For surface-knots, branched covers with degree 3 have simplifying numbers <3.
problem Understanding numerical invariants of branched covering surface-knots.
method Analyzing numerical invariant (simplifying number) of branched covering surface-knots with degree 3.
result Branched covering surface-knots with degree 3 have simplifying numbers less than 3.
Criteria found for knots not determined by their double branched covers.
problem Determining knots from their double branched covers.
method Criteria based on tunnel number one knots and their double branched covers.
result Provided criteria for knots not determined by their double branched covers.
Cyclic covers of knots uniquely determine the original knot.
problem Determining knots from their branched covers.
method Examining n-fold cyclic branched covers of alternating prime knots.
result The n-fold cyclic cover of an alternating prime knot uniquely determines the original knot.
Course on knots using branched coverings.
problem Determining knots using cyclic branched coverings.
method Cyclic branched coverings of knots.
result New insights into knot classification.
Simplifies surface-knots using chart moves involving black vertices.
problem Simplifying branched covering surface-knots.
method Chart moves involving black vertices to simplify surface-knots.
result Properties of simplified branched covering surface-knots with branch points.
New examples show transverse knots are determined by their branched covers.
problem Transverse knots and their isotopy classes.
method Constructing and analyzing non-isotopic transverse knots with contactomorphic cyclic branched covers.
result Transverse isotopy classes of many transverse knots are determined by the contactomorphism type of their cyclic branched covers.
The study counts SU(2) representations for torus-covering knots.
problem Counting irreducible metabelian SU(2) representations for torus-covering knots.
method Using Fox's p-colorability and knot determinant.
result Similar to classical knots, the number of representations is determined.
New knot invariants from covering involutions help deduce linear independence results.
problem Understanding the smooth concordance group of knots.
method Covering involutions and adapted ideas from previous research.
result Novel linear independence results in the smooth concordance group of knots.
Simplifies surface-knots by adding 1-handles with chart loops.
problem Complexity of branched covering surface-knots.
method Adding 1-handles with chart loops to simplify charts.
result Properties of simplified charts are investigated.
In this paper, we introduce a sequence of invariants of a knot K in S^3: the knot Floer homology groups of the preimage of K in the m-fold cyclic branched cover over K. We exhibit the knot Floer homology in the m-fold branched cover as the categorification of a multiple of the Turaev torsion in the case where the m-fol…
New findings on L-spaces and left-orderability for specific knot covers.
problem Understanding L-spaces and left-orderability in knot theory.
method Analyzing 3-fold and 5-fold cyclic branched covers of two-bridge knots.
result Fundamental groups of certain knot covers are not left-orderable.
Study on pretzel knots showing cyclic branched covers are L-spaces.
problem Understanding cyclic branched covers of pretzel knots and their properties.
method Analyzing pretzel knots Kk and their n-fold cyclic branched covers for all n≥1. result The n-fold cyclic branched covers of pretzel knots Kk are L-spaces for all n≥1. Study determines left-orderable properties of knot covers.
problem Determining left-orderability of knot covers.
method Analyzing 2-bridge knots, including double-twist knots, using cyclic branched covers.
result Identifies specific integers for left-orderability of knot covers.
Study negative definite spin fillings of knot covers.
problem Existence of negative definite spin fillings in branched double covers.
method Derive obstructions and characterize special knots.
result Characterization of special alternating knots.
New knot concordance invariants from cyclic covers of prime power.
problem Constructing new knot concordance invariants.
method Considering m-fold cyclic branched covers with m a prime power.
result Computations of new invariants for some families of knots.
Combinatorial proof for knot Floer homology in branched covers.
problem Proving the invariance of knot Floer homology in branched covers.
method Using a Heegaard diagram and grid diagram for a knot in its cyclic branched cover.
result Combinatorial proof of knot Floer homology invariance over Z.
In this article we examine the conjecture of Neumann and Reid that the only hyperbolic knots in the 3-sphere which admit hidden symmetries are the figure-eight knot and the two dodecahedral knots. Knots whose complements cover hyperbolic reflection orbifolds admit hidden symmetries, and we verify the Neumann-Reid con…
We show that, for any prime p, a knot K in the 3-sphere is determined by its p-fold cyclic unbranched covering. We also investigate when the m-fold cyclic unbranched covering of a knot coincides with the n-fold cyclic unbranched covering of another knot, for different coprime integers m and n.
Study eigenvalues of knot covers to bound knot properties.
problem Bounding knot genera, dealternating number, and alternation number.
method Minimal positive eigenvalues of double branched covers over spanning surfaces.
result Eigenvalue bounds provide estimates for knot properties.
For any alternating knot, it is known that the double branched cover of the 3-sphere branched over the knot is an L-space. We show that the three-fold cyclic branched cover is also an L-space for any genus one alternating knot.
We prove that a prime knot K is not determined by its p-fold cyclic branched cover for at most two odd primes p. Moreover, we show that for a given odd prime p, the p-fold cyclic branched cover of a prime knot K is the p-fold cyclic branched cover of at most one more knot K' non equivalent to K. To prove the main theor…
Study shows certain knots can't be sliced using 2-fold branched covers.
problem Determining which algebraically slice knots are actually slice.
method Used d invariants of 2-fold branched covers to show nonsliceness.
result Shows nonsliceness of a set of algebraically slice knots.
The paper details folding of branched covers of the 3-sphere over knots.
problem Understanding folding of branched covers of the 3-sphere over knots.
method Presenting detailed foldings of dihedral covers of the 3-sphere branched over torus knots.
result Detailed foldings of dihedral covers of the 3-sphere branched over torus knots are presented.
We consider a surface link in the 4-space which can be presented by a simple branched covering over the standard torus, which we call a torus-covering link. Torus-covering links include spun T2-knots and turned spun T2-knots. In this paper we braid a torus-covering link over the standard 2-sphere. This gives an u…
Study on left orderability of specific knot covers.
problem Determining left orderability of knot covers.
method Computing nonabelian SL2(C) character varieties and analyzing real points.
result Left orderability of fundamental groups of cyclic branched covers.
The paper proves a covering theorem for virtual knots.
problem Covering of virtual knots and their properties.
method Using Gauss diagrams, the paper defines and analyzes r-coverings of virtual knots. result For any finite set of virtual knots, there exists a virtual knot whose r-coverings match the set. This paper introduces a new method to create mod m almost classical links from virtual knots.
problem Creating mod m almost classical links from virtual knots.
method Using knots in S_g × S^1 to construct m-fold coverings.
result A new family of m-fold coverings over virtual knots are mod m almost classical.
Simple test for whether knots are amphichiral.
problem Determining if knots are amphichiral.
method Using the homology of 2-fold branched covers.
result A straightforward method to identify amphichiral knots.
The paper studies random covers of torus knot complements and their statistical properties.
problem Understanding the statistical behavior of finite covers of torus knot complements.
method Asymptotic subgroup growth analysis and Benjamini-Schramm limit theorems.
result Determination of the linear growth rate of Betti numbers for random covers of torus knot complements.
We introduce a new technique for studying classical knots with the methods of virtual knot theory. Let K be a knot and J a knot in the complement of K with lk(J,K)=0. Suppose there is covering space πJ:Σ×(0,1)→S3\V(J)ˉ, where V(J) is a regular neighborhood of J satisfyin…
It is known that any surface knot can be transformed to an unknotted surface knot or a surface knot which has a diagram with no triple points by a finite number of 1-handle additions. The minimum number of such 1-handles is called the unknotting number or the triple point cancelling number, respectively. In this paper,…
New signatures for knotted graphs linked to classical knot signatures.
problem Defining invariants for knotted trivalent graphs.
method Using branched covers to define and relate new signatures to classical knot signatures.
result Computable invariants for Kinoshita's knotted theta graph.
We study contact manifolds that arise as cyclic branched covers of transverse knots in the standard contact 3-sphere. We discuss properties of these contact manifolds and describe them in terms of open books and contact surgeries. In many cases we show that such branched covers are contactomorphic for smoothly isotopic…
In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually S3), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of …
Study proves non-left-orderability of 3-manifolds derived from specific knots.
problem Proving non-left-orderability of knot groups in 3-manifolds.
method Analyzing fundamental groups of cyclic branched covers of pretzel knots.
result Non-left-orderability of fundamental groups of n-fold cyclic branched covers of P(3,−3,−2k−1) for all integers k and n≥1. Knots generating infinite subgroup bound rational homology balls.
problem Understanding knots that bound rational homology balls.
method Cyclic branched covers and rational homology balls.
result Infinite two-torsion subgroup in knot concordance group.
Research on knots and their 4-manifold covers.
problem Determining if knots are slice.
method Analyzing double branched covers and computer searches.
result Methods to decide if knots are slice.
We study the connections among the mapping class group of the twice punctured torus, the cyclic branched coverings of (1,1)-knots and the cyclic presentations of groups. We give the necessary and sufficient conditions for the existence and uniqueness of the n-fold strongly-cyclic branched coverings of (1,1)-knots, thro…
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 symbolic dynamics.
A transverse knot exists in S^3 that covers all contact 3-manifolds.
problem Existence of a universal transverse knot covering all contact 3-manifolds.
method Contact covering via branched map φ: M → S^3.
result There exists a transverse knot K in S^3 such that every contact 3-manifold can be obtained as a contact covering branched along K.
We discuss 3-manifolds which are cyclic coverings of the 3-sphere, branched over 2-bridge knots and links. Different descriptions of these manifolds are presented: polyhedral, Heegaard diagram, Dehn surgery and coloured graph constructions. Using these descriptions, we give presentations for their fundamental groups, w…
Algorithm calculates ribbon obstructions for colored knots.
problem Computing ribbon obstructions for colored knots.
method Algorithm based on dihedral branched covers and colored knot diagrams.
result Algorithm successfully computes ribbon obstructions for colored knots.
We are interested in finite groups acting orientation-preservingly on 3-manifolds (arbitrary actions, ie not necessarily free actions). In particular we consider finite groups which contain an involution with nonempty connected fixed point set. This condition is satisfied by the isometry group of any hyperbolic cyclic …
Formula for Lefschetz number of knot branched covers.
problem Calculating Lefschetz numbers for knot branched covers.
method Analysis of Seiberg-Witten equations on 3-orbifolds, η-invariants.
result Formula for Lefschetz number of covering translations.
Hempel has shown that the fundamental groups of knot complements are residually finite. This implies that every nontrivial knot must have a finite-sheeted, noncyclic cover. We give an explicit bound, Φ(c), such that if K is a nontrivial knot in the three-sphere with a diagram with c crossings and a particularly s…
Study shows (2,1)-cable of figure-eight knot can't be smoothly sliced.
problem Determining if a knot can be smoothly sliced.
method Showed that the branched double cover of the (2,1)-cable of the figure-eight knot bounds no equivariant homology ball. result The (2,1)-cable of the figure-eight knot is not smoothly slice. Algorithms compute invariants of 4-manifolds as branched covers.
problem Computing invariants of 4-manifolds as branched covers.
method Diagrammatic algorithms using tri-plane diagrams and permutations.
result Automated algorithm for computing Kjuchukova's homotopy-ribbon obstruction.
Let K be a knot in S^3, and M and M' be distinct Dehn surgeries along K. We investigate when M covers M'. When K is a torus knot, we provide a complete classification of such covers. When K is a hyperbolic knot, we provide partial results in the direction of the conjecture that M never covers M'.