Once we simplify the knot so that we cannot remove any further crossings, the knot is classified by the number of crossings that remain. This means we use the Reidemeister moves (from above) to get as few crossings in the knot as possible. To be able to classify knots, it is easiest if we work with only one projection (or representation) of each knot to avoid duplication.įirst, we must “simplify” the knot as much as possible. One of the first objectives mathematicians wished to achieve in working with knots was formulating tables of distinct knots. TheActivities Page contains knot activities that have you use these three moves to find out if two knots are indeed equivalent. These three types of moves are called Reidemeister moves. Slide a strand from one side of a crossing to the other: Add or remove two crossings (lay one strand over another):ģ. Take out (or put in) a simple twist in the knot:Ģ. In 1926, Kurt Reidemeister (ride-a-my-stir) proved that if we have different representations (or projections) of the same knot, we can get one to look like the other using just three simple types of moves.ġ. The next section will give us the simple tools we need to begin working with knots in a mathematical context.ģ) How do we work with knots? (The Reidemeister moves) What was needed was a simple set of rules for working with knots.įinally, German mathematician Kurt Reidemeister (1893-1971) proved that all the different transformations on knots could be described in terms of three simple moves. Mathematicians were perplexed at the seemingly unending number of ways a knot could be shaped and turned. When we actually start trying to untangle and rearrange knots to look like one another, we begin what can seem like a very complicated process. Is it possible to transform this knot so that it looks like the unknot? Tie a trefoil knot yourself and see if you can untangle it to form a simple circular loop. But, take for example the trefoil knot animated at the top of the page. The central problem of Knot Theory is determining whether two knots can be rearranged (without cutting) to be exactly alike.Ī special case of this problem is one of the fundamental questions of Knot Theory: Given a knot, is it the unknot? Now, for a simple loop, that’s an easy question. The unknot can be drawn with no crossings, and is also called a trivial knot. In fact this knot has a special name: the unknot. What does that mean? Well, a loop like the one at the left is considered a knot in mathematical knot theory (it is a simple closed curve in 3-dimensional space). Definition:Ī knot is a simple closed curve in 3-dimensional space. The kinds of knots mathematicians work with are always formed on a closed loop (no loose ends). The last step, joining the ends of the rope, is what differentiates mathematical knots from regular knots.

A “mathematical” knot is just slightly different fr om the knots we see and use every day.įirst, take a piece of string or rope. In order to get started working with knots, we need to understand what mathematicians mean by the term “knots”.

Knot knotes