knot in S 3 . Though t ( K ⋆ ) ≤ t ( K ) + 1 , Example 1.1 in Section 1 shows that in some case, t ( K ⋆ ) ≤ t ( K ) . So it is interesting to know when t ( K ⋆ ) = t ( K ) + 1 . After using some combinatorial techniques, we prove that (1) for any nontrivial cable knot K ⋆ and its companion K, t ( K ⋆ ) ≥ t ( K ) ; (2) if either K admits a high distance Heegaard splitting or p / q is far away from a fixed subset in the Farey graph, then t ( K ⋆ ) = t ( K ) + 1 . Using the second conclusion, we construct a satellite knot and its companion so that the difference between their tunnel numbers is arbitrary large.

We study the knot invariant called trunk, as defined by Ozawa, and the relation of the trunk of a satellite knot with the trunk of its companion knot. Our first result is ${\rm trunk}(K) \geq n \cdot {\rm trunk}(J)$ where ${\rm trunk}(\cdot)$ denotes the trunk of a knot, $K$ is a satellite knot with companion $J$, and $n$ is the winding number of $K$. To upgrade winding number to wrapping number, which we denote by $m$, we must include an extra factor of $\frac{1}{2}$ in our second result ${\rm trunk}(K) > \frac{1}{2} m\cdot {\rm trunk}(J)$ since $m \geq n$. We also discuss generalizations of the second result., 21 pages, 5 figures

Since its inception, the notion of thin position has played an important role in low-dimensional topology. Thin position for knots in the 3-sphere was first introduced by David Gabai in order to prove the Property R Conjecture. In addition, this theory factored into Cameron Gordon and John Luecke’s proof of the knot complement problem and revolutionized the study of Heegaard splittings upon its adaptation by Martin Scharlemann and Abigail Thompson. Let h : S → R be a Morse function with two critical points. Loosely, thin position of a knot K in S is a particular embedding of K which minimizes the total number of intersections with a maximal collection of regular level sets, where this number of intersections is called the width of the knot. Although not immediately obvious, it has been demonstrated that there is a close relationship between a thin position of a knot K and essential meridional planar surfaces in its exterior E(K). In this thesis, we study the nature of thin position under knot companionship; namely, for several families of knots we establish a lower bound for the width of a satellite knot based on the width of its companion and the wrapping or winding number of its pattern. For one such class of knots, cable knots, in addition to finding thin position for these knots, we establish a criterion under which non-minimal bridge positions of cable knots are stabilized. Finally, we exhibit an embedding of the unknot whose width must be increased before it can be simplified to thin position.

In this paper, we give a proof of a conjecture which says that [math] , where [math] is the width of a knot, [math] is a satellite knot with [math] as its companion, and [math] is the winding number of the pattern. We also show that equality holds if [math] is a satellite knot with braid pattern.

We show that the problem of recognizing that a knot diagram represents a specific torus knot, or any torus knot at all, is in the complexity class ${\sf NP} \cap {\sf co\text{-}NP}$, assuming the generalized Riemann hypothesis. We also show that satellite knot detection is in ${\sf NP}$ under the same assumption, and that cabled knot detection and composite knot detection are unconditionally in ${\sf NP}$. Our algorithms are based on recent work of Kuperberg and of Lackenby on detecting knottedness., 23 pages; v2: reorganized section 2, other minor changes

We prove that links with meridional rank 3 whose 2-fold branched covers are graph manifolds are 3-bridge links. This gives a partial answer to a question by S. Cappell and J. Shaneson on the relation between the bridge numbers and meridional ranks of links. To prove this, we also show that the meridional rank of any satellite knot is at least 4., Comment: 16 pages, 15 figures

Let L = K1 ∪ K2 be a 2-component link in the 3-sphere such that K1 is a trivial knot. In this paper, we introduce a new bridge index, denoted by bK1 = 1([L]), for L. Roughly speaking, bK1 = 1([L]) is the minimum of the bridge numbers of the links ambient isotopic to L under the constraint that all of the bridge numbers of the components corresponding to K1 are 1. We provide a lower bound estimate of bK1 = 1([L]) in the case when L is a non-split satellite link. By using this result, we show that for each integer n(≥ 2), there exists a link Ln = K1n ∪ K2n with K1n a trivial knot such that bK1n = 1([Ln])−b([Ln]) = n−1, where b([Ln]) is the bridge index of Ln.

本文は「URL」欄から参照してください。

This paper generalizes Van Cott's bounds on the Heegaard–Floer τ-invariant and Rasmussen's s-invariant of cabled knots to apply to all satellite knots.

Let [Formula: see text] be a nontrivial knot in [Formula: see text]. It was conjectured that there exists a Neuwirth surface for [Formula: see text]. That is, a closed surface in [Formula: see text] containing the knot [Formula: see text] as a nonseparating curve and such that every compressing disk for the surface intersects the knot in at least two points. We provide explicit constructions of Neuwirth surfaces for a family of satellite knots, which do not depend on the existence of nonorientable algebraically incompressible and [Formula: see text]-incompressible spanning surfaces for these knots.

In a recent work "Arc-presentation of links: Monotonic simplification", Dynnikov shows that each rectangular diagram of the unknot, composite link, or split link can be monotonically simplified into a trivial, composite, or split diagram, respectively. The following natural question arises: Is it always possible to simplify monotonically a rectangular diagram of a satellite knot or link into one where the satellite structure is seen? Here we give a negative answer to that question both for knot and link cases. An example of a torus embedding that cannot be obtained from ordinary "thin" torus by methods of paper [2] is also constructed.

A torti-rational knot, denoted by K(2α, β|r), is a knot obtained from the 2-bridge link B(2α, β) by applying Dehn twists an arbitrary number of times, r, along one component of B(2α, β). We determine the genus of K(2α, β|r) and solve a question of when K(2α, β|r) is fibered. In most cases, the Alexander polynomials determine the genus and fiberedness of these knots. We develop both algebraic and geometric techniques to describe the genus and fiberedness by means of continued fraction expansions of β/2α. Then, we explicitly construct minimal genus Seifert surfaces. As an application, we solve the same question for the satellite knots of tunnel number one.

We show that there exist infinitely many pairs of distinct knots in the 3-sphere such that each pair can yield homeomorphic lens spaces by the same Dehn surgery. Moreover, each knot of the pair can be chosen to be a torus knot, a satellite knot or a hyperbolic knot, except that both cannot be satellite knots simultaneously. This exception is shown to be unavoidable by the classical theory of binary quadratic forms., 21 pages, 11 figures. Some errors in References are revised in version 2

We give sufficient conditions for a satellite knot to admit an L-space surgery, and use this result to give new infinite families of patterns which produce satellite L-space knots., 8 pages, 2 figures

Any knot in a solid torus, called a pattern or satellite operator, acts on knots in the 3-sphere via the satellite construction. We introduce a generalization of satellite operators which form a group (unlike traditional satellite operators), modulo a generalization of concordance. This group has an action on the set of knots in homology spheres, using which we recover the recent result of Cochran and the authors that satellite operators with strong winding number $\pm 1$ give injective functions on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4--dimensional Poincare Conjecture. The notion of generalized satellite operators yields a characterization of surjective satellite operators, as well as a sufficient condition for a satellite operator to have an inverse. As a consequence, we are able to construct infinitely many non-trivial satellite operators P such that there is a satellite operator $\overline{P}$ for which $\overline{P}(P(K))$ is concordant to K (topologically as well as smoothly in a potentially exotic $S^3\times [0,1]$) for all knots K; we show that these satellite operators are distinct from all connected-sum operators, even up to concordance, and that they induce bijective functions on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4--dimensional Poincare Conjecture., 20 pages, 9 figures; in the second version, we have added several new results about surjectivity of satellite operators, and inverses of satellite operators, and the exposition and structure of the paper have been improved

Let sQ be the satellite operation on knots defined by a pattern (V, Q), where V is a standard solid torus in S3 and Q ⊂ V is a knot that is geometrically essential in V. It is known (Kuperberg [5]) that if v is any knot invariant of order n ≥ 0, then v ◦ sQ is also a knot invariant of order ≤ n. We show that if the knot Q has the winding number zero in V, then the satellite map [Formula: see text] passes n-equivalent knots into (n + 1)-equivalent ones. Kalfagianni [4] has defined for each nonnegative integer n surgery n-trivial knots and studied their properties. It is known that for each n every surgery n-trivial knot is n-trivial. We show that for each n there are n-trivial knots which do not admit a non-unitary n-trivializer that show they to be surgery n-trivial. Przytycki showed [12] that if a knot Q is trivial in S3 and is embedded in V in such a way that it is k-trivial inside V and if a knot [Formula: see text] is m-trivial, then the satellite knot [Formula: see text] is (k + m + 1)-trivial. We establish a version of aforementioned Przytycki's result for surgery n-triviality, refining thus a construction for surgery n-trivial knots suggested by Kalfagianni.

Let K be a knot in the 3-sphere S 3 , and a disk in S 3 meeting K transversely in the interior. For non-triviality we assume that | \ K| 2 over all isotopies of K in S 3 @ . Let K ,n( S 3 ) be a knot obtained from K by n twistings along the disk . If the original knot is unknotted in S 3 , we call K ,n a twisted knot. We describe for which pair (K,) and an integer n, the twisted knot K ,n is a torus knot, a satellite knot or a hyperbolic knot.

Let V be the standard solid torus in S3. Let Kp, 2 be the (p, 2)-torus knot in V such that Kp, 2 meets a meridian disk D of V in two points with the winding number zero and the 2-string tangle TKp, 2 obtained by cutting along D is a rational tangle. We compute the Casson invariant of the cyclic covering space of S3 branched over a satellite knot whose companion is any 2-bridge knot D(b1,…,b2m) and pattern is (V, Kp, 2).

We consider compact 3-manifolds M having a submersion h to R in which each generic point inverse is a planar surface. The standard height function on a submanifold of the 3-sphere is a motivating example. To (M, h) we associate a connectivity graph G. For M in the 3-sphere, G is a tree if and only if there is a Fox reimbedding of M which carries horizontal circles to a complete collection of complementary meridian circles. On the other hand, if the connectivity graph of the complement of M is a tree, then there is a level-preserving reimbedding of M so that its complement is a connected sum of handlebodies. Corollary: The width of a satellite knot is no less than the width of its pattern knot. In particular, the width of K_1 # K_2 is no less than the maximum of the widths of K_1 and K_2., Comment: 23 pages, 13 figures

Let K be a knot in a closed orientable irreducible 3-manifold M. Suppose M admits a genus 1 Heegaard splitting and we denote by H the splitting torus. We say H is a 1-genus 1-bridge splitting of (M, K) if H intersects K transversely in two points, and divides (M, K) into two pairs of a solid torus and a boundary parallel arc in it. It is known that a 1-genus 1-bridge splitting of a satellite knot admits a satellite diagram disjoint from an essential loop on the splitting torus. If M = S 3 and the slope of the loop is longitudinal in one of the solid tori, then K is obtained by twisting a component of a 2-bridge link along the other component. We give a criterion for determining whether a given 1-genus 1-bridge splitting of a knot admits a satellite diagram of a given slope or not. As an application, we show there exist counter examples for a conjecture of Ait Nouh and Yasuhara.

Let K be a knot in the 3-sphere S3, and D a disk in S3 meeting K transversely in the interior. For non-triviality we assume that |D∩K|⩾2 over all isotopies of K in S3−∂D. Let KD,n(⊂S3) be the knot obtained from K by n twisting along the disk D. If the original knot is unknotted in S3, we call KD,n a twisted unknot. We describe for which pairs (K,D) and integers n, the twisted unknot KD,n is a torus knot, a satellite knot or a hyperbolic knot. To cite this article: M. Ait Nouh et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Dean introduced twisted torus knots, which are obtained from a torus knot and a torus link by splicing them together along a number of adjacent strands of each of them. We study the knot types of these knots.

We characterize satellite knots of free genus one, and know that the set of satellite knots of free genus one coincides with that of satellite knots of tunnel number one.

The notion of an (n, r)-coloring for a link diagram generalizes the idea of an n-coloring introduced by R. H. Fox. For any positive integer n the various (n, r)-colorings of a diagram for an oriented link l correspond in a natural way to the periodic points of the representation shift ΦZ/n(l) of the link. The number of (n, r)-colorings of a diagram for a satellite knot is determined by the colorings of its pattern and companion knots together with the winding number.

Let $P(K)$ be a satellite knot where the pattern, $P$, is a Berge-Gabai knot (i.e., a knot in the solid torus with a non-trivial solid torus Dehn surgery), and the companion, $K$, is a non-trivial knot in $S^3$. We prove that $P(K)$ is an L-space knot if and only if $K$ is an L-space knot and $P$ is sufficiently positively twisted relative to the genus of $K$. This generalizes the result for cables due to Hedden and the first author., 14 pages, 2 figures

In this paper we establish a relation between an appropriate version of the Casson-Gordon invariant of a satellite knot and those of its orbit and companion. We note that in some cases the contribution from, the companion falls. This gives a way to construct algebraically but not smoothly slice knots.

We evaluate all the primitive divergences contributing to the 7--loop $\beta$\/--function of $\phi^4$ theory, i.e.\ all 59 diagrams that are free of subdivergences and hence give scheme--independent contributions. Guided by the association of diagrams with knots, we obtain analytical results for 56 diagrams. The remaining three diagrams, associated with the knots $10_{124}$, $10_{139}$, and $10_{152}$, are evaluated numerically, to 10 sf. Only one satellite knot with 11 crossings is encountered and the transcendental number associated with it is found. Thus we achieve an analytical result for the 6--loop contributions, and a numerical result at 7 loops that is accurate to one part in $10^{11}$. The series of `zig--zag' counterterms, $\{6\zeta_3,\,20\zeta_5,\, \frac{441}{8}\zeta_7,\,168\zeta_9,\,\ldots\}$, previously known for $n=3,4,5,6$ loops, is evaluated to 10 loops, corresponding to 17 crossings, revealing that the $n$\/--loop zig--zag term is $4C_{n-1} \sum_{p>0}\frac{(-1)^{p n - n}}{p^{2n-3}}$, where $C_n=\frac{1}{n+1}{2n \choose n}$ are the Catalan numbers, familiar in knot theory. The investigations reported here entailed intensive use of REDUCE, to generate ${\rm O}(10^4)$ lines of code for multiple precision FORTRAN computations, enabled by Bailey's MPFUN routines, running for ${\rm O}(10^3)$ CPUhours on DecAlpha machines., Comment: 6 pages plain LaTex

For certain knots J in S 1 × D 2 , the dual knot J ∗ in S 1 × D 2 is defined. Let J ( O ) be the satellite knot of the unknot O with pattern J , and K be the satellite of J ( O ) with pattern J ∗ . The knot K then bounds a smooth disk in a 4-ball, but is not obviously a ribbon knot. We show that K is, in fact, ribbon. We also show that the connected sum J(O) # J ∗ (O) is a nonribbon knot for which all known algebraic obstructions to sliceness vanish.

Let P be a knot in a solid torus, K a knot in 3-space and P(K) the satellite knot of K with pattern P. This defines an operator on the set of knot types and induces a satellite operator P:C--> C on the set of smooth concordance classes of knots. There has been considerable interest in whether certain such functions are injective. For example, it is a famous open problem whether the Whitehead double operator is weakly injective (an operator is called weakly injective if P(K)=P(0) implies K=0 where 0 is the class of the trivial knot). We prove that, modulo the smooth 4-dimensional Poincare Conjecture, any strong winding number one satellite operator is injective on C. More precisely, if P has strong winding number one and P(K)=P(J), then K is smoothly concordant to J in S^3 x [0,1] equipped with a possibly exotic smooth structure. We also prove that any strong winding number one operator is injective on the topological knot concordance group. If P(0) is unknotted then strong winding number one is the same as (ordinary) winding number one. More generally we show that any satellite operator with non-zero winding number n induces an injective function on the set of Z[1/n]-concordance classes of knots. We extend some of our results to links., 16 pages; in second version we have added some results on operators on links and string links and added some references to connections with Mazur manifolds and corks; third version has only very minor changes and will appear in Journal of Topology

We show that if K is a satellite knot which admits a generalized cosmetic crossing change of order q with |q| \geq 6, then K admits a pattern knot with a generalized cosmetic crossing change of the same order. As a consequence of this, we find that any prime satellite knot which admits a pattern knot that is fibered cannot admit a generalized cosmetic crossing changes of order q with |q| \geq 6. We also show that if there is any knot admitting a generalized cosmetic crossing change of order q with |q| \geq 6, then there must be such a knot which is hyperbolic., Comment: 13 pages, 4 figures, a correction was made on page 12