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Arunima Ray

Publications and source records attributed to Arunima Ray.

32 records · Page 2Linked to original sources
arXiv2019

Null, recursively starlike-equivalent decompositions shrink

A subset $E$ of a metric space $X$ is said to be starlike-equivalent if it has a neighbourhood which is mapped homeomorphically into $\mathbb{R}^n$ for some $n$, sending $E$ to a starlike set. A subset $E\subset X$ is said to be recursively starlike-equivalent if it can be expressed as a finite nested union of closed subsets $\{E_i\}_{i=0}^{N+1}$ such that $E_{i}/E_{i+1}\subset X/E_{i+1}$ is starlike-equivalent for each $i$ and $E_{N+1}$ is a point. A decomposition $\mathcal{D}$ of a metric space $X$ is said to be recursively starlike-equivalent, if there exists $N\geq 0$ such that each element of $\mathcal{D}$ is recursively starlike-equivalent of filtration length $N$. We prove that any null, recursively starlike-equivalent decomposition $\mathcal{D}$ of a compact metric space $X$ shrinks, that is, the quotient map $X\to X/\mathcal{D}$ is the limit of a sequence of homeomorphisms. This is a strong generalisation of results of Denman-Starbird and Freedman and is applicable to the proof of Freedman's celebrated disc embedding theorem. The latter leads to a multitude of foundational results for topological $4$-manifolds, including the $4$-dimensional Poincaré conjecture.

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arXiv2019

Average four-genus of two-bridge knots

We prove that the expected value of the ratio between the smooth four-genus and the Seifert genus of two-bridge knots tends to zero as the crossing number tends to infinity.

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arXiv2018

On the Upsilon invariant and satellite knots

We study the effect of satellite operations on the Upsilon invariant of Ozsvath-Stipsicz-Szabo. We obtain results concerning when a knot and its satellites are independent; for example, we show that the set $\{D_{2^i,1}\}_{i=1}^\infty$ is a basis for an infinite rank summand of the group of smooth concordance classes of topologically slice knots, for D the positive clasped untwisted Whitehead double of any knot with positive tau-invariant, e.g. the right-handed trefoil. We also prove that the image of the Mazur satellite operator on the smooth knot concordance group contains an infinite rank subgroup of topologically slice knots.

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arXiv2018

Linear independence of cables in the knot concordance group

We produce infinite families of knots $\{K^i\}_{i\geq 1}$ for which the set of cables $\{K^i_{p,1}\}_{i,p\geq 1}$ is linearly independent in the knot concordance group. We arrange that these examples lie arbitrarily deep in the solvable and bipolar filtrations of the knot concordance group, denoted by $\{F_n\}$ and $\{B_n\}$ respectively. As a consequence, this result cannot be reached by any combination of algebraic concordance invariants, Casson-Gordon invariants, and Heegaard-Floer invariants such as tau, epsilon, and Upsilon. We give two applications of this result. First, for any n>=0, there exists an infinite family $\{K^i\}_{i\geq 1}$ such that for each fixed i, $\{K^i_{2^j,1}\}_{j\geq 0}$ is a basis for an infinite rank summand of $F_n$ and $\{K^i_{p,1}\}_{i, p\geq 1}$ is linearly independent in $F_{n}/F_{n.5}$. Second, for any n>=1, we give filtered counterexamples to Kauffman's conjecture on slice knots by constructing smoothly slice knots with genus one Seifert surfaces where one derivative curve has nontrivial Arf invariant and the other is nontrivial in both $F_n/F_{n.5}$ and $B_{n-1}/B_{n+1}$. We also give examples of smoothly slice knots with genus one Seifert surfaces such that one derivative has nontrivial Arf invariant and the other is topologically slice but not smoothly slice.

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arXiv2017

Concordance of knots in $S^1\times S^2$

We establish a number of results about smooth and topological concordance of knots in $S^1\times S^2$. The winding number of a knot in $S^1\times S^2$ is defined to be its class in $H_1(S^1\times S^2;\mathbb{Z})\cong \mathbb{Z}$. We show that there is a unique smooth concordance class of knots with winding number one. This improves the corresponding result of Friedl-Nagel-Orson-Powell in the topological category. We say a knot in $S^1\times S^2$ is slice (resp. topologically slice) if it bounds a smooth (resp. locally flat) disk in $D^2\times S^2$. We show that there are infinitely many topological concordance classes of non-slice knots, and moreover, for any winding number other than $\pm 1$, there are infinitely many topological concordance classes even within the collection of slice knots. Additionally we demonstrate the distinction between the smooth and topological categories by constructing infinite families of slice knots that are topologically but not smoothly concordant, as well as non-slice knots that are topologically slice and topologically concordant, but not smoothly concordant.

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arXiv2017

4-dimensional analogues of Dehn's lemma

We investigate certain $4$-dimensional analogues of the classical $3$-dimensional Dehn's lemma, giving examples where such analogues do or do not hold, in the smooth and topological categories. In particular, we show that an essential $2$-sphere $S$ in the boundary of a simply connected $4$-manifold $W$ such that $S$ is null-homotopic in $W$ need not extend to an embedding of a ball in $W$. However, if $W$ is simply connected (or more generally a $4$-manifold with abelian fundamental group) with boundary a homology sphere, then $S$ bounds a topologically embedded ball in $W$. Moreover, we give examples where such an $S$ does not bound any smoothly embedded ball in $W$. In a similar vein, we construct incompressible tori $T\subseteq \partial W$ where $W$ is a contractible $4$-manifold such that $T$ extends to a map of a solid torus in $W$, but not to any embedding of a solid torus in $W$. Moreover, we construct an incompressible torus $T$ in the boundary of a contractible $4$-manifold $W$ such that $T$ extends to a topological embedding of a solid torus in $W$ but no smooth embedding. As an application of our results about tori, we address a question posed by Gompf about extending certain families of diffeomorphisms of $3$-manifolds which he has recently used to construct infinite corks.

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arXiv2016

A new family of links topologically, but not smoothly, concordant to the Hopf link

We give new examples of 2-component links with linking number one and unknotted components that are topologically concordant to the positive Hopf link, but not smoothly so - in fact they are not smoothly concordant to the positive Hopf link with a knot tied in the first component. Such examples were previously constructed by Cha-Kim-Ruberman-Strle; we show that our examples are distinct from theirs.

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arXiv2015

Shake slice and shake concordant knots

A crucial step in the surgery-theoretic program to classify smooth manifolds is that of representing a middle--dimensional homology class by a smoothly embedded sphere. This step fails even for the simple 4-manifolds obtained from the 4-ball by adding a 2-handle with framing r along some knot K in S^3. An r-shake slice knot is one for which a generator of the second homology of this 4-manifold can be represented by a smoothly embedded 2-sphere. It is not known whether there exist 0-shake slice knots that are not slice. We define a relative notion of shake sliceness of knots, which we call shake concordance, which is easily seen to be a generalization of classical concordance, and we give the first examples of knots that are 0-shake concordant but not concordant; these may be chosen to be topologically slice. Additionally, for each r we completely characterize r-shake slice and r-shake concordant knots in terms of concordance and satellite operators. Our characterization allows us to construct new families of possible r-shake slice knots that are not slice.

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arXiv2015

Satellite operators as group actions on knot concordance

Any knot in a solid torus, called a pattern or satellite operator, acts on knots in the 3-sphere via the satellite construction. We introduce a generalization of satellite operators which form a group (unlike traditional satellite operators), modulo a generalization of concordance. This group has an action on the set of knots in homology spheres, using which we recover the recent result of Cochran and the authors that satellite operators with strong winding number $\pm 1$ give injective functions on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4--dimensional Poincare Conjecture. The notion of generalized satellite operators yields a characterization of surjective satellite operators, as well as a sufficient condition for a satellite operator to have an inverse. As a consequence, we are able to construct infinitely many non-trivial satellite operators P such that there is a satellite operator $\overline{P}$ for which $\overline{P}(P(K))$ is concordant to K (topologically as well as smoothly in a potentially exotic $S^3\times [0,1]$) for all knots K; we show that these satellite operators are distinct from all connected-sum operators, even up to concordance, and that they induce bijective functions on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4--dimensional Poincare Conjecture.

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arXiv2014

Casson towers and filtrations of the smooth knot concordance group

The n-solvable filtration $\{\mathcal{F}_n\}_{n=0}^\infty$ of the smooth knot concordance group (denoted by $\mathcal{C}$), due to Cochran-Orr-Teichner, has been instrumental in the study of knot concordance in recent years. Part of its significance is due to the fact that certain geometric characterizations of a knot imply membership in various levels of the filtration. We show the counterpart of this fact for two new filtrations of $\mathcal{C}$ due to Cochran-Harvey-Horn, the positive and negative filtrations, denoted by $\{\mathcal{P}_n\}_{n=0}^\infty$ and $\{\mathcal{N}_n\}_{n=0}^\infty$ respectively. In particular, we show that if a knot K bounds a Casson tower of height n+2 in the 4-ball with only positive (resp. negative) kinks in the base-level kinky disk, then K is in $\mathcal{P}_n$ (resp. $\mathcal{N}_n$). En route to this result we show that if a knot K bounds a Casson tower of height n+2 in the 4-ball, it bounds an embedded (symmetric) grope of height n+2, and is therefore, n-solvable (this also implies that topologically slice knots bound arbitrarily tall gropes in the 4-ball). We also define a variant of Casson towers and show that if K bounds a tower of type (2,n) in the 4-ball, it is n-solvable. If K bounds such a tower with only positive (resp. negative) kinks in the base-level kinky disk then K is in $\mathcal{P}_n$ (resp. $\mathcal{N}_n$). Our results show that either every knot which bounds a Casson tower of height three is topologically slice or there exists a knot which is not topologically slice but lies in each $\mathcal{F}_n$. We also give a 3-dimensional characterization, up to concordance, of knots which bound kinky disks in the 4-ball with only positive (resp. negative) kinks; such knots form a subset of $\mathcal{P}_0$ (resp. $\mathcal{N}_0$).

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arXiv2014

Satellite operators with distinct iterates in smooth concordance

Let P be a knot in an unknotted solid torus (i.e. a satellite operator or pattern), K a knot in S^3 and P(K) the satellite of K with pattern P. For any satellite operator P, this correspondence gives a function P : C -> C on the set of smooth concordance classes of knots. We give examples of winding number one satellite operators P and a class of knots K, such that the iterated satellites P^i(K) are distinct as smooth concordance classes, i.e. if i=/=j>0, P^i(K)=/=P^j(K), where each P^i is unknotted when considered as a knot in S^3. This implies that the operators P^i give distinct functions on C, providing further evidence for the fractal nature of C. There are several other applications of our result, as follows. By using topologically slice knots K, we obtain infinite families {P^i(K)} of topologically slice knots that are distinct in smooth concordance. We can also obtain infinite families of 2-component links (with unknotted components and linking number one) which are not smoothly concordant to the positive Hopf link. For a large class of L-space knots K (including the positive torus knots), we obtain infinitely many prime knots {P^i(K)} which have the same Alexander polynomial as K but are not themselves L-space knots.

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arXiv2013

Slice knots which bound punctured Klein bottles

We investigate the properties of knots in S^3 which bound Klein bottles, such that a pushoff of the knot has zero linking number with the knot, i.e. has zero framing. This is motivated by the many results in the literature regarding slice knots of genus one, for example, the existence of homologically essential zero self-linking simple closed curves on genus one Seifert surfaces for algebraically slice knots. Given a knot K bounding a Klein bottle F with zero framing, we show that J, the core of the orientation-preserving band in any disk-band form of F, has zero self-linking. We prove that such a K is slice in a Z[1/2]-homology 4-ball if and only if J is as well, a stronger result than what is currently known for genus one slice knots. As an application, we prove that given knots K and J and any odd integer p, the (2, p) cables of K and J are Z[1/2]-concordant if and only if K and J are Z[1/2]-concordant. In particular, if the (2,1)-cable of a knot K is slice, K is slice in a Z[1/2]-homology ball.

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arXiv2013

Injectivity of satellite operators in knot concordance

Let P be a knot in a solid torus, K a knot in 3-space and P(K) the satellite knot of K with pattern P. This defines an operator on the set of knot types and induces a satellite operator P:C--> C on the set of smooth concordance classes of knots. There has been considerable interest in whether certain such functions are injective. For example, it is a famous open problem whether the Whitehead double operator is weakly injective (an operator is called weakly injective if P(K)=P(0) implies K=0 where 0 is the class of the trivial knot). We prove that, modulo the smooth 4-dimensional Poincare Conjecture, any strong winding number one satellite operator is injective on C. More precisely, if P has strong winding number one and P(K)=P(J), then K is smoothly concordant to J in S^3 x [0,1] equipped with a possibly exotic smooth structure. We also prove that any strong winding number one operator is injective on the topological knot concordance group. If P(0) is unknotted then strong winding number one is the same as (ordinary) winding number one. More generally we show that any satellite operator with non-zero winding number n induces an injective function on the set of Z[1/n]-concordance classes of knots. We extend some of our results to links.

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