Invertible Knots Research Articles

Strongly Invertible Knots, Invariant Surfaces, and the Atiyah–Singer Signature Theorem

Strongly Invertible Knots, Invariant Surfaces, and the Atiyah–Singer Signature Theorem

Cosmetic operations and Khovanov multicurves

We prove an equivariant version of the Cosmetic Surgery Conjecture for strongly invertible knots. Our proof combines a recent result of Hanselman with the Khovanov multicurve invariants Kh~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\widetilde{{{\\,\ extrm{Kh}\\,}}}}$$\\end{document} and BN~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\widetilde{{{\\,\ extrm{BN}\\,}}}}$$\\end{document}. We apply the same techniques to reprove a result of Wang about the Cosmetic Crossing Conjecture and split links. Along the way, we show that Kh~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\widetilde{{{\\,\ extrm{Kh}\\,}}}}$$\\end{document} and BN~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\widetilde{{{\\,\ extrm{BN}\\,}}}}$$\\end{document} detect if a Conway tangle is split.

Strongly invertible knots, equivariant slice genera, and an equivariant algebraic concordance group

AbstractWe use the Blanchfield form to obtain a lower bound on the equivariant slice genus of a strongly invertible knot. For our main application, let be a strongly invertible genus one slice knot with nontrivial Alexander polynomial. We show that the equivariant slice genus of an equivariant connected sum is at least . We also formulate an equivariant algebraic concordance group, and show that the kernel of the forgetful map to the classical algebraic concordance group is infinite rank.

A refinement of Khovanov homology

We refine Khovanov homology in the presence of an involution on the link. This refinement takes the form of a triply-graded theory, arising from a pair of filtrations. We focus primarily on strongly invertible knots and show, for instance, that this refinement is able to detect mutation.

F -distance of knotoids and protein structure

Recent studies classify the topology of proteins by analysing the distribution of their projections using knotoids. The approximation of this distribution depends on the number of projection directions that are sampled. Here, we investigate the relation between knotoids differing only by small perturbations of the direction of projection. Since such knotoids are connected by at most a single forbidden move, we characterize forbidden moves in terms of equivariant band attachment between strongly invertible knots and of strand passages between θ -curves. This allows for the determination of the optimal sample size needed to produce a well-approximated knotoid distribution. Based on that and on topological properties of the distribution, we probe the depth of knotted proteins with the trefoil as the predominant knot type without using subchain analysis.

Khovanov homology and the symmetry group of a knot

We introduce an invariant of tangles in Khovanov homology by considering a natural inverse system of Khovanov homology groups. As application, we derive an invariant of strongly invertible knots; this invariant takes the form of a graded vector space that vanishes if and only if the strongly invertible knot is trivial. While closely tied to Khovanov homology — and hence the Jones polynomial — we observe that this new invariant detects non-amphicheirality in subtle cases where Khovanov homology fails to do so. In fact, we exhibit examples of knots that are not distinguished by Khovanov homology but, owing to the presence of a strong inversion, may be distinguished using our invariant. This work suggests a strengthened relationship between Khovanov homology and Heegaard Floer homology by way of two-fold branched covers that we formulate in a series of conjectures.

Seifert fibered surgeries on strongly invertible knots without primitive/Seifert positions

We find an infinite family of Seifert fibered surgeries on strongly invertible knots which do not have primitive/Seifert positions. Each member of the family is obtained from a trefoil knot after alternate twists along a pair of seiferters for a Seifert fibered surgery on a trefoil knot.

Surgery obstructions from Khovanov homology

For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S 3, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundamental group. These obstructions are based on homological width in Khovanov homology, and in the case of finite fundamental group depend on a calculation of the homological width for a family of Montesinos links.

Strongly Invertible Knots, Rational-Fold Branched Coverings, and Hyperbolic Spatial Graphs

A construction of a spatial graph from a strongly invertible knot was developed by the second author, and a necessary and sufficient condition for the given spatial graph to be hyperbolic was provided as well. The condition is improved in this paper. This enable us to show that certain classes of knots can yield hyperbolic spatial graphs via the construction.

Polynomial representation of strongly-invertible knots and strongly-negative-amphicheiral knots

It is shown that the symmetric behaviour of certain class of knots can be realized by their polynomial representations. We prove that every strongly invertible knot (open) can be represented by a polynomial embedding $t\mapsto (f(t),g(t),h(t))$ of $\mathbb{R}$ in $\mathbb{R}^{3}$ where among the polynomials $f(t)$, $g(t)$ and $h(t)$ two of them are odd polynomials and one is an even polynomial. We also prove that a subclass of strongly negative amphicheiral knots can be represented by a polynomial embedding $t\mapsto(f(t),g(t),h(t))$ of $\mathbb{R}$ in $\mathbb{R}^{3}$ where all three polynomials $f(t)$, $g(t)$ and $h(t)$ are odd polynomials.

Hyperbolic spatial graphs arising from strongly invertible knots

Spatial graphs in the three-dimensional sphere are constructed from strongly invertible knots. Such a graph is proved to be hyperbolic, which means that its exterior admits a hyperbolic structure with totally geodesic boundary, if the exterior has no equivalent essential torus, or a pair of tori, with respect to the involution.

Dehn surgeries on strongly invertible knots which yield lens spaces

In this article we show no Dehn surgery on nontrivial strongly invertible knots can yield the lens space L ( 2 p , 1 ) L(2p,1) for any integer p p . In order to do that, we determine band attaches to ( 2 , 2 p ) (2,2p) -torus links producing the trivial knot.

Crossing changes

This is an expository survey article about the role that the simple operation of changing a crossing has played in knot theory. Topics include: the connections between unknotting number, tunnel number, and crossing number; connections with Dehn surgery and sutured maniford theory; nullifying crossings and the Conway skein trees; generalized crossing changes; crossing changes and strongly invertible knots; and connections with 4-manifold topology.

Arf invariants of strongly invertible knots obtained from unknotting number one knots

Two techniques, branched covering space and Dehn surgery, are known as popular methods by which we obtain new 3-manifolds from a link in S. In 1977, Montesinos [8] showed the following relationship between the 2-fold cyclic branched covering of S branched over a link and the closed, orientable 3-manifold which is obtained by doing surgery on a link in S. A link in S is called strongly invertible if there is an orientation preserving involution of S which induces in each component of L an involution with two fixed points.

Band Sums of Links which Yield Composite Links. The Cabling Conjecture for Strongly Invertible Knots

We consider composite links obtained by bandings of another link.It is shown that if a banding of a split link yields a composite knot then there is a decomposing sphere crossing the band in one arc, unless there is such a sphere disjoint from the band. We also prove that if a banding of the trivial knot yields a composite knot or link then there is a decomposing sphere crossing the band in one arc. The last theorem implies, via double branche covers, that the only way we can get a reducible manifold by surgery on a strongly invertible knot is when the knot is cabled and the surgery is via the slope of the cabling annulus

Band sums of links which yield composite links. The cabling conjecture for strongly invertible knots

We consider composite links obtained by bandings of another link. It is shown that if a banding of a split link yields a composite knot then there is a decomposing sphere crossing the band in one arc, unless there is such a sphere disjoint from the band. We also prove that if a banding of the trivial knot yields a composite knot or link then there is a decomposing sphere crossing the band in one arc. The last theorem implies, via double branched covers, that the only way we can get a reducible manifold by surgery on a strongly invertible knot is when the knot is cabled and the surgery is via the slope of the cabling annulus.

A projective plane in [formula omitted]4 with three critical points is standard. Strongly invertible knots have property P

A projective plane in [formula omitted]4 with three critical points is standard. Strongly invertible knots have property P

Surgery on double knots and symmetries

W. Whitten conjectured [Pacific J. Math. 97 (1981), no. 1, 209–216] that no 3-manifold obtained by a nontrivial surgery on a double of a noninvertible knot is a 2-fold branched covering of S3. The authors give counterexamples to this conjecture and determine the exact range of validity of the conjecture. More generally, they consider closed, orientable 3-manifolds obtained by nontrivial Dehn surgery on a double of a non-strongly invertible knot and study the symmetries of such manifolds, i.e. the homeomorphisms of finite order on these manifolds. They show that, except for a finite number of surgeries, these manifolds admit no (nontrivial) symmetry.

Polynomials of invertible knots

Polynomials of invertible knots

On prime noninvertible links

It is the purpose of this paper to exhibit an infinite collection of prime, noninvertible links; each link in the collection is the union of two invertible knots. In fact, we present two such families of links. Our examples heuristically indicate that collections of links of this type exist in many varieties.