A knot k in a closed orientable 3-manifold is called nonsimple if the exterior of k possesses a properly embedded essential surface of nonnegative Euler characteristic. We show that if k is a nonsimple prime tunnel number one knot in a lens space M (where M does not contain any embedded Klein bottles), then k is a (1,1) knot. Elements of the proof include handle addition and Dehn filling results/techniques of Jaco, Eudave-Munoz and Gordon as well as structure results of Schultens on the Heegaard splittings of graph manifolds.
The genus of satellite tunnel number one knots and torti-rational knots is computed using the tools introduced by Floyd and Hatcher. An implementation of an algorithm is given to compute genus and slopes of minimal genus Seifert surfaces for such knots.
We give lower bounds for the tunnel number of knots and handlebody-knots. We also give a lower bound for the cutting number, which is a "dual" notion to the tunnel number in the handlebody-knot theory. We provide necessary conditions for constituent handlebody-knots by using $G$-family of quandles colorings. The above two evaluations are obtained as the corollaries. Furthermore, we construct handlebody-knots with arbitrary tunnel number and cutting number.
这是三篇论文中的第二篇，它们完善和扩展了我们早期预印本的部分内容，“结隧道的深度”。“他们一起重新制作了整个预印本。我们在“The tree of knot tunnels”中介绍的1号隧道结理论产生了一个参数化，其中每个隧道由一个有理参数的有限序列和一个0和1的有限序列唯一描述，它们共同编码了一个构造结和隧道的过程。在本文中，我们计算这些不变量的所有隧道的环面结。
We provide criteria ensuring that a tunnel number one knot $K$ is not determined by its double branched cover, in the sense that the double branched cover is also the double branched cover of a knot $K'$ not equivalent to $K$.
This is the first of three papers that refine and extend portions of our earlier preprint, "Depth of a knot tunnel." Together, they rework the entire preprint. H. Goda, M. Scharlemann, and A. Thompson described a general construction of all tunnels of all tunnel number 1 knots using "tunnel moves". We apply the theory that we introduced in "The tree of knot tunnels" to study this construction. In particular, we use it to calculate the number of distinct minimal sequences of tunnel moves that can produce a given tunnel. As a consequence, we see that for a sparse infinite set of tunnels, the min…
We calculate the twisted Alexander polynomials of all tunnel number one Montesinos knots associated to their $SL_2(\mathbb{C})$-representations and obtain their leading coefficients and degrees. As a corollary, we get some interesting examples, that is, nonfibered knots with monic Alexander polynomials which have non-monic twisted Alexander polynomials.
Using exhaustive techniques and results from Lackenby and many others, we compute the tunnel number of all 1655 alternating 11 and 12 crossing knots and of 881 non-alternating 11 and 12 crossing knots. We also find all 5525 Montesinos knots with 14 crossings or fewer.
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。
We exhibit an infinite family of knots in the Poincare homology sphere with tunnel number 2 that have a lens space surgery. Notably, these knots are not doubly primitive and provide counterexamples to a few conjectures. In the appendix, it is shown that hyperbolic knots in the Poincare homology sphere with a lens space surgery has either no symmetries or just a single strong involution.
Let $K$ be a nontrivial knot in $S^{3}$ and $t(K)$ its tunnel number. For any $(p\geq 2,q)$-slope in the torus boundary of a closed regular neighborhood of $ K$ in $S^{3}$, denoted by $K^{\star}$, it is a nontrivial cable knot in $S^{3}$. Though $t(K^{\star})\leq t(K)+1$, Example 1.1 in Section 1 shows that in some case, $ t(K^{\star})\leq t(K)$. So it is interesting to know when $t(K^{\star})= t(K)+1$. After using some combinatorial techniques, we prove that (1) for any nontrivial cable knot $K^{\star}$ and its companion $K$, $t(K^{\star})\geq t(K)$; (2) if either $K$ admits a high distance H…
As one of the background papers of the classification project of hyperbolic primitive/Seifert knots in $S^3$ whose complete list is given in [BK20], this paper classifies all possible R-R diagrams of two disjoint simple closed curves $R$ and $β$ lying in the boundary of a genus two handlebody $H$ up to equivalence such that $β$ is a proper power curve and a 2-handle addition $H[R]$ along $R$ embeds in $S^3$ so that $H[R]$ is the exterior of a tunnel-number-one knot. As a consequence, if $R$ is a nonseparating simple closed curve on the boundary of a genus two handlebody such that $H[R]$ embeds…
Suppose $α$ and $R$ are disjoint simple closed curves in the boundary of a genus two handlebody $H$ such that $H[R]$ embeds in $S^3$ as the exterior of a hyperbolic knot $k$(thus, $k$ is a tunnel-number-one knot), and $α$ is Seifert in $H$(i.e., a 2-handle addition $H[α]$ is a Seifert-fibered space) and not the meridian of $H[R]$. Then for a slope $γ$ of $k$ represented by $α$, $γ$-Dehn surgery $k(γ)$ is a Seifert-fibered space. Such a construction of Seifert-fibered Dehn surgeries generalizes that of Seifert-fibered Dehn surgeries arising from primtive/Seifert positions of a knot, which was i…
University of Miami（迈阿密大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Humboldt-Universität zu Berlin（柏林洪堡大学）；Temple University（天普大学）；University of California, Berkeley（加州大学伯克利分校）；Hacettepe University（哈塞特佩大学）；University of Maryland（马里兰大学）；Dartmouth College（达特茅斯学院）；University of Melbourne（墨尔本大学）；University of Nebraska-Lincoln（内布拉斯加大学林肯分校）
Connected sum and trivalent vertex sum are natural operations on genus 2 spatial graphs and, as with knots, tunnel number behaves in interesting ways under these operations. We prove sharp Scharlemann-Schultens type bounds for the tunnel number of a composite genus 2 spatial graph. For the tunnel number of a composite Brunnian $θ$-curve, our result implies that the tunnel number is at least the number of summands, as in the knot case. We also prove a version of a theorem of Morimoto for knots: the tunnel number of a composite m-small genus 2 spatial graph is at least the sum of the tunnel numb…
We define two new families of invariants for (3-manifold, graph) pairs which detect the unknot and are additive under connected sum of pairs and (-1/2)-additive under trivalent vertex sum of pairs. The first of these families is closely related to both bridge number and tunnel number. The second of these families is a variation and generalization of Gabai's width for knots in the 3-sphere. We give applications to the tunnel number and higher genus bridge number of connected sums of knots.
We describe the genus two knots which admit a genus one, one bridge position. These are divided into several families, one consists of vertical bandings of two genus one $(1,1)$-knots, other consists of vertical bandings of two cross cap number two 2-bridge knots, and the last one consists of genus two tunnel number one satellite knots.
