By Lynn Arthur Steen, J. Arthur Seebach Jr.

Over a hundred and forty examples, preceded via a succinct exposition of common topology and simple terminology. every one instance taken care of as an entire. Over 25 Venn diagrams and charts summarize homes of the examples, whereas discussions of normal tools of building and alter provide readers perception into developing counterexamples. comprises difficulties and routines, correlated with examples. Bibliography. 1978 version.

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**Sample text**

For example, the cut and paste procedure for the connected sum becomes a simple concatenation. 4. Gauss diagrams and virtual knots. Plane knot diagrams are convenient for presenting knots graphically, but for other purposes, such as coding knots in a computer-recognizable form, Gauss diagrams are suited better. Definition. A Gauss diagram is an oriented circle with a distinguished set of distinct points divided into ordered pairs, each pair carrying a sign ±1. Graphically, an ordered pair of points on a circle can be represented by a chord with an arrow connecting them and pointing, say, to the second point.

Knot invariants Now, for any diagram D of an oriented link L we put F (L) := a−w(D) Λ(D). It turns out that this polynomial is equivalent to the collection of the quantum invariants associated with the Lie algebra soN and its standard N dimensional representation for all values of N (see [Tur3]). 2. 7. 1. Kauffman polynomials of knots with 8 crossings 62 2. 1. 7. Two-variable link polynomials As in the previous section, we conclude with a series of exercises with additional information on the Kauffman polynomial.

It follows that the minimal number of crossings required to draw a diagram of a nontrivial knot is at least 3. A little later we shall see that the trefoil knot is indeed nontrivial. Obviously, c(K) is a knot invariant taking values in the set of nonnegative integers. 2. Unknotting number. Another integer-valued invariant of knots which admits a simple definition is the unknotting number. Represent a knot by a plane diagram. The diagram can be transformed by plane isotopies, Reidemeister moves and crossing changes: 41 42 2.

### Counterexamples in Topology (Dover Books on Mathematics) by Lynn Arthur Steen, J. Arthur Seebach Jr.

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