Related papers: Infinite Order Amphicheiral Knots
Every knot can be embedded in the union of finitely many half planes with a common boundary line in such a way that the portion of the knot in each half plane is a properly embedded arc. The minimal number of such half planes is called the…
We prove that the irreducible affine Coxeter groups are first-order rigid and deduce from this that they are profinitely rigid in the absolute sense. We then show that the first-order theory of any irreducible affine Coxeter group does not…
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…
For n >1, if the Seifert form of a knotted 2n-1 sphere K in S^{2n+1} has a metabolizer, then the knot is slice. Casson and Gordon proved that this is false in dimension three (n = 1). However, in the three dimensional case it is true that…
The authors conjectured previously that a knot is nonfibered if and only if its infinite cyclic cover has uncountably many finite covers. We prove the conjecture for a class of knots that includes all knots of genus 1, using techniques from…
We show that the class of multigraphs with at most $p$ connected components and bonds of size at most $k$ is well-quasi-ordered by edge contraction for all positive integers $p,k$. (A bond is a minimal non-empty edge cut.) We also…
Shake slice generalizes the notion of a slice link, naturally extending the notion of shake slice knots to links. There is also a relative version, shake concordance, that generalizes link concordance. We show that if two links are shake…
Greene-Jabuka and Lecuona confirmed the slice-ribbon conjecture for 3-stranded pretzel knots except for an infinite family $P(a,-a-2,-\frac{(a+1)^2}{2})$ where $a$ is an odd integer greater than $1$. Lecuona and Miller showed that…
We find a formula for the L2 signature of a (p,q) torus knot, which is the integral of the omega-signatures over the unit circle. We then apply this to a theorem of Cochran-Orr-Teichner to prove that the n-twisted doubles of the unknot, for…
It is proved that every finitely generated profinite group with fewer than $2^{\aleph_0}$ conjugacy classes of elements of infinite order is finite
A lattice-ordered group (an $\ell$-group) $G(\oplus, \vee, \wedge)$ can be naturally viewed as a semiring $G(\vee,\oplus)$. We give a full classification of (abelian) $\ell$-groups which are finitely generated as semirings, by first showing…
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…
Given a positive integer $n$, we say that two knots are $V_n$-equivalent if they have the same Vassiliev invariants of order $\le n$. We showed that the $V_n$-equivalence classes of ribbon knots form a group, the operation being induced by…
Double twist knots $K_{m, n}$ are known to be rationally slice if $mn = 0$, $n = -m\pm 1$, or $n = -m$. In this paper, we prove the converse. It is done by showing that infinitely many prime power-fold cyclic branched covers of the other…
Either fibered knots supporting the tight contact structure are unique in their smooth concordance class or there exists a fibered counterexample to the Slice-Ribbon Conjecture.
The concordance genus of a knot is the least genus of any knot in its concordance class. It is bounded above by the genus of the knot, and bounded below by the slice genus, two well-studied invariants. In this paper we consider the…
We construct the first example of a finitely-presented, residually-finite group that contains an infinite sequence of non-isomorphic finitely-presented subgroups such that each of the inclusion maps induces an isomorphism of profinite…
Using the classification and description of the structure of bisimple monogenic orthodox semigroups obtained in \cite{key10}, we prove that every bisimple orthodox semigroup generated by a pair of mutually inverse elements of infinite order…
We define an obstruction for a knot to be Z[Z]-homology ribbon, and use this to provide restrictions on the integers that can occur as the triple linking numbers of derivative links of knots that are either homotopy ribbon or doubly slice.…
We discuss an obstruction to a knot being smoothly slice that comes from minimum-genus bounds on smoothly embedded surfaces in definite 4-manifolds. As an example, we provide an alternate proof of the fact that the (2,1)-cable of the figure…