Exact Triangle Research Articles

A surgery triangle for lattice cohomology

Lattice cohomology, defined by N\'emethi in (arXiv:0709.0841), is an invariant of negative definite plumbed 3-manifolds which conjecturally computes the Heegaard Floer homology HF^+. We prove a surgery exact triangle for the lattice cohomology analogous to the one for HF^+. This is a step towards comparing these two invariants.

A combinatorial spanning tree model for knot Floer homology

We iterate Manolescu’s unoriented skein exact triangle in knot Floer homology with coefficients in the field of rational functions over Z/2Z. The result is a spectral sequence which converges to a stabilized version of δ-graded knot Floer homology. The (E2,d2) page of this spectral sequence is an algorithmically computable chain complex expressed in terms of spanning trees, and we show that there are no higher differentials. This gives the first combinatorial spanning tree model for knot Floer homology.

A link surgery spectral sequence in monopole Floer homology

To a link L ⊂ S 3 , we associate a spectral sequence whose E 2 page is the reduced Khovanov homology of L and which converges to a version of the monopole Floer homology of the branched double cover. The pages E k for k ⩾ 2 depend only on the mutation equivalence class of L. We define a mod 2 grading on the spectral sequence which interpolates between the δ-grading on Khovanov homology and the mod 2 grading on Floer homology. We also derive a new formula for link signature that is well adapted to Khovanov homology. More generally, we construct new bigraded invariants of a framed link in a 3-manifold as the pages of a spectral sequence modeled on the surgery exact triangle. The differentials count monopoles over families of metrics parameterized by permutohedra. We utilize a connection between the topology of link surgeries and the combinatorics of graph-associahedra. This also yields simple realizations of permutohedra and associahedra, as refinements of hypercubes.

The new exact solutions of Volterra difference-differential equation and KdV difference-differential equation

Based on the auxiliary equation method and the trial function method, a method for combining function transformation with auxiliary equation is proposed. And the method is applied to construct new exact solitary wave solutions and triangle function solutions of Volterra and KdV difference-differential equations with the help of symbolic computation system Mathematica. Our method can also be applied to other nonlinear difference-differential equations.

An unoriented skein exact triangle for knot Floer homology

Given a crossing in a planar diagram of a link in the three-sphere, we show that the knot Floer homologies of the link and its two resolutions at that crossing are related by an exact triangle. As a consequence, we deduce that for any quasi-alternating link, the total rank of its knot Floer homology is equal to the determinant of the link.

Singularities in derived categories

Let A be a finite dimensional algebra over an algebraically closed field k and let M and N be two complexes in the bounded derived category Db(A) of finitely generated A-modules. Together with Alexander Zimmermann we have defined a notion of degeneration for derived categories. We say that M degenerates to N if there is a complex Z and an exact triangle N→M⊕Z→Z→N[1]. In this paper we define and study the type of singularity at every degeneration in the bounded derived categrory.

Rohlin?s invariant and gauge theory, I. Homology 3-tori

This is the first in a series of papers exploring the relationship between the Rohlin invariant and gauge theory. We discuss a Casson-type invariant of a 3-manifold Y with the integral homology of the 3-torus, given by counting projectively flat U(2)-connections. We show that its mod 2 evaluation is given by the triple cup product in cohomology, and so it coincides with a certain sum of Rohlin invariants of Y. Our counting argument makes use of a natural action of H^1 (Y;Z_2) on the moduli space of projectively flat connections; along the way we construct perturbations that are equivariant with respect to this action. Combined with the Floer exact triangle, this gives a purely gauge-theoretic proof that Casson's homology sphere invariant reduces mod 2 to the Rohlin invariant

Improved tanh-function method and the new exact solutions for the general variab le coefficient KdV equation and MKdV equation

In this paper, by using the extended tanh-function method,the general variable coefficient KdV and MKdV equations are reduced to first-order variable coefficient nonlinear ordinary differential equations,and then the new exact solutions for these equations,which include exact soliton-like,rational formal and triangle function solutions,are obtained through solving these ordinary differential equations-

A GENERALIZATION OF HOMOLOGICAL ALGEBRA

Our aim in this paper is to introduce a generalization of some notions in homological algebra. We define the concepts of chain U-complex, U-homology, chain (U, U')-map, chain (U, U')-homotopy and <TEX>$\mu$</TEX>-functor. We also obtain some interesting results. We use these results to find a generalization of Lambek Lemma, Snake Lemma, Connecting Homomorphism and Exact Triangle.

Casson's knot invariant and gauge theory

It is known that twice the Casson invariant for integral homology 3 spheres is equal to the Euler characteristic of the Floer homology group of them. Here we show that a similar result holds in case of the Casson invariant for knots in integral homology 3 spheres. This result is obtained as a corollary of Floer's exact triangle. But we give a more elementary proof here. We also show that a similar result holds in case of the Casson–Walker invariant for null homologous knots in rational homology 3 spheres. This result is not obtained as a corollary of Floer's exact triangle, and so is new. These results will serve as a starting point to obtain the Dehn surgery formula for the Floer homology groups of general 3-dimensional manifolds.

The Exactness Theorem for Floer Homology

In this article, we study a long exact sequence relating the mstanton homology of two homology 3-spheres which are obtained from each other by ± 1 -surgery. We prove that the long exact sequence, via a short exact sequence of chain complex, is the same as the sequence defined by the exact triangle of cobordisms introduced by Floer.

Triangle amplitude with off-shell Coulomb T matrix for exchange reactions in atomic and nuclear physics.

The lowest-order rescattering contribution (triangle amplitude) in three-body models of exchange reactions with charged particles contains the off-shell two-body T matrix describing the intermediate-state Coulomb scattering of charged subsystems. General properties of the exact exchange triangle amplitude, when the incoming and outgoing particles are on the energy shell, are derived. This includes the analytic behavior, i.e., the positions and characters of its leading singularities, in the cos\ensuremath{\vartheta} plane, where \ensuremath{\vartheta} is the scattering angle, in the vicinity of the forward- and backward-scattering directions. Since for computational reasons the Coulomb T matrix is usually replaced by the Coulomb potential, the effects of such an approximation on the analytic properties are investigated. The theoretically established behavior of the exact and the approximate exchange triangle amplitudes is then illustrated by numerical calculations, for both atomic and nuclear reactions, for energies below and above the corresponding three-body dissociation thresholds, for elastic and inelastic exchange. \textcopyright{} 1996 The American Physical Society.