Higher-dimensional Knots Research Articles

Doubly slice knots and metabelian obstructions

An [Formula: see text]-dimensional knot [Formula: see text] is called doubly slice if it occurs as the cross section of some unknotted [Formula: see text]-dimensional knot. For every [Formula: see text] it is unknown which knots are doubly slice, and this remains one of the biggest unsolved problems in high-dimensional knot theory. For [Formula: see text], we use signatures coming from [Formula: see text]-cohomology to develop new obstructions for [Formula: see text]-dimensional knots with metabelian knot groups to be doubly slice. For each [Formula: see text], we construct an infinite family of knots on which our obstructions are nonzero, but for which double sliceness is not obstructed by any previously known invariant.

Formally integrable complex structures on higher dimensional knot spaces

Formally integrable complex structures on higher dimensional knot spaces

Formal Global AKSZ Gauge Observables and Generalized Wilson Surfaces

We consider a construction of observables by using methods of supersymmetric field theories. In particular, we give an extension of AKSZ-type observables constructed in Mnev (Lett Math Phys 105:1735–1783, 2015) using the Batalin–Vilkovisky structure of AKSZ theories to a formal global version with methods of formal geometry. We will consider the case where the AKSZ theory is “split” which will give an explicit construction for formal vector fields on base and fiber within the formal global action. Moreover, we consider the example of formal global generalized Wilson surface observables whose expectation values are invariants of higher-dimensional knots by using BF field theory. These constructions give rise to interesting global gauge conditions such as the differential quantum master equation and further extensions.

Double L–groups and doubly slice knots

We develop a theory of chain complex double-cobordism for chain complexes equipped with Poincar\'{e} duality. The resulting double-cobordism groups are a refinement of Ranicki's torsion algebraic $L$-groups for localisations of a commutative ring with involution. The refinement is analogous to the difference between metabolic and hyperbolic linking forms. We apply the double $L$-groups in high-dimensional knot theory to define an invariant for doubly-slice $n$-knots. We prove that the "stably doubly-slice implies doubly-slice" property holds (algebraically) for Blanchfield forms, Seifert forms and for the Blanchfield complexes of $n$-knots for $n\geq 1$.

Counterexamples to Kauffman's conjectures on slice knots

In the late 1960s Jerome Levine classified the odd high-dimensional knot concordance groups in terms of a linking matrix associated to an arbitrary bounding manifold for the knot. His proof fails for classical knots in S3. Yet this philosophy has remained the only known strategy for understanding the classical knot concordance group. We show that this strategy is fundamentally flawed. Specifically, in 1982, in support of Levine's philosophy, Louis Kauffman conjectured that if a knot in S3 is a slice knot then on any Seifert surface for that knot there exists a homologically essential simple closed curve of self-linking zero which is itself a slice knot, or at least has Arf invariant zero. Since that time, considerable evidence has been amassed in support of this conjecture. In particular, many invariants that obstruct a knot from being a slice knot have been explicitly expressed in terms of invariants of such curves on its Seifert surface. We give counterexamples to Kauffman's conjecture, that is, we exhibit (smoothly) slice knots that admit (unique minimal genus) Seifert surfaces on which every homologically essential simple closed curve of self-linking zero has non-zero Arf invariant and non-zero signatures.

Quandles and symmetric quandles for higher dimensional knots

A symmetric quandle is a quandle with a good involution. For a knot in $\mathbb R^3$, a knotted surface in $\mathbb R^4$ or an $n$-manifold knot in $\mathbb R^{n+2}$, the knot symmetric quandle is defined. We introduce the notion of a symmetric qua

Homology equivalences of manifolds and zero-in-the-spectrum examples

Working with group homomorphisms, a construction of manifolds is introduced to preserve homology groups. The construction gives as special cases Qullien's plus construction with handles obtained by Hausmann, the existence of one-sided $h$-cobordism of Guilbault and Tinsley, the existence of homology spheres and higher-dimensional knots proved by Karvaire. We also use it to get counter-examples to the zero-in-the-spectrum conjecture found by Farber-Weinberger, and by Higson-Roe-Schick.

THE ALGEBRA OF RACK AND QUANDLE COHOMOLOGY

This paper presents the first complete calculation of the cohomology of any nontrivial quandle, establishing that this cohomology exhibits a very rich and interesting algebraic structure. Rack and quandle cohomology have been applied in recent years to attack a number of problems in the theory of knots and their generalizations like virtual knots and higher-dimensional knots. An example of this is estimating the minimal number of triple points of surface knots [E. Hatakenaka, An estimate of the triple point numbers of surface knots by quandle cocycle invariants, Topology Appl139(1–3) (2004) 129–144.]. The theoretical importance of rack cohomology is exemplified by a theorem [R. Fenn, C. Rourke and B. Sanderson, James bundles and applications, Proc. London Math. Soc. (3)89(1) (2004) 217–240] identifying the homotopy groups of a rack space with a group of bordism classes of high-dimensional knots. There are also relations with other fields, like the study of solutions of the Yang–Baxter equations.

Unsolvable problems about higher-dimensional knots and related groups

Unsolvable problems about higher-dimensional knots and related groups

LOCAL MOVE IDENTITIES FOR THE ALEXANDER POLYNOMIALS OF HIGH-DIMENSIONAL KNOTS AND INERTIA GROUPS

As analogues of the well-known skein relations for the Alexander and the Jones polynomials for classical links, we present three relations that hold among invariants of high dimensional knots differing by "local moves". Two are for the Alexander polynomials and the other is for the Arf-invariants, the inertia group and the bP-subgroup.

KNOT GROUPS AND SLICE CONDITIONS

We introduce the notions of "k-connected-slice" and "π1-slice", interpolating between "homotopy ribbon" and "slice". Every high-dimensional knot group π is the group of an (n - 1)-connected-slice n-knot, for all n ≥ 3. However, if π is the group of an n-connected-slice n-knot, the augmentation ideal I(π) has deficiency 1 as a module, while (n + 1)-connected-slice n-knots are trivial. If π is the group of a π1-slice 2-knot and π' is finitely generated, then π' is free, and so def(π) = 1.

A New Route to the Interpretation of Hopf Invariant

We discuss an object from algebraic topology, Hopf invariant, and reinterpret it in terms of the ϕ-mapping topological current theory. The main purpose of this paper is to present a new theoretical framework, which can directly give the relationship between Hopf invariant and the linking numbers of the higher dimensional submanifolds of Euclidean space R2n-1. For the sake of this purpose we introduce a topological tensor current, which can naturally deduce the (n-1)-dimensional topological defect in R2n-1 space. If these (n-1)-dimensional topological defects are closed oriented submanifolds of R2n-1, they are just the (n - 1)-dimensional knots. The linking number of these knots is well defined. Using the inner structure of the topological tensor current, the relationship between Hopf invariant and the linking numbers of the higher-dimensional knots can be constructed.

Topological triviality of smoothly knotted surfaces in $4$-manifolds

Some generalizations and variations of the Fintushel-Stern rim surgery are known to produce smoothly knotted surfaces. We show that if the fundamental groups of their complements are cyclic, then these surfaces are topologically unknotted. Using a twist-spinning construction from high-dimensional knot theory, we construct examples of knotted surfaces whose complements have cyclic fundamental groups.

Configuration space integral for longn–knots and the Alexander polynomial

There is a higher dimensional analogue of the perturbative Chern–Simons theory in the sense that a similar perturbative series as in 3 dimensions, which is computed via configuration space integral, yields an invariant of higher dimensional knots (Bott– Cattaneo–Rossi invariant). This invariant was constructed by Bott for degree 2 and by Cattaneo–Rossi for higher degrees. However, its feature is yet unknown. In this paper we restrict the study to long ribbon n–knots and characterize the Bott–Cattaneo–Rossi invariant as a finite type invariant of long ribbon n–knots introduced by Habiro– Kanenobu–Shima [10]. As a consequence, we obtain a nontrivial description of the Bott–Cattaneo–Rossi invariant in terms of the Alexander polynomial.

Higher dimensional knot spaces for manifolds with vector cross products

Vector cross product structures on manifolds include symplectic, volume, G 2 - and Spin ( 7 ) -structures. We show that the knot spaces of such manifolds have natural symplectic structures, and relate instantons and branes in these manifolds to holomorphic disks and Lagrangian submanifolds in their knot spaces. For the complex case, the holomorphic volume form on a Calabi–Yau manifold defines a complex vector cross product structure. We show that its isotropic knot space admits a natural holomorphic symplectic structure. We also relate the Calabi–Yau geometry of the manifold to the holomorphic symplectic geometry of its isotropic knot space.

Certain Cyclically Presented Groups with the Same Polynomial

We consider three infinite families of cyclic presentations of groups, depending on a finite set of integers and having the same polynomial. Then we prove that the corresponding groups with the same parameters are isomorphic, and that the groups are almost all infinite. Finally, we completely compute the maximal Abelian quotients of such groups, and show that their HNN extensions are high-dimensional knot groups. Our results contain as particular cases the main theorems obtained in two nice articles: Johnson et al. (1999) and Havas et al. (2001).

Knot and braid invariants from contact homology II

We present a topological interpretation of knot and braid contact homology in degree zero, in terms of cords and skein relations. This interpretation allows us to extend the knot invariant to embedded graphs and higher-dimensional knots. We calculate the knot invariant for two-bridge knots and relate it to double branched covers for general knots. In the appendix we show that the cord ring is determined by the fundamental group and peripheral structure of a knot and give applications.

Wilson Surfaces and Higher Dimensional Knot Invariants

An observable for nonabelian, higher-dimensional forms is introduced, its properties are discussed and its expectation value in BF theory is described. This is shown to produce potential and genuine invariants of higher-dimensional knots.

PULL BACK RELATION FOR NON-SPHERICAL KNOTS

We introduce a new relation for high dimensional non-spherical knots, which is motivated by the codimension two surgery theory: a knot is a pull back of another knot if the former is obtained as the inverse image of the latter by a certain degree one map between the ambient spheres. We show that this relation defines a partial order for (2n-1)-dimensional simple fibered knots for n≥3. We also give some related results concerning cobordisms and isotopies of knots together with several important explicit examples.

Elementary moves for higher dimensional knots

For smooth knottings of compact (not necessarily orientable) $n$-dimensional manifolds in ${\mathbb R}^{n+2}$ (or ${\mathbb S}^{n+2}$ ), we generalize the notion of knot moves to higher dimensions. This reproves and generalizes the Reidemeister moves of c