Skein Theory
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Skein relations are a mathematical tool used to study
knots A knot is a fastening in rope or interwoven lines. Knot or knots may also refer to: Other common meanings * Knot (unit), of speed * Knot (wood), a timber imperfection Arts, entertainment, and media Films * ''Knots'' (film), a 2004 film * ''Kn ...
. A central question in the mathematical theory of knots is whether two
knot diagram In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, the simplest k ...
s represent the same knot. One way to answer the question is using
knot polynomial In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot. History The first knot polynomial, the Alexander polynomial, was introdu ...
s, which are invariants of the knot. If two diagrams have different
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s, they represent different knots. However, the converse is not true. Skein relations are often used to give a simple definition of knot polynomials. A skein relation gives a linear relation between the values of a knot polynomial on a collection of three links which differ from each other only in a small region. For some knot polynomials, such as the
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,
Alexander Alexander () is a male name of Greek origin. The most prominent bearer of the name is Alexander the Great, the king of the Ancient Greek kingdom of Macedonia who created one of the largest empires in ancient history. Variants listed here ar ...
, and
Jones polynomial In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polyno ...
s, the relevant skein relations are sufficient to calculate the polynomial
recursively Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in m ...
.


Definition

A skein relationship requires three link diagrams that are identical except at one crossing. The three diagrams must exhibit the three possibilities that could occur for the two line segments at that crossing, one of the lines could pass ''under,'' the same line could be ''over'' or the two lines might not cross at all. Link diagrams must be considered because a single skein change can alter a diagram from representing a knot to one representing a link and vice versa. Depending on the knot polynomial in question, the links (or tangles) appearing in a skein relation may be oriented or unoriented. The three diagrams are labelled as follows. Turn the three link diagram so the directions at the crossing in question are both roughly northward. One diagram will have northwest over northeast, it is labelled ''L''. Another will have northeast over northwest, it's ''L''+. The remaining diagram is lacking that crossing and is labelled ''L''0. : (The labelling is independent of direction insofar as it remains the same if all directions are reversed. Thus polynomials on undirected knots are unambiguously defined by this method. However, the directions on ''links'' are a vital detail to retain as one recurses through a polynomial calculation.) It is also sensible to think in a generative sense, by taking an existing link diagram and "patching" it to make the other two—just so long as the patches are applied with compatible directions. To recursively define a knot (link) polynomial, a function ''F'' is fixed and for any triple of diagrams and their polynomials labelled as above, :F\Big(L_-,L_0,L_+\Big)=0 or more pedantically :F\Big(L_-(x),L_0(x),L_+(x),x\Big)=0 for all x (Finding an ''F'' which produces polynomials independent of the sequences of crossings used in a recursion is no trivial exercise.) More formally, a skein relation can be thought of as defining the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of a
quotient map In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
from the
planar algebra In mathematics, planar algebras first appeared in the work of Vaughan Jones on the Subfactor#Standard invariant, standard invariant of a II-1 subfactor, II1 subfactor. They also provide an appropriate algebraic framework for many knot invariants ( ...
of tangles. Such a map corresponds to a knot polynomial if all closed diagrams are taken to some (polynomial) multiple of the image of the empty diagram.


Example

Sometime in the early 1960s,
Conway Conway may refer to: Places United States * Conway, Arkansas * Conway County, Arkansas * Lake Conway, Arkansas * Conway, Florida * Conway, Iowa * Conway, Kansas * Conway, Louisiana * Conway, Massachusetts * Conway, Michigan * Conway Townshi ...
showed how to compute the Alexander polynomial using skein relations. As it is
recursive Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in m ...
, it is not quite so direct as Alexander's original
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method; on the other hand, parts of the work done for one knot will apply to others. In particular, the network of diagrams is the same for all skein-related polynomials. Let function ''P'' from link diagrams to
Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansio ...
in \sqrt x be such that P()=1 and a triple of skein-relation diagrams (L_-, L_0, L_+) satisfies the equation :P(L_-) = (x^-x^)P(L_0) + P(L_+) Then ''P'' maps a knot to one of its Alexander polynomials. In this example, we calculate the Alexander polynomial of the cinquefoil knot (), the
alternating knot In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link. A link is alternating if it has an alternating diagram. Many of the knots with crossing ...
with five crossings in its minimal diagram. At each stage we exhibit a relationship involving a more complex link and two simpler diagrams. Note that the more complex link is on the right in each step below except the last. For convenience, let ''A'' = ''x''−1/2−x1/2. To begin, we create two new diagrams by patching one of the cinquefoil's crossings (highlighted in yellow) so :''P''() = ''A'' × ''P''() + ''P''() The second diagram is actually a trefoil; the first diagram is two unknots with four crossings. Patching the latter :''P''() = ''A'' × ''P''() + ''P''() gives, again, a trefoil, and two unknots with ''two'' crossings (the
Hopf link In mathematics, mathematical knot theory, the Hopf link is the simplest nontrivial link (knot theory), link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf. Geometric realizat ...
br>
. Patching the trefoil :''P''() = ''A'' × ''P''() + ''P''() gives the unknot and, again, the Hopf link. Patching the Hopf link :''P''() = ''A'' × ''P''() + ''P''() gives a link with 0 crossings (unlink) and an unknot. The unlink takes a bit of sneakiness: :''P''() = ''A'' × ''P''() + ''P''()


Computations

We now have enough relations to compute the polynomials of all the links we've encountered, and can use the above equations in reverse order to work up to the cinquefoil knot itself. The calculation is described in the table below, where ? denotes the unknown quantity we are solving for in each relation: Thus the Alexander polynomial for a cinquefoil is P(x) = x−2 -x−1 +1 -x +x2.


Etymology

In knot theory, the term Hank_(unit_of_measure), skein appears to have been coined by John Conway around 1979, and refers to the unit of measure of yarn in the textiles industry.


Sources

*American Mathematical Society
Knots and Their Polynomials
Feature Column. * *. {{Knot theory Knot theory Diagram algebras