Connected sum

In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces.

More generally, one can also join manifolds together along identical submanifolds; this generalization is often called the fiber sum. There is also a closely related notion of a connected sum on knots, called the knot sum or composition of knots.

Connected sum at a point

A connected sum of two m-dimensional manifolds is a manifold formed by deleting a ball inside each manifold and gluing together the resulting boundary spheres.

If both manifolds are oriented, there is a unique connected sum defined by having the gluing map reverse orientation. Although the construction uses the choice of the balls, the result is unique up to homeomorphism. One can also make this operation work in the smooth category, and then the result is unique up to diffeomorphism. There are subtle problems in the smooth case: not every diffeomorphism between the boundaries of the spheres gives the same composite manifold, even if the orientations are chosen correctly. For example, Milnor showed that two 7-cells can be glued along their boundary so that the result is an exotic sphere homeomorphic but not diffeomorphic to a 7-sphere. However there is a canonical way to choose the gluing which gives a unique well defined connected sum. This uniqueness depends crucially on the disc theorem, which is not at all obvious.

Knot

A knot is a method of fastening or securing linear material such as rope by tying or interweaving. It may consist of a length of one or several segments of rope, string, webbing, twine, strap, or even chain interwoven such that the line can bind to itself or to some other object (the "load"). Knots have been the subject of interest for their ancient origins, their common uses, and the area of mathematics known as knot theory.

Use

There is a large variety of knots, each with properties that make it suitable for a range of tasks. Some knots are used to attach the rope (or other knotting material) to other objects such as another rope, cleat, ring, or stake. Some knots are used to bind or constrict objects. Decorative knots usually bind to themselves to produce attractive patterns.

Teaching

While some people can look at diagrams or photos and tie the illustrated knots, others learn best by watching how a knot is tied. Knot tying skills are often transmitted by sailors, scouts, climbers, cavers, arborists, rescue professionals, stagehands, fishermen, linemen and surgeons. The International Guild of Knot Tyers is an organisation dedicated to the Promotion of Knot tying.

Glossary of graph theory

This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by edges.

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κ(G) is the size of the maximum clique in G; see clique.

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Knot (papermaking)

Knots are unwanted, large, dark aggregates of wood fibres when making chemical pulp.

Knots are incompletely cooked wood chips coming out of the digester. Their origin is often dense parts of branches, such as compression wood or timber knots – hence the name. Knots can also stem from large/oversized wood chips due to insufficient impregnation with cooking liquors. The content of lignin is very high. In kraft pulping knots are typically 0.5–3.0 % of the digester throughput. The knots are screened from the pulp, because if left in the pulp they may damage washing equipment and consume large amounts of bleaching chemicals. They are normally sent back to the digester and re-cooked so that their fibres are not wasted.

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References