Cable Knot Research Articles

Geometry of knots in real projective 3-space

This paper discusses some geometric ideas associated with knots in real projective 3-space [Formula: see text]. These ideas are borrowed from classical knot theory. Since knots in [Formula: see text] are classified into three disjoint classes: affine, class-[Formula: see text] non-affine and class-[Formula: see text] knots, it is natural to wonder in which class a given knot belongs to. In this paper we attempt to answer this question. We provide a structure theorem for these knots which helps in describing their behavior near the projective plane at infinity. We propose a procedure called space bending surgery, on affine knots to produce several examples of knots. We later show that this operation can be extended on an arbitrary knot in [Formula: see text]. We then study the notion of companionship of knots in [Formula: see text] and using it we provide geometric criteria for a knot to be affine. We also define a notion of “genus” for knots in [Formula: see text] and study some of its properties. We prove that this genus detects knottedness in [Formula: see text] and gives some criteria for a knot to be affine and of class-[Formula: see text]. We also prove a “non-cancellation” theorem for space bending surgery using the properties of genus. Then we show that a knot can have genus 1 if and only if it is a cable knot with a class-1 companion. We produce examples of class-[Formula: see text] non-affine knots with genus [Formula: see text]. Thus we highlight that, [Formula: see text] admits a knot theory with a truly different flavor than that of [Formula: see text] or [Formula: see text].

Cable knots obtained by positively twisting torus knots

The twisted torus knot [Formula: see text] is a knot obtained from the torus knot [Formula: see text] by twisting [Formula: see text] adjacent stands fully [Formula: see text] times. If [Formula: see text] or [Formula: see text] is a multiple of [Formula: see text], then [Formula: see text] is known to be a cable of a nontrivial knot or a torus knot. Assuming that [Formula: see text] and [Formula: see text] is not a multiple of [Formula: see text], we determine the parameters [Formula: see text] for which the positively twisted torus knot [Formula: see text] is a cable knot.

A Cable Knot and BPS-Series

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 figure eight knot. This relation provides an alternative simple method of finding the former series invariant.

Development of virtual ophthalmic surgical skills training

BackgroundThis study aims to assess whether ophthalmic surgical skills can be taught successfully online to a diverse international and interprofessional student group.MethodsMixed methods study involving 20 students and 5 instructors. Each student completed a pre-session and post-session questionnaire to assess their perceptions regarding online instruction. Changes in questionnaire responses were analysed using Wilcoxon signed rank (SPSS 25). Semi-structured interviews were conducted to assess instructor perceptions towards virtual surgical skills teaching. Thematic analysis was undertaken using NVivo 12.0 software.ResultsThere was a 100% completion rate of pre- and post-session questionnaires. Prior to the session, lack of instructor supervision and inability to provide constructive feedback were emergent themes from students. Pre-session concerns regarding online delivery: 40% of students thought their view of skills demonstration would be negatively impacted, 60% their level of supervision and 55% their interaction with instructors. Following the session 10%, 15% and 5% held this view respectively. All students were ‘satisfied’ or ‘very satisfied’ regarding the ‘Surgeon’s View’ camera angle as well as the use of breakout rooms. 75% perceived an improvement in their confidence in instrument handling, 80% in cable knot tying and 70% in suture tying. Overall student rating for the virtual surgical skills session was 8.85 (±1.19) out of 10 (10 being most satisfied).ConclusionsWe demonstrate that successful delivery of a virtual ophthalmic surgical skills course is feasible. We were able to widen accessibility and participation through virtual delivery, which has future implications for ophthalmic surgical teaching and its reach.

Vassiliev knot invariants derived from cable Γ-polynomials

For coprime integers p(>0) and q, the (p,q)-cable Γ-polynomial of a knot K is the Γ-polynomial of the (p,q)-cable knot of K, where the Γ-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. In this paper, we give some results on Vassiliev knot invariants derived from the cable Γ-polynomials. In particular, we show that all Vassiliev knot invariants of order ≤4 are determined by the cable Γ-polynomials.

On tunnel numbers of a cable knot and its companion

Let K be a nontrivial knot in S3 and t(K) its tunnel number. For any (p≥2,q)-slope in the torus boundary of a closed regular neighborhood of K in S3, denoted by K⋆, it is a nontrivial cable knot in S3. 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.

Immersed Möbius bands in knot complements

We study the 3-dimensional immersed crosscap number of a knot, which is a nonorientable analogue of the immersed Seifert genus. We study knots with immersed crosscap number 1, and show that a knot has immersed crosscap number 1 if and only if it is a nonntrivial $(2p,q)$-torus or $(2p,q)$-cable knot. We show that unlike in the orientable case the immersed crosscap number can differ from the embedded crosscap number by arbitrarily large amounts, and that it is neither bounded below nor above by the 4-dimensional crosscap number. We then use these constructions to find, for any $n\geq 2$, an oriented 3-manifold $Y_n$ and class $\alpha_n \in H_2(Y_n;\mathbb{Z}_2)$ such that $\alpha_n$ can be represented by an immersed $\mathbb{RP}^2$, but any embedded representative of $\alpha_n$ has a component $S$ with $\chi(S) \leq 1-n$.

Representativity and waist of cable knots

We study the essential surfaces in the exterior of a cable knot to compute the representativity and waist of most cable knots. Our computation answers Ozawa’s question [5] about the relationship between the representativity and the waist of a knot in the negative.

The (2,1)-cable Γ-polynomials of knots up to ten crossings

For coprime integers [Formula: see text] and [Formula: see text], the [Formula: see text]-cable [Formula: see text]-polynomial of a knot is the [Formula: see text]-polynomial of the [Formula: see text]-cable knot of the knot, where the [Formula: see text]-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. Since it is known that the [Formula: see text]-polynomial is computable in polynomial time, the [Formula: see text]-cable [Formula: see text]-polynomial is also computable in polynomial time. In this paper, we show that the [Formula: see text]-cable [Formula: see text]-polynomial completely classifies the unoriented knots up to ten crossings including the chirality information.

Infinitely many knots with the trivial (2,1)-cable Γ-polynomial

For coprime integers [Formula: see text] and [Formula: see text], the [Formula: see text]-cable [Formula: see text]-polynomial of a knot is the [Formula: see text]-polynomial of the [Formula: see text]-cable knot of the knot, where the [Formula: see text]-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. In this paper, we show that there exist infinitely many knots with the trivial [Formula: see text]-cable [Formula: see text]-polynomial, that is, the [Formula: see text]-cable [Formula: see text]-polynomial of the trivial knot. Moreover, we see that the knots have the trivial [Formula: see text]-polynomial, the trivial first coefficient HOMFLYPT and Kauffman polynomials.

Bounds on alternating surgery slopes

We show that if $p/q$-surgery on a nontrivial knot $K$ yields the branched double cover of an alternating knot or link, then $|p/q|\leq 4g(K)+3$. This generalises a bound for lens space surgeries first established by Rasmussen. We also show that all surgery coefficients yielding the double branched cover of an alternating knot or link must be contained in an interval of width two and this full range can be realised only if $K$ is a cable knot. The work of Greene and Gibbons shows that if $S^3_{p/q}(K)$ bounds a sharp 4-manifold $X$, then the intersection form of $X$ takes the form of a changemaker lattice. We extend this to show that the intersection form is determined uniquely by the knot $K$, the slope $p/q$ and the Betti number $b_2(X)$.

Knot contact homology detects cabled, composite, and torus knots

Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot. Further, if the knot contact homology of a knot is isomorphic to that of a cable (respectively composite) knot, then the knot is a cable (respectively composite).

ON KILLERS OF CABLE KNOT GROUPS

A killer of a group$G$is an element that normally generates$G$. We show that the group of a cable knot contains infinitely many killers such that no two lie in the same automorphic orbit.

On links with locally infinite Kakimizu complexes

We show that the Kakimizu complex of a knot may be locally infinite, answering a question of Przytycki--Schultens. We then prove that if a link $L$ only has connected Seifert surfaces and has a locally infinite Kakimizu complex then $L$ is a satellite of either a torus knot, a cable knot or a connected sum, with winding number 0.

DEHN SURGERIES ON 2-BRIDGE LINKS WHICH YIELD REDUCIBLE 3-MANIFOLDS

We completely determine which Dehn surgeries on 2-bridge links yield reducible 3-manifolds. Further, we consider which surgery on one component of a 2-bridge link yields a torus knot, a cable knot and a satellite knot in this paper.

Mutant knots with symmetry

AbstractMutant knots, in the sense of Conway, are known to share the same Homfly polynomial. Their 2-string satellites also share the same Homfly polynomial, but in general theirm-string satellites can have different Homfly polynomials form> 2. We show that, under conditions of extra symmetry on the constituent 2-tangles, the directedm-string satellites of mutants share the same Homfly polynomial form< 6 in general, and for all choices ofmwhen the satellite is based on a cable knot pattern.We give examples of mutants with extra symmetry whose Homfly polynomials of some 6-string satellites are different, by comparing their quantumsl(3) invariants.

SATELLITE DOUBLE TORUS KNOTS

We characterize satellite double torus knots. Especially, if a satellite double torus knot is not a cable knot, then it has a torus knot companion. This answers Question 12 (a) raised by Hill and Murasugi in [4].

Plan projectif créé par chirurgie de Dehn

This Note concerns Dehn surgeries on a knot K in S 3 that produce a projective plane. The first result is the existence of an edge, said to be projective, which allows studying the intersection graphs obtained in the usual manner. Then we prove that K cannot be a Conway-reducible or a 2-bridge knot, except if K is a ( 1,2)-cable knot.

Two-Generator Cable Knots are Tunnel One

A two-generator cable knot exterior is a genus two handlebody with a single two-handle attached.

Two-generator cable knots are tunnel one

A two-generator cable knot exterior is a genus two handlebody with a single two-handle attached.