This is the second of three papers that refine and extend portions of our earlier preprint, "The depth of a knot tunnel." Together, they rework the entire preprint.
The theory of tunnel number 1 knots that we introduced in "The tree of knot tunnels" yields a parameterization in which each tunnel is described uniquely by a finite sequence of rational parameters and a finite sequence of 0's and 1's, that together encode a procedure for constructing the knot and tunnel. In this paper we calculate these invariants for all tunnels of torus knots.
The 2-loop polynomial is a polynomial presenting the 2-loop part of the Kontsevich invariant of knots. We show a cabling formula for the 2-loop polynomial of knots. In particular, we calculate the 2-loop polynomial for torus knots.
We continue our study of the degree of the colored Jones polynomial under knot cabling started in "Knot Cabling and the Degree of the Colored Jones Polynomial" (arXiv:1501.01574). Under certain hypothesis on this degree, we determine how the Jones slopes and the linear term behave under cabling. As an application we verify Garoufalidis' Slope Conjecture and a conjecture of the authors for cables of a two-parameter family of closed 3-braids called 2-fusion knots.
The fusion number of a ribbon knot is the minimal number of 1-handles needed to construct a ribbon disk. The strong homotopy fusion number of a ribbon knot is the minimal number of 2-handles in a handle decomposition of a ribbon disk complement. We demonstrate that these invariants behave completely differently under cabling by showing that the (p,1)-cable of any ribbon knot with fusion number one has strong homotopy fusion number one and fusion number p. Our main tools are Juhász-Miller-Zemke's bound on fusion number coming from the torsion order of knot Floer homology and Hanselman-Watson's …
In this note we study Legendrian and transverse knots in the knot type of a (p,q)-cable of a knot K in 3-sphere. We give two structural theorems that describe when the (p,q)-cable of a Legendrian simple knot type K is also Legendrian simple.
We define a nontrivial mod 2 valued additive concordance invariant defined on the torsion subgroup of the knot concordance group using involutive knot Floer package. For knots not contained in its kernel, we prove that their iterated $(\text{odd},1)$-cables have infinite order in the concordance group and, among them, infinitely many are linearly independent. Furthermore, by taking $(2,1)$-cables of the aforementioned knots, we present an infinite family of knots which are strongly rationally slice but not slice.
We say that a given knot $J\subset S^3$ is detected by its knot Floer homology and $A$-polynomial if whenever a knot $K\subset S^3$ has the same knot Floer homology and the same $A$-polynomial as $J$, then $K=J$. In this paper we show that every torus knot $T(p,q)$ is detected by its knot Floer homology and $A$-polynomial. We also give a one-parameter family of infinitely many hyperbolic knots in $S^3$ each of which is detected by its knot Floer homology and $A$-polynomial. In addition we give a cabling formula for the A-polynomials of cabled knots in $S^3$, which is of independent interest. I…
Let $k\subset S^3$ be a nontrivial knot. The Cabling Conjecture of Francisco González-Acuña and Hamish Short posits that $π$-Dehn surgery on $k$ produces a reducible manifold if and only if $k$ is a $(p,q)$-cable knot and the surgery slope $π$ equals $pq$. We extend the work of James Allen Hoffman to prove the Cabling Conjecture for knots with bridge number up to $5$.
Closed 3-string braids admit many bandings to two-bridge links. By way of the Montesinos Trick, this allows us to construct infinite families of knots in the connected sum of lens spaces L(r,1) # L(s,1) that admit a surgery to a lens space for all pairs of integers (r,s) except (0,0). These knots are typically hyperbolic. We also demonstrate that the previously known two families of examples of hyperbolic knots in non-prime manifolds with lens space surgeries of Eudave-Munoz--Wu and Kang all fit this construction. As such, we propose a generalization of the Cabling Conjecture of Gonzales-Acuna…
We classify which positive integral surgeries on positive torus knots bound rational homology balls. Additionally, for a given knot K we consider which cables K(p,q) admit integral surgeries that bound rational homology balls. For such cables, let S(K) be the set of corresponding rational numbers q/p. We show that S(K) is bounded for each K. Moreover, if n-surgery on K bounds a rational homology ball then n is an accumulation point for S(K).
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.
We prove that if positive integer p-surgery along a knot K \subset S^3 produces an L-space and it bounds a sharp 4-manifold, then the knot genus obeys the bound 2g(K) -1 \leq p - \sqrt{3p+1}. Moreover, there exists an infinite family of pairs (K_n,p_n) attaining this bound, where K_n denotes an n-fold iterated cable of the unknot and p_n \to \infty. In particular, the stated bound applies when the knot surgery produces a lens space or a connected sum thereof. Combined with work of Gordon-Luecke, Hoffman, and Matignon-Sayari, it follows that if surgery along a knot produces a connected sum of l…
We study the behavior of the degree of the colored Jones polynomial and the boundary slopes of knots under the operation of cabling. We show that, under certain hypothesis on this degree, if a knot $K$ satisfies the Slope Conjecture then a $(p, q)$-cable of $K$ satisfies the conjecture, provided that $p/q$ is not a Jones slope of $K$. As an application we prove the Slope Conjecture for iterated cables of adequate knots and for iterated torus knots. Furthermore we show that, for these knots, the degree of the colored Jones polynomial also determines the topology of a surface that satisfies the …
We examine geometric properties of a knot J that are unchanged by taking a (p,q)-cable K of J. Specifically, we relate w(K) to w(J), where w(K) is the width of K in the sense of Gabai. We use this information to demonstrate that thin position is a minimal bridge position of J if and only if the same is true for K, and more generally we show that any thin position of K is an "obvious" cabling of a thin position of J. We conclude by proving that J is meridionally small (mp-small) if and only if K is meridionally small (mp-small), and if J is mp-small and every non-minimal bridge position of J is…
We continue our study of the knot Floer homology invariants of cable knots. For large |n|, we prove that many of the filtered subcomplexes in the knot Floer homology filtration associated to the (p,pn+1) cable of a knot, K, are isomorphic to those of K. This result allows us to obtain information about the behavior of the Ozsvath-Szabo concordance invariant under cabling, which has geometric consequences for the cabling operation. Applications considered include quasipositivity in the braid group, the knot theory of complex curves, smooth concordance, and lens space (or, more generally, L-spac…
The Heegaard Floer correction term ($d$-invariant) is an invariant of rational homology 3-spheres equipped with a Spin$^c$ structure. In particular, the correction term of 1-surgeries along knots in $S^3$ is a ($2\mathbb{Z}$-valued) knot concordance invariant $d_1$. In this paper, we estimate $d_1$ for the $(2,q)$-cable of any knot $K$. This estimate does not depend on the knot type of $K$. If $K$ belongs to a certain class which contains all negative knots, then equality holds. As a corollary, we show that the relationship between $d_1$ and the Heegaard Floer $τ$-invariant is very weak in gen…
In this paper we will show how to classify Legendrian and transverse knots in the knot type of "sufficiently positive" cables of a knot in terms of the classification of the underlying knot. We will also completely explain the phenomena of "Legendrian large" cables. These are Legendrian representatives of cables that have Thurston-Bennequin invariant larger that the framing coming from the cabling torus. Such examples have only recently, and unexpectedly, been found. We will also give criteria that determines the classification of Legendrian and transverse knots the the knot type of negative c…
We study Legendrian knots in a cabled knot type. Specifically, given a topological knot type K, we analyze the Legendrian knots in knot types obtained from K by cabling, in terms of Legendrian knots in the knot type K. As a corollary of this analysis, we show that the (2,3)-cable of the (2,3)-torus knot is not transversely simple and moreover classify the transverse knots in this knot type. This is the first classification of transverse knots in a non-transversely-simple knot type. We also classify Legendrian knots in this knot type and exhibit the first example of a Legendrian knot that does …
We prove an explicit cabling formula for the colored Jones polynomial. As an application we prove the volume conjecture for all zero volume knots and links, i.e. all knots and links that are obtained from the unknot by repeated cabling and connected sum.
We prove that the (p,q)-cable of a knot K in S^3 admits a positive L-space surgery if and only if K admits a positive L-space surgery and q/p \geq 2g(K)-1, where g(K) is the Seifert genus of K. The "if" direction is due to Hedden.
This paper is devoted to the study of the knot Floer homology groups HFK(S^3,K_{2,n}), where K_{2,n} denotes the (2,n) cable of an arbitrary knot, K. It is shown that for sufficiently large |n|, the Floer homology of the cabled knot depends only on the filtered chain homotopy type of CFK(K). A precise formula for this relationship is presented. In fact, the homology groups in the top 2 filtration dimensions for the cabled knot are isomorphic to the original knot's Floer homology group in the top filtration dimension. The results are extended to (p,pn+-1) cables. As an example we compute HFK((T…
We prove that the class of topological knot types that are both Legendrian simple and satisfy the uniform thickness property (UTP) is closed under cabling. An immediate application is that all iterated cabling knot types that begin with negative torus knots are Legendrian simple. We also examine, for arbitrary numbers of iterations, iterated cablings that begin with positive torus knots, and establish the Legendrian simplicity of large classes of these knot types, many of which also satisfy the UTP. In so doing we obtain new necessary conditions for both the failure of the UTP and Legendrian n…
We determine the relationship between the contact structure induced by a fibered knot, K, in the three-sphere and the contact structures induced by its various cables. Understanding this relationship allows us to classify fibered cable knots which bound a properly embedded complex curve in the four-ball satisfying a genus constraint. This generalizes the well-known classification of links of plane curve singularities.
We prove that an iterated torus knot type fails the uniform thickness property (UTP) if and only if all of its iterations are positive cablings, which is precisely when an iterated torus knot type supports the standard contact structure. We also show that all iterated torus knots that fail the UTP support cabling knot types that are transversely non-simple.
This is the third of three papers that refine and extend portions of our earlier preprint, "The depth of a knot tunnel." Together, they rework the entire preprint. In this paper, we use the theory of tunnel number 1 knots that we introduced in "The tree of knot tunnels" to strengthen the Tunnel Leveling Theorem of H. Goda, M. Scharlemann, and A. Thompson. This yields considerable information about bridge numbers of tunnel number 1 knots. In particular, we calculate the minimum bridge number of a knot as a function of the maximum depth invariant d of its tunnels. The growth of this value is on …
We compute the Kauffman bracket skein module of the complement of a twist knot, finding that it is free and infinite dimensional. The basis consists of cables of a two-component link, one component of which is a meridian of the knot. The cabling of the meridian can be arbitrarily large while the cabling of the other component is limited to the number of twists.
The theory of tunnel number 1 knots detailed in our previous paper, The tree of knot tunnels, provides a non-negative integer invariant called the depth of the tunnel. We give various results related to the depth invariant. Noting that it equals the minimum number of Goda-Scharlemann-Thompson tunnel moves needed to construct the tunnel, we calculate the number of distinct minimal sequences of tunnel moves that can produce a given tunnel. Next, we give a recursion that tells the minimum bridge number of a knot having a tunnel of depth D. The rate of growth of this value improves the known estim…
We present a new theory which describes the collection of all tunnels of tunnel number 1 knots in the 3-sphere (up to orientation-preserving equivalence in the sense of Heegaard splittings) using the disk complex of the genus-2 handlebody and associated structures. It shows that each knot tunnel is obtained from the tunnel of the trivial knot by a uniquely determined sequence of simple cabling constructions. A cabling construction is determined by a single rational parameter, so there is a corresponding numerical parameterization of all tunnels by sequences of such parameters and some addition…
A consequence of the Cabling Conjecture of Gonzalez-Acuña and Short is that Dehn surgery on a knot in $S^3$ cannot produce a manifold with more than two connected summands. In the event that some Dehn surgery produces a manifold with three or more connected summands, then the surgery parameter is bounded in terms of the bridge number by a result of Sayari. Here this bound is sharpened, providing further evidence in favour of the Cabling Conjecture.
In this paper, we study the behavior of $Υ_K(t)$ under the cabling operation, where $Υ_K(t)$ is the knot concordance invariant defined by Ozsváth, Stipsicz, and Szabó, associated to a knot $K\subset S^3$. The main result is an inequality relating $Υ_K(t)$ and $Υ_{K_{p,q}}(t)$, which generalizes the inequalities of Hedden and Van Cott on the Ozsváth-Szabó $τ$-invariant. As applications, we give a computation of $Υ_{(T_{2,-3})_{2,2n+1}}(t)$ for $n\geq 8$, and we also show that the set of iterated $(p,1)$-cables of $Wh^{+}(T_{2,3})$ for any $p\geq 2$ span an infinite-rank summand of topologically…
This paper gives a complete classification of all alternating knots with tunnel number one, and all their unknotting tunnels. We prove that the only such knots are two-bridge knots and certain Montesinos knots.
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.
We show that twisted torus knots $T(p,q,3,s)$ are tunnel number one. A short spanning arc connecting two adjacent twisted strands is an unknotting tunnel.
In this paper we classify Legendrian and transverse knots in the knot types obtained from positive torus knots by cabling. This classification allows us to demonstrate several new phenomena. Specifically, we show there are knot types that have non-destabilizable Legendrian representatives whose Thurston-Bennequin invariant is arbitrarily far from maximal. We also exhibit Legendrian knots requiring arbitrarily many stabilizations before they become Legendrian isotopic. Similar new phenomena are observed for transverse knots. To achieve these results we define and study "partially thickenable" t…
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…
In a previous paper the authors defined the growth rate of the tunnel number of knots, an invariant that measures that asymptotic behavior of the tunnel number under connected sum. In this paper we calculate the growth rate of the tunnel number of m-small knots in terms of their bridge indices.
Let K be a knot that has an unknotting tunnel tau. We prove that K admits a strong involution that fixes tau pointwise if and only if K is a two-bridge knot and tau its upper or lower tunnel.
Let $K$ be a tunnel number two knot. Then, by considering the $(g, b)$-decompositions, $K$ is one of (3, 0)-, (2, 1)-, (1, 2)- or (0, 3)-knots. In the present paper, we analyze the connected sum summands of composite tunnel number two knots and give a complete table of those summands from the point of view of $(g, b)$-decompositions.
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.
The only knots that are tunnel number one and genus one are those that are already known: 2-bridge knots obtained by plumbing together two unknotted annuli and the satellite examples classified by Eudave-Munoz and by Morimoto-Sakuma. This confirms a conjecture first made by Goda and Teragaito.
In this paper, we show that any unknotting tunnel for a two bridge knot is isotopic to either one of known ones. This together with Morimoto-Sakuma's result gives the complete classification of unknotting tunnels for two bridge knots up to isotopies and homeomorphisms.
We study 2-string free tangle decompositions of knots with tunnel number two. As an application, we construct infinitely many counter-examples to a conjecture in the literature stating that the tunnel number of the connected sum of prime knots doesn't degenerate by more than one.
We determine the genus one fibered knots in lens spaces that have tunnel number one. We also show that every tunnel number one, once-punctured torus bundle is the result of Dehn filling a component of the Whitehead link in the 3-sphere.
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.
For a genus-1 1-bridge knot in the 3-sphere, that is, a (1,1)-knot, a middle tunnel is a tunnel that is not an upper or lower tunnel for some (1,1)-position. Most torus knots have a middle tunnel, and non-torus-knot examples were obtained by Goda, Hayashi, and Ishihara. In a previous paper, we generalized their construction and calculated the slope invariants for the resulting examples. We give an iterated version of the construction that produces many more examples, and calculate their slope invariants. If one starts with the trivial knot, the iterated constructions produce all the 2-bridge k…
