arXiv research

Obstructs 2-torsion in rational knot concordance group.

New knots not rationally concordant to their reverses found.

New knots show linear independence in slice concordance.

The following is a long-standing open question: "If the zero-framed surgeries on two knots in the 3-sphere are integral homology cobordant, are the knots themselves concordant?" We show that an obvious rational version of this question has a negative answer. Namely, we give examples of knots whose zero-framed surgeries…

The paper proves that rational concordance of double twist knots is reciprocal.

We study the group of rational concordance classes of codimension two knots in rational homology spheres. We give a full calculation of its algebraic theory by developing a complete set of new invariants. For computation, we relate these invariants with limiting behaviour of the Artin reciprocity over an infinite tower…

New method uses rational Witt span to bound concordance crosscap number of knots.

Knots generating infinite subgroup bound rational homology balls.

Study calculates homotopy groups and derivatives for disc diffeomorphisms.

A theory of signatures for odd-dimensional links in rational homology spheres is studied via their generalized Seifert surfaces. The jump functions of signatures are shown invariant under appropriately generalized concordance and a special care is given to accommodate 1-dimensional links with mutual linking. Furthermor…

New findings on knots that are both topologically and rationally slice.

Ribbon cobordism forms a partial order in 3-manifolds.

We prove that certain fibered, $-$amphicheiral knots are rationally slice. Moreover, we show that the concordance invariants $ν^+$ and $Υ(t)$ from Heegaard Floer homology vanish for a class of knots that includes rationally slice knots.

Khovanov homology shows (4,5) torus knot is a summand in its concordance class.

We show that there are links whose individual components are concordant to the unknot, but which are not concordant to any link with unknotted components. We give examples in the topological category, and examples in the smooth category which are topologically slice. We also give generalizations regarding components of…

We study two homomorphisms to the rational homology sphere group. If $ψ$ denotes the inclusion homomorphism from the integral homology sphere group, then using work of Lisca we show that the image of $ψ$ intersects trivially with the subgroup of the rational homology sphere group generated by lens spaces. As corollarie…

We classify connected sums of three-dimensional lens spaces which smoothly bound rational homology balls. We use this result to determine the order of each lens space in the group of rational homology 3-spheres up to rational homology cobordisms, and to determine the concordance order of each 2-bridge knot.

New invariant detects infinite order cabled knots.

Alternating links bound rational homology balls if their chessboard lattice is cubiquitous.

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

New knots not slice in rational 4-balls found.

New example shows figure eight knot not smoothly concordant but homology cobordant.

This paper provides a rational model for fiberwise THH transfer using A-infinity algebras.

Under the relation of $0$-concordance, the set of knotted 2-spheres in $S^4$ forms a commutative monoid $\mathcal{M}_0$ with the operation of connected sum. Sunukjian has recently shown that $\mathcal{M}_0$ contains a submonoid isomorphic to $\mathbb{Z}^{\ge 0}$. In this note, we show that $\mathcal{M}_0$ contains a su…

Positive knots are minimal in a specific knot ordering.

We extend the construction of upsilon-type invariants to null-homologous knots in rational homology three-spheres. By considering $m$-fold cyclic branched covers with $m$ a prime power, this extension provides new knot concordance invariants $Υ_m^C (K)$ of knots in $S^3$. We give computations of these invariants for so…

We give simple homological conditions for a rational homology 3-sphere Y to have infinite order in the rational homology cobordism group, and for a collection of rational homology spheres to be linearly independent. These translate immediately to statements about knot concordance when Y is the branched double cover of …

For all n > 0 there is a homomorphism from the smooth concordance group of knots in dimension 2n + 1 to an algebraically defined group called the rational algebraic concordance group. This algebraic concordance group splits as a direct sum of groups indexed by polynomials. For n > 1 the homomorphism is injective. This …

Let C_T be the subgroup of the smooth knot concordance group generated by topologically slice knots and let C_D be the subgroup generated by knots with trivial Alexander polynomial. We prove the quotient C_T/C_D is infinitely generated, and uncover similar structure in the 3-dimensional rational spin bordism group. Our…

A polynomial f(t) with rational coefficients is strongly irreducible if f(t^k) is irreducible for all positive integers k. Likewise, two polynomials f and g are strongly coprime if f(t^k) and g(t^l) are relatively prime for all positive integers k and l. We provide some sufficient conditions for strong irreducibility a…

In his pioneering work from 1969, Jerry Levine introduced a complete set of invariants of algebraic concordance of knots. The evaluation of these invariants requires a factorization of the Alexander polynomial of the knot, and is therefore in practice often hard to realize. We thus propose the study of an alternative s…

The study shows knots from 3-braids cannot be concordant to a specific Legendrian unknot.

New knots found that are 4-genus minimal.

We consider the question of when a rational homology 3-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology embeds in any other …

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

New invariants from framed instanton homology for knot concordance.

Study of knots with generalized Mazur patterns and their invariants.

The theory of signature invariants of links in rational homology spheres is applied to covering links of homology boundary links. From patterns and Seifert matrices of homology boundary links, an explicit formula is derived to compute signature invariants of their covering links. Using the formula, we produce fused bou…

Survey of homology cobordism group with open problems.

We investigate rational homology cobordisms of 3-manifolds with non-zero first Betti number. This is motivated by the natural generalization of the slice-ribbon conjecture to multicomponent links. In particular we consider the problem of which rational homology $S^1\times S^2$'s bound rational homology $S^1\times D^3$'…

Levine defined the rational algebraic knot concordance group and proved that each nontrivial element is of order two, of order four, or of infinite order. The determination of the order of an element depends on a p-adic analysis for all primes p. Here we develop effective means to determine the order of any element tha…

Every negative amphichiral knot is rationally slice.

Formula calculates knot Floer complexes for specific cable knots.

We define homotopy-theoretic invariants of knots in prime 3-manifolds. Fix a knot J in a prime 3-manifold M. Call a knot K in M concordant to J if it cobounds a properly embedded annulus with J in MxI, and call K J-characteristic if there is a degree-one map f:M --> M throwing K onto J and mapping M-K to M-J. These inv…

We obtain new invariants of topological link concordance and homology cobordism of 3-manifolds from Hirzebruch-type intersection form defects of towers of iterated p-covers. Our invariants can extract geometric information from an arbitrary depth of the derived series of the fundamental group, and can detect torsion wh…

Determines conditions for ribbon cobordisms between lens spaces.

The smooth rational homology cobordism group of rational homology three spheres, T, contains subgroups T_p generated by 3-manifolds with first homology p-torsion, where p is a prime. Rochlin's theorem and gauge theoretic methods show that the inclusion of the direct sum of the T_p into T has infinitely generated kernel…

In 1997 Cochran-Orr-Teichner introduced a natural filtration, called the n-solvable filtration, of the smooth knot concordance group, C. Its terms {F_n} are indexed by half integers. We show that each associated graded abelian group G_n=F_n/F_{n.5}, n>1, contains infinite linearly independent sets of elements of order …