Take a loop of rope, tie a knot, glue the ends together, and hold it to a mirror: for some knots, the reflection is a genuinely different knot. This article walks from the plain circle to the trefoil, the figure-eight, the cinquefoil, and finally the Perko pair, a famous case where one knot sat in the tables twice for decades.
What Counts as a Knot?
Two knots count as the same if one can be stretched, squashed, and rearranged into the other without cutting the rope or passing it through itself. With a rope whose loose ends can move freely, this rule fails, because every knot can be untied into the same straight piece, so all knots would be equal. Mathematicians therefore glue the two ends together. The result is a closed loop, called a knot, and its study is called knot theory. This is a central example of topology, which compares shapes through continuous deformation.
Knots live in three-dimensional space, but we usually study them through knot diagrams, flattened pictures in which the strand passing underneath at a crossing is drawn with a small break. The fewest crossings in any diagram of a knot is its crossing number, a simple measure of complexity [1]. Tables of knots are arranged by this number [1].
The Unknot
The unknot is a loop with no knotting at all. It can be drawn as a circle, or as any shape that can be deformed into a circle. A loop with a single twist has one crossing, yet a single flip removes it, so it is still the unknot. Its crossing number is 0.
Not every tangled diagram is easy to simplify. So-called hard unknots can be straightened into a plain circle, but some require a sequence of moves that first makes the diagram more complicated before it simplifies. The distinction is between the underlying knot, which is still the unknot, and the difficulty of finding a simplifying sequence of diagram moves.
The Trefoil Knot
Tie an overhand knot, then join the loose ends without adding crossings. The result looks like a three-leaf clover, hence the name trefoil. It is the unique prime knot with three crossings when mirror images are counted together [2], and showing that its crossing number really is 3 is equivalent to showing that it is genuinely knotted [3].
The trefoil differs from its mirror image, a fact first proved by Dehn in 1914 [2]. A knot that is not equivalent to its mirror image is called chiral, and one that is, achiral or amphicheiral [4]. The two trefoils are named the right-handed and left-handed trefoil. The trefoil is also alternating: tracing it, you pass over and under in turn [3]. It is often called the simplest non-trivial knot, where trivial means the case with no actual knotting.
The Figure-Eight Knot
Tie an overhand knot again, but add an extra half twist to the loop before the working end passes through. Joining the ends gives the figure-eight knot. It is the unique knot with crossing number four [5]. Unlike the trefoil, it is equal to its mirror image, so it is achiral [3]. It is alternating as well [5].
The Cinquefoil Knot
A double overhand knot, with its ends joined, gives the cinquefoil, named for a five-leaved plant. Its crossing number is five. It can be drawn neatly as a pentagram, and tracing the drawing alternates over and under at every crossing. Strictly, the pentagram is a diagram rather than the knot itself.
The trefoil is the torus knot T(2,3) and the cinquefoil is T(2,5) [3]. Odd-numbered relatives continue the family, each drawable with no crossings on the surface of a donut, winding around the ring and through the hole a fixed number of times. Exactly two prime knots have crossing number five: the cinquefoil and another one, which the video calls the three-twist knot. Prime knot counts then grow quickly, with 165 prime knots at exactly ten crossings when mirror images are counted together [6]. The torus is one of the surfaces explored in complex geometry shapes.
The Perko Pair
The last knot is the one listed twice. In Rolfsen's table the entries are labeled 10₁₆₁ and 10₁₆₂ [7]. They appeared separately in classical tables going back to Little, and Conway missed the duplication in his own table before it reached Rolfsen's [7]. In 1973, Kenneth Perko, a lawyer who had studied mathematics at Princeton, was working through the ten-crossing tables and noticed the two entries were the same knot [7]. He published the result in 1974 [8].
The writhe of an oriented diagram is its number of positive crossings minus its number of negative crossings [9]. Little had claimed that minimal-crossing diagrams of one knot always share the same writhe [9]. That holds for alternating knots, but the Perko pair shows it fails for non-alternating ones [10]. The two diagrams are the same knot yet have different writhes [8].
The popular story says Perko worked with rope on his living room floor, but he has called that internet nonsense and says he used diagrams on a legal pad [11]. The duplicated entry is another reminder that small errors can survive repeated checking, a theme explored in small mathematical mistakes with large consequences.
Sources
- The Jones polynomial for dummies. Vaughan Jones2 February 12, 2014
- Trefoil Knot — from Wolfram MathWorld
- KNOTS KNOTES Contents 1. Motivation, basic definitions …
- KNOT THEORY AND APPLICATIONS: LECTURE NOTES
- Figure-eight knot (mathematics)
- Prime Knot — from Wolfram MathWorld
- Perko pair
- Perko Pair — from Wolfram MathWorld
- Tait's conjectures and odd crossing number amphicheiral knots
- Tait conjectures
- The revenge of the Perko pair
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