Related papers: A construction of slice knots via annulus twists
The stable Kauffman conjecture posits that a knot in $S^3$ is slice if and only if it admits a slice derivative. We prove a related statement: A knot is handle-ribbon (also called strongly homotopy-ribbon) in a homotopy 4-ball $B$ if and…
In [D.A. Fedoseev, V.O. Manturov, A sliceness criterion for odd free knots,arXiv:1707.04923], the authors proved a sliceness criterion for odd free knots: free knots with odd chords. In the present paper we give a similar criterion for…
A knotted ribbon is one of physical aspect of a knot. A folded ribbon knot is a depiction of a knot obtained by folding a long and thin rectangular strip to become flat. The ribbonlength of a knot type can be defined as the minimum length…
We present a short proof of a theorem of Tanaka that if a composite ribbon knot admits a symmetric union presentation with one twisting region, then it has a non-trivial knot and its mirror image as connected summands.
We construct ribbon surfaces of Euler characteristic one for several infinite families of alternating 3-braid closures. We also use a twisted Alexander polynomial obstruction to conclude the classification of smoothly slice knots which are…
We give a necessary, and in some cases sufficient, condition for sliceness inside the family of pretzel knots $P (p_1,...,p_n)$ with one $p_i$ even. The three stranded case yields two interesting families of examples: the first consists of…
There are 352.2 million prime knots in the 3-sphere with at most 19 crossings. We study which of these knots are slice, in both the smooth and topological categories. While no algorithm is known for deciding whether a given knot is slice in…
We use recently introduced Rasmussen invariant to find knots that are topologically locally-flatly slice but not smoothly slice. We note that this invariant can be used to give a combinatorial proof of the slice-Bennequin inequality.…
A knot in $S^3$ is rationally slice if it bounds a disk in a rational homology ball. We give an infinite family of rationally slice knots that are linearly independent in the knot concordance group. In particular, our examples are all…
It is known that the linking form on the 2-cover of slice knots has a metabolizer. We show that several weaker conditions, or some other conditions related to sliceness, do not imply the existence of a metabolizer. We then show how the…
In answer to a question of Long, Flapan constructed an example of a prime strongly positive amphicheiral knot that is not slice. Long had proved that all such knots are algebraically slice. Here we show that the concordance group of…
The ribbonlength Rib$(K)$ of a knot $K$ is the infimum of the ratio of the length of any flat knotted ribbon with core $K$ to its width. A twisted torus knot $T_{p,q;r,s}$ is obtained from the torus knot $T_{p,q}$ by twisting $r$ adjacent…
An open question asks if every knot of 4-genus g_s can be changed into a slice knot by g_s crossing changes. A counterexample is given.
In the 60's Levine proved that if $R$ is a slice knot, then on any genus $g$ Seifert surface for $R$ there is a $g$ component link $J$, called a derivative of $R$, on which the Seifert form vanishes. Many subsequent obstructions to $R$…
We take a close look at a classical magic trick performed with a string, where a trivial knot is seemingly isotoped into a trefoil, and generalize it to a family of magic tricks for transforming the unknot into other knots. We encode such a…
We prove that an odd pretzel knot is doubly slice if it has $2n+1$ twist parameters consisting of $n+1$ copies of $a$ and $n$ copies of $-a$ for some odd integer $a$. Combined with the work of Issa and McCoy, it follows that these are the…
Given a simply-connected 4-manifold with boundary the 3-sphere, this paper establishes sufficient conditions for a knot in the boundary to be sliced by a locally flat disc in the 4-manifold, whose complement has finite cyclic fundamental…
We study the relationship between fibered ribbon 1-knots and fibered ribbon 2-knots by studying fibered slice disks with handlebody fibers. We give a characterization of fibered homotopy-ribbon disks and give analogues of the Stallings…
For $\ell >1$, we develop $L^{(2)}$-signature obstructions for $(4\ell-3)$-dimensional knots with metabelian knot groups to be doubly slice. For each $\ell>1$, we construct an infinite family of knots on which our obstructions are non-zero,…
A knot in $S^3$ is topologically slice if it bounds a locally flat disk in $B^4$. A knot in $S^3$ is rationally slice if it bounds a smooth disk in a rational homology ball. We prove that the smooth concordance group of topologically and…