Download Algebraic Topology (Colloquium Pbns. Series, Vol 27) by Solomon Lefschetz PDF

By Solomon Lefschetz

ISBN-10: 0821810278

ISBN-13: 9780821810279

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Extra info for Algebraic Topology (Colloquium Pbns. Series, Vol 27)

Example text

Fox and J. Milnor suggested looking at a knot from a 4-dimensional point of view which led to the slice knot [Fox 1962]. During the last decades geometric methods have gained importance in knot theory – but they are, as a rule, no longer elementary. 1. Show that the trefoil is symmetric, and that the four-knot is both symmetric and amphicheiral. 2. Let p(k) ⊂ E 2 be a regular projection of a link k. The plane E 2 can be coloured with two colours in such a way that regions with a common arc of p(k) in their boundary obtain different colours (chess-board colouring).

Fox and J. Milnor introduced the notion of a slice knot. It arises from the study of embeddings S 2 ⊂ S 4 [Fox 1962]. 15 Definition (Slice knot). A knot k ⊂ R3 is called a slice knot if it can be obtained as a cross section of a locally flat 2-sphere S 2 in R4 by a hyperplane R3 . ) The local flatness is essential: Any knot k ⊂ R3 ⊂ R4 is a cross section of a 2-sphere S 2 embedded in R4 . Choose the double suspension of k with suspension points P+ and P− respectively in the halfspace R4+ and R4− defined by R3 .

From this it follows that l is not nullhomologous on ∂V (k). C Companion Knots and Product Knots Another important idea was added by H. Schubert [1949]: the product of knots. 7 Definition (Product of knots). Let an oriented knot k ⊂ R3 meet a plane E in two points P and Q. The arc of k from P to Q is closed by an arc in E to obtain a knot k1 ; the other arc (from Q to P ) is closed in the same way and so defines a knot k2 . 6. k is also called a composite knot when both knots k1 and k2 are non-trivial.

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