arXiv research

Polynomially parameterizes knots and spheres, proving analogous results.

Any knot in $S^3$ may be reduced to a slice knot by crossing changes. Indeed, this slice knot can be taken to be the unknot. In this paper we study the question of when the same holds for knots in homology spheres. We show that a knot in a homology sphere is nullhomotopic in a smooth homology ball if and only if that k…

L-space knots lack essential Conway spheres, proven with Floer theory.

Polynomially parametrize interesting knotted surfaces.

The paper constructs many knotted and linked objects in higher dimensions.

Study cosmetic surgeries on knots in homology spheres using Casson-Walker invariant.

The paper generates triangulations of 2-knot complements via spinning 1-knots.

For any knot, a 3-sphere triangulation exists with a knotted edge.

We give examples of non-fibered hyperbolic knot complements in homology spheres that are not commensurable to fibered knot complements in homology spheres. In fact, we give many examples of knot complements in homology spheres with the property that every commensurable knot complement in a homology sphere has non-monic…

The paper proves unique factorization of knotted handlebodies and examines handlebody-knot symmetry.

Calegari's 4-spheres from fibered knots are proven standard.

Cyclic covers of knots uniquely determine the original knot.

Study shows knots in homology spheres can be topologically equivalent to those in $S^3$.

Smoothly knotted 5RP^2 found in 4-sphere.

Classifies knots in the Poincaré sphere, using fixed points and folding automata.

Extends exotic embeddings of RP^2 to a larger family and produces homotopy spheres.

The paper characterizes when a 2-sphere can be embedded in a knot trace.

Using spinning we analyze in a geometric way Haefliger's smoothly knotted (4k-1)-spheres in the 6k-sphere. Consider the 2-torus standardly embedded in the 3-sphere, which is further standardly embedded in the 6-sphere. At each point of the 2-torus we have the normal disk pair: a 4-dimensional disk and a 1-dimensional p…

In 2016 Levine showed that there exists a knot in a homology 3-sphere which is not smoothly concordant to any knot in the 3-sphere where one allows concordances in any smooth homology cobordism. Whether the same is true if one allows topological concordances is not known. One might hope that such an example might be de…

New invariant fully describes finite type invariants of knots in homology 3-spheres.

Surgery on knots can produce non-separating spheres, using Heegaard Floer homology.

We show that a handlebody-knot whose exterior is boundary-irreducible has a unique maximal unnested set of knotted handle decomposing spheres up to isotopies and annulus-moves. As an application, we show that the handlebody-knots $6_{14}$ and $6_{15}$ are not equivalent. We also show that some genus two handlebody-knot…

The study confirms conjectures about slopes of knots using knot Floer homology.

We show that there exists an infinite family of knots, each of which has, for each integer k>=0, a destabilized (2k+5)-bridge sphere. We also show that, for each integer n>=4, there exists a knot with a destabilized 3-bridge sphere and a destabilized n-bridge sphere.

Kirby and Lickorish showed that every knot in the 3-sphere is concordant to a prime knot, equivalently, every concordance class contains a prime knot. We prove here that their result can be strengthened: Every knot in the 3-sphere is invertibly concordant to a prime knot. A consequence is that every double concordance …

Local knots can't bound smaller surfaces in rational homology 3-spheres.

The study determines lens spaces that can be obtained from surgeries on knots in the Poincaré homology sphere.

In 1965, E. C. Zeeman proved that the (+/-)-twist spin of any knotted sphere in (n-1)-space is unknotted in the n-sphere. In 1991, Y. Marumoto and Y. Nakanishi gave an alternate proof of Zeeman's theorem by using the moving picture method. In this paper, we define a knotted 2-dimensional foam which is a generalization …

Survey of invariants for knotted 2-spheres in 4-space.

We study cosmetic contact surgeries along transverse knots in the standard contact 3-sphere, i.e. contact surgeries that yield again the standard contact 3-sphere. The main result is that we can exclude non-trivial cosmetic contact surgeries along all transverse knots not isotopic to the transverse unknot with self-lin…

Perelman's proof confirmed, new method uses 4D topology.

The paper details folding of branched covers of the 3-sphere over knots.

It is known that if any prime power branched cyclic cover of a knot in the 3-sphere is a homology sphere, then the knot has vanishing Casson-Gordon invariants. We construct infinitely many examples of (topologically) non-slice knots in the 3-sphere whose prime power branched cyclic covers are homology spheres. We show …

In this paper we look at the knot complement problem for L-space $\mathbb{Z}$-homology spheres. We show that an L-space $\mathbb{Z}$-homology sphere $Y$ cannot be obtained as a non-trivial surgery along a knot $K\subset Y$. As a consequence, we prove that knots in an L-space $\mathbb{Z}$-homology sphere are determined …

Paper proves every stable 4-sphere has a unique diffeomorphism class.

Examples are given to show that some compact contractible 4-manifolds can be knotted in the 4-sphere. It is then proved that any finitely presented perfect group with a balanced presentation is a knot group for an embedding of some contractible 4-manifold in the 4-sphere.

In the present study we consider knotted spheres in Euclidean $4$-space $\mathbb{E}^{4}$. Firstly, we give some basic curvature properties of knotted spheres in $\mathbb{E}^{4}$. Further, we obtained some results related with the conjugate nets and Laplace transforms of these kind of surfaces.

New method finds non-orientable knotted surfaces in 4D.

Study shows volume and genus unrelated for hyperbolic fibred knots.

We show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere L-spaces has Heegaard Floer homology rank strictly greater than one. In particular, splicing the complements of nontrivial knots in the 3-sphere never produces an L-space. The proof uses bordered Flo…

We examine surgery on a knot in $S^3$ to determine surgery obstructions to Seifert fibered integral homology spheres. We find such surgery obstructions using Heegaard Floer, Knot Floer homology and the mapping cone formula for computing Heegaard Floer homology of surgery on a knot. Here however, we take a different app…

Hedden defined two knots in each lens space that, through analogies with their knot Floer homology and doubly pointed Heegaard diagrams of genus one, may be viewed as generalizations of the two trefoils in S^3. Rasmussen shows that when the `left-handed' one is in the homology class of the dual to a Berge knot of type …

Study extends contact cosmetic surgeries to non-trivial Legendrian knots in L-spaces.

We exhibit an infinite family of knots in the Poincare homology sphere with tunnel number 2 that have a lens space surgery. Notably, these knots are not doubly primitive and provide counterexamples to a few conjectures. In the appendix, it is shown that hyperbolic knots in the Poincare homology sphere with a lens space…

Surgery obstructions extended to integer homology spheres using Heegaard Floer homology.

Standardizes Dunfield-Gong's 4-sphere, solves knot sliceness problem.

We show that if K is a non-trivial knot inside a homology sphere X, the rank of the knot Floer homology group associated with K is strictly bigger than the rank of the Heegaard Floer homology group associated with X.

Gluck twisting certain knots results in standard 4-spheres.