Related papers: Non-slice 3-stranded pretzel knots
The knot quandle is an invariant of $n$-knots. In this note, we study the knot quandles of Suciu's ribbon $n$-knots, an infinite family of knots with isomorphic knot groups. We prove that their knot quandles are mutually non-isomorphic.…
We use twisted Alexander polynomials to show that certain algebraically slice 2-bridge knots are not topologically slice, even though all prime power Casson-Gordon signatures vanish. We also provide some computations indicating the efficacy…
The second author and Powell asked whether there exist knots bounding infinitely many slice disks that remain pairwise nonisotopic, even after local knotting. We answer this question in the affirmative, giving many classes of examples…
The 3-strand pretzel knots and links are a well-studied source of examples in knot theory. However, while there have been computations of the Khovanov homology of some sub-families of 3-strand pretzel knots, no general formula has been…
An $n$-strand braid is order-preserving if its action on the free group $F_n$ preserves some bi-order of $F_n$. A braid $\beta$ is order-preserving if and only if the link $L$ obtained as the union of the closure of $\beta$ and its axis has…
We show that the family of smoothly non-isotopic Legendrian pretzel knots from the work of Cornwell-Ng-Sivek that all have the same Legendrian invariants as the standard unknot have front-spuns that are Legendrian isotopic to the front-spun…
The lattice stick number of a knot type is defined to be the minimal number of straight line segments required to construct a polygon presentation of the knot type in the cubic lattice. In this paper, we mathematically prove that the…
We give examples of a linear combination of algebraic knots and their mirrors that are algebraically slice, but whose topological and smooth four-genus is two. Our examples generalize an example of non-slice algebraically slice linear…
The AJ conjecture relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. It has been verified for some classes of knots, including all torus knots, most double twist knots, (-2,3,6n \pm 1)-pretzel knots, and…
The slicing number of a knot, $u_s(K)$, is the minimum number of crossing changes required to convert $K$ to a slice knot. This invariant is bounded above by the unknotting number and below by the slice genus $g_s(K)$. We show that for many…
We provide explicit formulas for the Alexander polynomial of pretzel knots and establish several immediate corollaries, including the characterization of pretzel knots with a trivial Alexander polynomial. As an application, we construct a…
In the early 1980's Mike Freedman showed that all knots with trivial Alexander polynomial are topologically slice (with fundamental group Z). This paper contains the first new examples of topologically slice knots. In fact, we give a…
The twisting number of a ribbon knot $K$ is the minimal number of tangle replacements on the symmetry axis of $J \# -J$ for any knot $J$ that is required to produce a symmetric union diagram of $K$. We prove that the twisting number is…
Given a thin strip of paper, tie a knot, connect the ends, and flatten into the plane. This is a physical model of a folded ribbon knot in the plane, first introduced by Louis Kauffman. We study the folded ribbonlength of these folded…
If there are any 2-component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such…
Here we discuss $r-$shake slice knots, and their relation to corks, we then prove that $0$-shake slice knots are slice.
We introduce a new technique for showing classical knots and links are not slice. As one application we resolve a long-standing question as to whether certain natural families of knots contain topologically slice knots. We also present a…
Let K be a hyperbolic (-2,3,n) pretzel knot and M = S^3 K its complement. For these knots, we verify a conjecture of Reid and Walsh: there are at most three knot complements in the commensurability class of M. Indeed, if n \neq 7, we show…
As a corollary of work of Ozsvath and Szabo [math.GT/0301149], it is shown that the classical concordance group of algebraically slice knots has an infinite cyclic summand and in particular is not a divisible group.
A knot in the three-sphere is doubly slice if it is the cross-section of an unknotted two-sphere in the four-sphere. For low-crossing knots, the most complete work to date gives a classification of doubly slice knots through 9 crossings. We…