Related papers: Knot Concordance and Higher-Order Blanchfield Dual…
Let T denote the group of smooth concordance classes of topologically sice knots. We show that the first quotient in the bipolar filtration of T (i.e. 0-bipolar knots modulo 1-bipolar knots) has infinite rank, even modulo Alexander…
We define an algebraic group comprising symmetric chain complexes which captures the first two stages of the Cochran-Orr-Teichner solvable filtration of the knot concordance group in a single invariant. To achieve this we impose additional…
We address the primary decomposition of the knot concordance group in terms of the solvable filtration and higher-order von Neumann $\rho$-invariants by Cochran, Orr, and Teichner. We show that for a nonnegative integer n, if the connected…
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…
We construct many examples of non-slice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional…
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…
We address primary decomposition conjectures for knot concordance groups, which predict direct sum decompositions into primary parts. We show that the smooth concordance group of topologically slice knots has a large subgroup for which the…
Let $\mathcal{T}$ be the group of smooth concordance classes of topologically slice knots, and $\{0\}\subset\cdots\subset \mathcal{T}_{n+1}\subset\mathcal{T}_{n}\subset \cdots\subset \mathcal{T}_{0}\subset \mathcal{T}$ be the bipolar…
We provide new information about the structure of the abelian group of topological concordance classes of knots in $S^3$. One consequence is that there is a subgroup of infinite rank consisting entirely of knots with vanishing Casson-Gordon…
We study knots of order 2 in the grope filtration $\{\G_h\}$ and the solvable filtration $\{\F_h\}$ of the knot concordance group. We show that, for any integer $n\ge4$, there are knots generating a $\Z_2^\infty$ subgroup of…
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…
It is known that each of the successive quotient groups of the grope and solvable filtrations of the knot concordance group has an infinite rank subgroup. The generating knots of these subgroups are constructed using iterated doubling…
By studying the Heegaard Floer homology of the preimage of a knot K in S^3 inside its double branched cover, we develop simple obstructions to K having finite order in the classical smooth concordance group. As an application, we prove that…
The concordance group of algebraically slice knots is the subgroup of the classical knot concordance group formed by algebraically slice knots. Results of Casson and Gordon and of Jiang showed that this group contains in infinitely…
We define a new smooth concordance homomorphism based on the knot Floer complex and an associated concordance invariant, epsilon. As an application, we show that an infinite family of topologically slice knots are independent in the smooth…
Let $\widehat{\mathcal{C}}_{\mathbb{Z}}$ denote the group of knots in homology spheres that bound homology balls, modulo smooth concordance in homology cobordisms. Answering a question of Matsumoto, the second author previously showed that…
The knot concordance group can be contextualized as organizing problems about 3- and 4-dimensional spaces and the relationships between them. Every 3-manifold is surgery on some link, not necessarily a knot, and thus it is natural to ask…
By a recent result of Livingston, it is known that if a knot has a prime power branched cyclic cover that is not a homology sphere, then there is an infinite family of non-concordant knots having the same Seifert form as the knot. In this…
The knot Floer complex together with the associated concordance invariant epsilon can be used to define a filtration on the smooth concordance group. We show that the indexing set of this filtration contains the natural numbers cross the…
In \cite{NST23}, Nozaki-Sato-Taniguchi defined a family of invariants $ r_s $ for integer homology spheres with filtered instanton homology \cite{FS92}. Coupling these with techniques in classical knot theory, we produce some results in the…