Knot Modules. I

Some Properties of I-Closed Modules

Patterns in Knot Cohomology, I

Cohomology theory of links, introduced in [Kh1], is combinatorial. Dror Bar-Natan recently wrote a program that found ranks of cohomology groups of all prime knots with up to 11 crossings [BN]. His surprising experimental data is discussed in this note.

Cocycle Knot Invariants from Quandle Modules and Generalized Quandle Cohomology

Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Graña. We specialize that theory to the case when there is a group action on the coefficients. First, quandle modules are used to generalize Burau representations and Alexander modules for classical knots. Second, 2-cocycles valued in non-abelian groups are us...

Surgery on a Knot in Surface × I

Suppose F is a compact orientable surface, K is a knot in F × I, and (F × I)surg is the 3-manifold obtained by some non-trivial surgery on K. If F × {0} compresses in (F × I)surg , then there is an annulus in F × I with one end K and the other end an essential simple closed curve in F ×{0}. Moreover, the end of the annulus at K determines the surgery slope. An application: suppose M is a compac...

FINITENESS PROPERTIES OF LOCALE COHOMOLOGY MODULES FOR (I;J)- MINIMAX MODULES

ABSTRACT. Let R be a commutative noetherian ring, I and J are two ideals of R. Inthis paper we introduce the concept of (I;J)- minimax R- module, and it is shown thatif M is an (I;J)- minimax R- module and t a non-negative integer such that HiI;J(M) is(I;J)- minimax for all i