We have all run into knots at one point in our lives or another. Tying our shoes, knotting rope, and even creating paracord bracelets all involve knots of one kind or another. However, mathematical knots don’t have ends meaning that the knots cannot be undone.
What is a Mathematical Knot?
The mathematical knot is the embedding of a circle in Euclidean space. These knots can be described in a number of different ways. There are usually knot diagrams and knot invariants used. A knot diagram is usually a two-dimensional drawing of the three-dimensional knot. A knot invariant is the quantity given to the knot and include knot polynomials, knot groups, and hyperbolic invariants. Brush up on math fundamentals with an online course.
As stated before, knot diagrams are two-dimensional drawings of the three-dimensional knots. These diagrams will reveal the crossings of the knots, and the over-strand has to be shown against the under-strand so the original knot can be recreated to its three-dimensional form. By creating a break, like the breaks shown in the image of a Celtic knot above, you can see which thread is over and which is under.
Equivalent knots will have equivalent knot invariants. That means that two equivalent knots could have the same knot groups, knot polynomials, and/or hyperbolic invariants. Knot groups are part of the knot complement.
Knot polynomials are exactly what the name implies – polynomials within the knots. There are several types of knot polynomials including Jones polynomials, Alexander polynomials, and Alexander-Conway polynomials. The third is a variant of the Alexander polynomial.
Hyperbolic invariants were discovered by William Thurston who proved a number of knots were hyperbolic knots. What this means is that the knot complement has a geometric structure from hyperbolic geometry. Using geometry, mathematicians can visualize the inside of a knot.
How are knots tabulated?
The knots are usually recorded using the crossing number. The crossing number is the number of times the knot crosses over itself. Knot tables don’t usually list all knots but rather the prime knots. There are several different notations available for listing the knots. Three are listed below – the Alexander-Briggs notation, the Dowker notation, and the Conway notation.
Knots have been recorded up to sixteen total crossings. Tabulation began with the unknot, and it has moved on with the help of computer searches to find over 100,000 different prime knots. There is a list of the most common prime knots below as well as their notation in the three different notation types listed.
Along with knowing the differnent ways to notate knots, it’s important to keep the Perko pair in mind. Discovered by Kenneth Perko in 1974, Perko theorized that entries within the classic knot tables could actually represent the same knot. These knots can be mirror images of each other. Take a class in GRE and GMAT math.
The Alexander-Briggs Notation
Designed by J. W. Alexander and G. Briggs in 1927, this particular notation of knots is the most traditional method. It involves organizing a knot by its crossing number. Knots that have the same crossing number are given a subscript to show their order in the table of knots.
The Dowker Notation
This particular notation is more complicated than the Alexander-Briggs notation. It involves labeling the knots with a sequence of even integers. Dowker notations involve crossings that have labelled pairs, which are then translated into the sequence. However, knot diagrams can have more than one Dowker notation making it difficult to decide which knot is being described.
The Conway Notation
John Horton Conway came up with a theory of tangles in the 1970s, and he created a notation for knots that reflects this theory. Conway’s notation actually involved properties of the knot giving a better description than the Dowker notation.
The Most Common Prime Knots and Their Notations
There are a number of prime knots, but only those that have six or fewer are listed by name as well. Here’s a table below with the most common knots involving six or less crossings. Learn differential equations with an online course.
|Name||Alexander-Briggs Notation||Dowker Notation||Conway Notation|
|Trefoil knot||31||4 6 2|||
|Figure-eight knot||41||4 6 8 2|||
|Cinquefoil knot||51||6 8 10 2 4|||
|Three-twist knot||52||4 8 10 2 6|||
|Stevedore knot||61||4 8 12 10 2 6|||
|62 knot||62||4 8 10 12 2 6|||
|63 knot||63||4 8 10 2 12 6|||
There are more prime knots with more crossings, but it wouldn’t make any sense to list them all here. Jim Hoste, Jeff Weeks, and Morwen Thistlethwaite used a special computer program to search for knots with sixteen crossings or fewer. The results were over 100,000 prime knots, and there could even be more.
Other Theories Building Off of Knot Theory or Relating to Knot Theory
There are a number of theories that have begun to build off of knot theory or relate to knot theory. One such theory is the theory of tangle by John Horton Conway discovered in the 1970s. Another theory is the braid theory designed by Emil Artin in 1925.
John Conway’s tangle theory describes the proper embedding of n arcs into a 3-ball. This particular theory also connects to link theory, which builds off of knot theory. Conway uses his theory regarding tangles to tabulate knots, which is described above in Conway’s notation.
Emil Artin’s braid theory involves knot diagrams that include n points connected to n strands to a second set of n points. These strands have to continue monotonically downward. A braid can be created using a special type of generator and composition operations.