By von Neumann J.

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**Additional resources for Collected works. Vol.4 Continuous geometry and other topics**

**Example text**

In our example, the conditional and contrapositive statements are true for all integers. We prove these statements using a logical argument or proof. The converse, inverse, and biconditional statements are not true—a single example is sufficient to show that these statements are not true. So make sure that you don't confuse your converse and your contrapositive! We can also form more complicated statements using and and or. The statement P and Q is denoted P ∧ Q. The statement P ∧ Q is true when both P and Q are true.

12 Computing ????a(L) We compute that ????1(L) = ????−1(L) = −1 and ????a(L) = 0 for all a > 1. Based on these calculations, the trefoil and the virtual figure 8 knot are not equivalent: L ≁ VT. We use our experience computing ????a(K) to prove that ????a(K) is a knot invariant. The proof is a direct proof and centers around showing that the diagrammatic moves do not change the value of ????a(K). 7. Let K be a knot and let a ≠ 0. Then ????a(K) is unchanged by the diagrammatic moves and ????a(K) is a knot invariant.

The book is designed to introduce key ideas and provide the background for undergraduate research on knot theory. Suggested readings from research papers are included in each chapter. The goal is for students to experience mathematical research. The proofs are written as simply as possible using combinatorial approaches, equivalence classes and linear algebra so that they are accessible to junior-senior level students. I enjoy explaining virtual knot theory to undergraduate students. In the combinatorial proofs, I find examples of many of the concepts that we to try to introduce students to in a course on proof writing.

### Collected works. Vol.4 Continuous geometry and other topics by von Neumann J.

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