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Corollary 8 Subcomplexes $Y_{1},Y_{2},Y_{3}$ are graded subcomplexes of the complex $\overline{C}(D_{2}).$

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

Mikhail Khovanov

Original source: arXiv:math/9908171v2

In this chapter

Chapter overview

1 Introduction
2 Preliminaries
2.1 The ring $R$
2.2 The algebra $A$
2.3 Algebra $A$ and (1+1)-dimensional cobordisms
2.4 Kauffman bracket
3 Cubes
3.1 Complexes of $R$ -modules
3.2 Commutative cubes
3.3 skew-commutative cubes
3.4 skew-commutative cubes and complexes
4 Diagrams
4.1 Reidemeister moves
4.2 Constructing cubes and complexes from plane diagrams
4.3 Surfaces and cube morphisms
5 Transformations
5.1 Left-twisted curl
5.2 Right-twisted curl
5.3 The tangency move
5.4 Triple point move
6 Properties of cohomology groups
6.1 Some elementary properties
6.2 Computational shortcuts and cohomology of $(2,n)$ torus links
6.3 Link cobordisms and maps of cohomology groups
7 Setting $c$ to $0$
7.1 Cohomology groups ${\cal H}^{i,j}$
7.2 Properties of ${\cal H}^{i,j}$ : Euler characteristic, change of orientation
7.3 Cohomology of the mirror image
7.4 Cohomology of the disjoint union and connected sum of knots
7.5 A spectral sequence
7.6 Examples
7.7 An application to the crossing number
8 Invariants of $(1,1)$ -tangles
8.1 Graded $A$ -modules
8.2 Non-closed (1+1)-cobordisms
8.3 $(1,1)$ -tangles
8.4 Invariants
References

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