Registered users receive a variety of benefits including the ability to customize email alerts, create favorite journals list, and save searches.
Please note that a Project Euclid web account does not automatically grant access to full-text content. An institutional or society member subscription is required to view non-Open Access content.
Contact email@example.com with any questions.
For each graph, we construct a bigraded chain complex whose graded Euler characteristic is a version of the Tutte polynomial. This work is motivated by earlier work of Khovanov, Helme-Guizon and Rong, and others.
For a knot in , let be the characteristic toric sub-orbifold of the orbifold as defined by Bonahon–Siebenmann. If has unknotting number one, we show that an unknotting arc for can always be found which is disjoint from , unless either is an EM–knot (of Eudave-Muñoz) or contains an EM–tangle after cutting along . As a consequence, we describe exactly which large algebraic knots (ie, algebraic in the sense of Conway and containing an essential Conway sphere) have unknotting number one and give a practical procedure for deciding this (as well as determining an unknotting crossing). Among the knots up to 11 crossings in Conway’s table which are obviously large algebraic by virtue of their description in the Conway notation, we determine which have unknotting number one. Combined with the work of Ozsváth–Szabó, this determines the knots with 10 or fewer crossings that have unknotting number one. We show that an alternating, large algebraic knot with unknotting number one can always be unknotted in an alternating diagram.
As part of the above work, we determine the hyperbolic knots in a solid torus which admit a non-integral, toroidal Dehn surgery. Finally, we show that having unknotting number one is invariant under mutation.
We study global fixed points for actions of Coxeter groups on nonpositively curved singular spaces. In particular, we consider property , an analogue of Serre’s property FA for actions on complexes. Property has implications for irreducible representations and complex of groups decompositions. In this paper, we give a specific condition on Coxeter presentations that implies and show that this condition is in fact equivalent to for and 2. As part of the proof, we compute the Gersten–Stallings angles between special subgroups of Coxeter groups.
We consider the existence of simple closed geodesics or “geodesic knots” in finite volume orientable hyperbolic 3-manifolds. Previous results show that a least one geodesic knot always exists [Bull. London Math. Soc. 31(1) (1999) 81–86], and that certain arithmetic manifolds contain infinitely many geodesic knots [J. Diff. Geom. 38 (1993) 545–558], [Experimental Mathematics 10(3) (2001) 419–436]. In this paper we show that all cusped orientable finite volume hyperbolic 3-manifolds contain infinitely many geodesic knots. Our proof is constructive, and the infinite family of geodesic knots produced approach a limiting infinite simple geodesic in the manifold.
Using stably free non-free relation modules we construct an infinite collection of 2–dimensional homotopy types, each of Euler-characteristic one and with trefoil fundamental group. This provides an affirmative answer to a question asked by Berridge and Dunwoody [J. London Math. Soc. 19 (1979) 433–436]. We also give new examples of exotic relation modules. We show that the relation module associated with the generating set for the Baumslag–Solitar group is stably free non-free of rank one.
There is an elegant relation found by Fabricius-Bjerre [Math. Scand 40 (1977) 20–24] among the double tangent lines, crossings, inflections points, and cusps of a singular curve in the plane. We give a new generalization to singular curves in . We note that the quantities in the formula are naturally dual to each other in , and we give a new dual formula.
Natural linear and coalgebra transformations of tensor algebras are studied. The representations of certain combinatorial groups are given. These representations are connected to natural transformations of tensor algebras and to the groups of the homotopy classes of maps from the James construction to loop spaces. Applications to homotopy theory appear in a sequel [Applications of combinatorial groups to Hopf invariant and the exponent problem, Algebr. Geom. Topol. 6 (2006) 2229-2255].
Combinatorial groups together with the groups of natural coalgebra transformations of tensor algebras are linked to the groups of homotopy classes of maps from the James construction to a loop space. This connection gives rise to applications to homotopy theory. The Hopf invariants of the Whitehead products are studied and a rate of exponent growth for the strong version of the Barratt Conjecture is given.
It is a classical observation that a based continuous functor from the category of finite CW–complexes to the category of based spaces that takes homotopy pushouts to homotopy pullbacks “represents” a homology theory—the collection of spaces obtained by evaluating on spheres yields an –prespectrum. Such functors are sometimes referred to as linear or excisive. The main theorem of this paper provides an equivariant analogue of this result. We show that a based continuous functor from finite –CW–complexes to based –spaces represents a genuine equivariant homology theory if and only if it takes –homotopy pushouts to –homotopy pullbacks and satisfies an additional condition requiring compatibility with Atiyah duality for orbit spaces .
Our motivation for this work is the development of a recognition principle for equivariant infinite loop spaces. In order to make the connection to infinite loop space theory precise, we reinterpret the main theorem as providing a fibrancy condition in an appropriate model category of spectra. Specifically, we situate this result in the context of the study of equivariant diagram spectra indexed on the category of based –spaces homeomorphic to finite –CW–complexes for a compact Lie group . Using the machinery of Mandell–May–Schwede–Shipley, we show that there is a stable model structure on this category of diagram spectra which admits a monoidal Quillen equivalence to the category of orthogonal –spectra. We construct a second “absolute” stable model structure which is Quillen equivalent to the stable model structure. There is a model-theoretic identification of the fibrant continuous functors in the absolute stable model structure as functors such that for the collection forms an ––prespectrum as varies over the universe . Thus, our main result provides a concrete identification of the fibrant objects in the absolute stable model structure.
This description of fibrant objects in the absolute stable model structure makes it clear that in the equivariant setting we cannot hope for a comparison between the category of equivariant continuous functors and equivariant –spaces, except when is finite. We provide an explicit analysis of the failure of the category of equivariant –spaces to model connective –spectra, even for .
If a closed, orientable hyperbolic –manifold has volume at most 1.22 then has dimension at most for every prime , and and have dimension at most . The proof combines several deep results about hyperbolic –manifolds. The strategy is to compare the volume of a tube about a shortest closed geodesic with the volumes of tubes about short closed geodesics in a sequence of hyperbolic manifolds obtained from by Dehn surgeries on .
It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links.
The Morton–Franks–Williams inequality gives a lower bound for braid index. And sharpness of the inequality on a knot type implies the truth of the conjecture for the knot type.
We prove that there are infinitely many examples of knots and links for which the inequality is not sharp but the conjecture is still true. We also show that if the conjecture is true for and then it is also true for the –cable of and for the connect sum of and
It is a conjecture that the signature of a positive link is bounded below by an increasing function of its negated Euler characteristic. In relation to this conjecture, we apply the generator description for canonical genus to show that the boundedness of the genera of positive knots with given signature can be algorithmically partially decided. We relate this to the result that the set of knots of canonical genus is dominated by a finite subset of itself in the sense of Taniyama’s partial order.
Bing doubling is an operation which produces a 2–component boundary link from a knot . If is slice, then is easily seen to be boundary slice. In this paper, we investigate whether the converse holds. Our main result is that if is boundary slice, then is algebraically slice. We also show that the Rasmussen invariant can tell that certain Bing doubles are not smoothly slice.
We consider surgery moves along –component Brunnian links in compact connected oriented –manifolds, where the framing of the components is in . We show that no finite type invariant of degree can detect such a surgery move. The case of two link-homotopic Brunnian links is also considered. We relate finite type invariants of integral homology spheres obtained by such operations to Goussarov–Vassiliev invariants of Brunnian links.
Given a connected, compact, totally geodesic submanifold of noncompact type inside a compact locally symmetric space of noncompact type , we provide a sufficient condition that ensures that ; in low dimensions, our condition is also necessary. We provide conditions under which there exist a tangential map of pairs from a finite cover to the nonnegatively curved duals .
In this article, we give a first prototype-definition of overtwistedness in higher dimensions. According to this definition, a contact manifold is called overtwisted if it contains a plastikstufe, a submanifold foliated by the contact structure in a certain way. In three dimensions the definition of the plastikstufe is identical to the one of the overtwisted disk. The main justification for this definition lies in the fact that the existence of a plastikstufe implies that the contact manifold does not have a (semipositive) symplectic filling.
Let be a finitely generated, amenable group. Using an idea of É Ghys, we prove that if has a nontrivial, orientation-preserving action on the real line, then has an infinite, cyclic quotient. (The converse is obvious.) This implies that if has a faithful action on the circle, then some finite-index subgroup of has the property that all of its nontrivial, finitely generated subgroups have infinite, cyclic quotients. It also means that every left-orderable, amenable group is locally indicable. This answers a question of P Linnell.