Related papers: Cosmetic crossings and Seifert matrices
This paper computes the Jones polynomial and the invariants obstructing cosmetic surgery which are derived from it for two infinite families of knots, proving they satisfy the Purely Cosmetic Surgery Conjecture. Both the method of…
Whitehead doubles provide a plethora of examples of knots that are topologically slice but not smoothly slice. We discuss the problem of the Whitehead double of the Figure 8 knot and survey commonly used techniques to obstructing sliceness.…
This elementary article introduces easy-to-manage invariants of genus one knots in homology 3-spheres. To prove their invariance, we investigate properties of an invariant of 3-dimensional genus two homology handlebodies called the…
We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify…
It has long been known that the quadratic term in the degree of the colored Jones polynomial of a knot is bounded above in terms of the crossing number of the knot. We show that this bound is sharp if and only if the knot is adequate. As an…
We present new obstructions for a knot K in S^3 to admit purely cosmetic surgeries, which arise from the study of Witten-Reshetikhin-Turaev invariants at fixed level. In particular, we strengthen a recent result of Hanselman, showing that…
A geometric argument is given to prove that the Seifert genus of a positive knot equals its slice genus. A combinatorial invariant, giving a lower bound for the slice genus, is formulated for arbitrary knots. Properties and applications of…
The structure of the first homology group of a cyclic covering of a knot is an important invariant well known in the knot theory. In the last century, H. Seifert developed a general approach to compute the homology group of the covering.…
We give examples of knots with some unusual properties of the crossing number of positive diagrams or strand number of positive braid representations. In particular we show that positive braid knots may not have positive minimal (strand…
There are infinitely many pretzel links with the same Alexander polynomial (actually with trivial Alexander polynomial). By contrast, in this note we revisit the Jones polynomial of pretzel links and prove that, given a natural number S,…
We analyze all monodromies of genus one fibered knots that possess clean or once-unclean arcs, and use this to determine all manifolds containing genus one fibered knots with generalized crossing changes resulting in another genus one…
We show that two Dehn surgeries on a knot $K$ never yield manifolds that are homeomorphic as oriented manifolds if $V_K''(1)\neq 0$ or $V_K'''(1)\neq 0$. As an application, we verify the cosmetic surgery conjecture for all knots with no…
A pair of surgeries on a knot is chirally cosmetic if they result in homeomorphic manifolds with opposite orientations. Using recent methods of Ichihara, Ito, and Saito, we show that, except for the (2,5) and (2,7)-torus knots, the genus 2…
It is known that knot Floer homology detects the genus and Alexander polynomial of a knot. We investigate whether knot Floer homology of $K$ detects more structure of minimal genus Seifert surfaces for $K$. We define an invariant of…
We define a filtration on the vector space spanned by Seifert matrices of knots related to Vassiliev's filtration on the space of knots. Further we show that the invariants of knots derived from the filtration can be expressed by…
In an earlier paper, we introduced a collection of graded Abelian groups $\HFKa(Y,K)$ associated to knots in a three-manifold. The aim of the present paper is to investigate these groups for several specific families of knots, including the…
We introduce a geometric invariant of knots in the three-sphere, called the first-order genus, that is derived from certain 2-complexes called gropes, and we show it is computable for many examples. While computing this invariant, we draw…
In this note, we prove a lower bound for the positive kinkiness of a closed braid which we then use to derive an estimate for the positive kinkiness of a link in terms of its Seifert system. As an application, we show that certain pretzel…
In this paper, we prove that there are no truly cosmetic surgeries on genus one classical knots. If the two surgery slopes have the same sign, we give the only possibilities of reflectively cosmetic surgeries. The result is an application…
We consider diagrams of links in $S^2$ obtained by projection from $S^3$ with the Hopf map and the minimal crossing number for such diagrams. Knots admitting diagrams with at most one crossing are classified. Some properties of these knots…