An easy way to make a knot is to take an (unclasped) necklace chain, make any loops and ties in it, and then clasp the ends together. This can also be done with a garden hose or piece of rope, where the final step is to glue the two ends together. Most necklaces are worn as a simple loop with no ties, which is a special case called an unknot.

Any knot that can be created from the unknot without tearing or cutting is also an unknot.

Try It Yourself: Starting with a rubber band or rope with ends glued together, which of the following knots can we make without tearing or cutting the rubber band? Solution: The first three knots can be made by making a single twist in the rubber band, which shows these knots are unknots. It seems that the fourth knot truly crosses itself and cannot be made from a simple rubber band without cutting, but how can we prove this? We will see later how to use knot colorings to answer this question.

Knots and links are studied in topology, which studies properties that are unchanged by continuous transformations. Knots are examples of embeddings, since they are loops living in in 3-dimensional space.

A knot is a closed loop of string in three dimensional space. Two knots are equivalent if one can be continuously transformed into the other without any cutting or gluing.

Note the difference between mathematical knots and knots you tie in a shoelace or rope:

mathematical knots are closed and there are no loose ends to tie or untie

for mathematical purposes, we imagine the string or rope as having no thickness

to get a mental picture of continuous transformation, imagine that the knot is made of an elastic material that can be moved around, stretched, or shrunken as much as you like without breaking or tearing.

Instead of studying knots in three dimensions, it is sometimes easier to study the two-dimensional shadows cast by these knots. A knot diagram captures the closed loop of a knot by drawing either an over strand or an under strand at the crossing positions. The simplest nontrivial knot is a circle that winds through itself, called the trefoil knot. It also comes in two forms: left-handed and right-handed configurations, which are mirror images of each other.

There may be more than one way to draw a knot using a knot diagram. We consider the following two questions:

Given a knot diagram, is the knot equivalent to the unknot? Given two knot diagrams, how can you tell if they represent the same knot?

To start with, we can describe some invariants of a knot, or properties that do not change even if drawn in multiple ways in different knot diagrams. One important indicator of knot complexity is the minimum number of crossings in any drawing of a given knot.

The crossing number of a knot is the minimum number of crossings in any drawing of the knot.



For example, the unknot has crossing number 0 and the trefoil knot has crossing number 3.