Algorithms for recognizing knots and 3-manifolds

Algorithms are of interest to geometric topologists for two reasons. First, they have bearing on the decidability of a problem. Certain topological questions, such as finding a classification of four dimensional manifolds, admit no solution. It is important to know if other problems fall into this category. Secondly, the discovery of a reasonably efficient algorithm can lead to a computer program which can be used to examine interesting examples. In this paper we will survey some topological algorithms, in particular those that relate to distinguishing knots. Our approach is somewhat informal, with many details omitted, but references are given to sources which develop these ideas in full depth. Given a question Q, a decision procedure for Q or an algorithm to decide Q can be thought of as a computer program which will produce an answer to Q in a finite amount of time. A formal description of an algorithm or a computer is given by the notion of a Turing machine. A Turing machine is a basic computational device that reads and writes onto a tape. The questions such a machine can decide are the same as those that can be decided by more complicated computers. The tape is divided into cells, which the Turing machine can read from and write to, one at a time. The tape has a leftmost cell, but is infinite to the right. A finite set of symbols can be written onto the tape the usual English alphabet if we wish. The Turing machine has a finite number of possible states, and its behavior is determined by its state. Initially, some finite number of cells on the tape contain symbols and the rest are blank. At each time interval, the Turing machine scans the symbol at the current tape location, and in a manner determined by the symbol and its current state it: 1. Changes to a new state. 2. Overwrites the symbol it has read with a new symbol. 3. Moves the tape one cell left or right. Some states are final. The computation ends when they occur. Q is called recursive if there is an algorithm that produces an answer in a finite amount of time. Showing that there is an algorithm to decide Q is equiv-