Richard H. Crowell

COMPUTERS WITH CALCULUS AT DARTMOUTH

ABSTRACT This note reports on an experiment done recently at Dartmouth College, in which the standard entry-level freshmen class in calculus was given in a new format making regular use of the Macintosh personal computer. The new format is described in some detail. The performance on a final examination of the experimental class is compared with the performance on the same examination of the corresponding non-experimental class of the previous year. Based on these data, a few tentative conclusions on the value of the computer in the calculus curriculum are suggested.

Calculation of Fundamental Groups

It was remarked in Chapter II that a rigorous calculation of the fundamental group of a space X is rarely just a straightforward application of the definition of π(X). At this point the collection of topological spaces whose fundamental groups the reader can be expected to know (as a result of the theory so far developed in this book) consists of spaces topologically equivalent to the circle or to a convex set. This is not a very wide range, and the purpose of this chapter is to do something about increasing it. The techniques we shall consider are aimed in two directions. The first is concerned with what we may call spaces of the same shape. Figures 14, 15, and 16 are examples of the sort of thing we have in mind. From an understanding of the fundamental group as formed from the set of classes of equivalent loops based at a point, it is geometrically apparent that the spaces shown in Figure 14 below have the same, or isomorphic, fundamental groups.

The Knot Polynomials

The underlying knot-theoretic structure developed in this book is a chain of successively weaker invariants of knot type. The sequence of knot polynomials, to which this chapter is devoted, is the last in the chain

Presentation of Groups

\begin{gathered} \quad \quad \quad \quad knot\,type\,of\,K \hfill \\ \quad \quad \quad \quad \quad \quad \downarrow \hfill \\ presentation\,type\;of\,\pi \left( {{R^3} - K} \right) \hfill \\ \quad \quad \quad \quad \quad \quad \downarrow \hfill \\ sequence\,of\,elementary\,ideals \hfill \\ \quad \quad \quad \quad \quad \;\;\; \downarrow \hfill \\ sequence\,of\,knot\,polynomials \hfill \\ \end{gathered}

The Fundamental Group

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Knots and Knot Types

In this chapter we return to knot theory. The major objective here is the description and verification of a procedure for deriving from any polygonal knot K in regular position two presentations of the group of K, which are called respectively the over and under presentations. The classical Wirtinger presentation is obtained as a special case of the over presentation. In a later section we calculate over presentations of the groups of four separate knots explicitly, and the final section contains a proof of the existence of nontrivial knots, in that it is shown that the clover-leaf knot can not be untied. The fact that our basic description in this chapter is concerned with a pair of group presentations represents a concession to later theory. It is of no significance at this stage. One presentation is plenty, and, for this reason, Section 4 is limited to examples of over presentations. The existence of a pair of over and under presentations is the basis for a duality theory which will be exploited in Chapter IX to prove one of the important theorems.

Characteristic Properties of the Knot Polynomials

In this chapter we give a firm foundation to the concept of defining a group by generators and relations. This is an important step; for example, if one is not careful to distinguish between the elements of a group and the words that describe these elements, utter confusion is likely to ensue.

Introduction to Knot Theory

In many applications of group theory, and specifically in our subsequent analysis of the fundamental groups of the complementary spaces of knots, the groups are described by “defining relations,” or, as we are going to say later, are “presented”. We have here another (and completely different) analogy with analytic geometry. In analytic geometry a coordinate system is selected, and the geometric configuration to be studied is defined by a set of one or more equations. In the theory of group presentations the role that is played in analytic geometry by a coordinate system is played by a free group. Therefore, the study of group presentations must begin with a careful description of the free groups.