We show that the 3-fold (resp. 6-fold) connected sum of the $(2,1)$-cable of the figure-eight knot cannot bound a smooth null-homologous disk in a punctured $S^2 \times S^2$ (resp. in a punctured $#_2 S^2 \times S^2$. This result is obtained using a real version of the $10/8$-inequality established by Konno, Miyazawa, and Taniguchi.
We give a formula of the Upsilon invariant of any L-space cable knot $K_{p,q}$ using $p,Υ_K$ and $Υ_{T_{p,q}}$. The integral value of the Upsilon invariant gives a ${\mathbb Q}$-valued knot concordance invariant. We compute the integral values for L-space iterated cable knots.
Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in $S^3$ do not admit purely cosmetic surgeries. In this article, we confirm this conjecture for cable knots.
We show that a $(p,q)$-cable of a non-trivial knot $K$ does not admit chirally cosmetic surgery for $q\neq 2$, or $q=2$ with additional assumptions. In particular, we show that $(p,q)$-cable of non-trivial knot $K$ does not admit chirally cosmetic surgery as long as the JSJ piece of knot exterior does not contain $(2,r)$-torus exterior. We also show that an iterated torus knot other than $(2,p)$-torus knot does not admit chirally cosmetic surgery.
We construct a filtered mapping cone formula that computes the knot Floer complex of the $(n,1)$--cable of the knot meridian in any rational surgery, generalizing Truong's result about the $(n,1)$--cable of the knot meridian in large surgery and Hedden-Levine's filtered mapping cone formula. As an application, we show that there exist knots in integer homology spheres with arbitrary $φ_{i,j}$ values for any $i>j\geq 0$, where $φ_{i,j}$ are the concordance homomorphisms defined by Dai-Hom-Stoffregen-Truong.
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…
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 study the asymptotic behavior of the $N$-dimensional colored Jones polynomial of a cable of the figure-eight knot, evaluated at $\exp(ξ/N)$ for a real number $ξ$. We show that if $ξ$ is sufficiently large, the colored Jones polynomial grows exponentially when $N$ goes to the infinity. Moreover the growth rate is related to the Chern-Simons invariant of the knot exterior associated with an $\mathrm{SL}(2;\mathbb{R})$ representation.
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.
Using Dowlin's spectral sequence from Khovanov homology to knot Floer homology, we prove that reduced Khovanov homology (over $\mathbb{Q}$) detects the figure-eight knot.
We produce infinite families of knots $\{K^i\}_{i\geq 1}$ for which the set of cables $\{K^i_{p,1}\}_{i,p\geq 1}$ is linearly independent in the knot concordance group. We arrange that these examples lie arbitrarily deep in the solvable and bipolar filtrations of the knot concordance group, denoted by $\{F_n\}$ and $\{B_n\}$ respectively. As a consequence, this result cannot be reached by any combination of algebraic concordance invariants, Casson-Gordon invariants, and Heegaard-Floer invariants such as tau, epsilon, and Upsilon. We give two applications of this result. First, for any n>=0, th…
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…
A series invariant of a complement of a knot was introduced recently. The invariant for several prime knots up to ten crossings have been explicitly computed. We present the first example of a satellite knot, namely, a cable of the figure eight knot, which has more than ten crossings. This cable knot result provides nontrivial evidence for the conjectures for the series invariant and demonstrates the robustness of integrality of the quantum invariant under the cabling operation. Furthermore, we observe a relation between the series invariant of the cable knot and the series invariant of the fi…
We show some computations on representations of the fundamental group in SL(2;C) and Reidemeister torsion for a homology 3-sphere obtained by Dehn surgery along the figure-eight knot. This is the second version. We recorrected several errors in the first version.
The $0$-surgeries of two knots $K_1$ and $K_2$ are homology cobordant rel meridians if there exists a $\mathbb{Z}$-homology cobordism $X$ between them such that the two knot meridians are in the same homology class in $H_{1}(X,\mathbb{Z})$. In this paper, we give a pair of rationally slice knots which are not smoothly concordant but whose $0$-surgeries are homology cobordant rel meridians. One knot in the pair is the figure eight knot, which has concordance order two; all previous examples of such pairs of knots are infinite order.
Suppose that every non-minimal bridge position of a knot $K$ is perturbed. We show that if $L$ is a $(2, 2q)$-cable link of $K$, then every non-minimal bridge position of $L$ is also perturbed.
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.
