Knot theory
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In mathematics, knot theory is the branch of topology that studies mathematical knots, which are defined as embeddings of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations (isotopies). This is basically equivalent to a conventional knot with the ends of the string joined together to prevent it from becoming undone.
Knots can be described in various ways, but the most common method is by planar diagrams. Given a method of description, a knot will have many descriptions, e.g., many diagrams, representing it. A fundamental problem in knot theory is determining when two descriptions represent the same knot. One way of distinguishing knots is by using a knot invariant, a "quantity" which remains the same even with different descriptions of a knot. Two knots can be shown to be different if an invariant takes different values on them; however, an invariant may take the same value on different knots.
The concept of a knot has been extended to higher dimensions by considering n-dimensional spheres in m-dimensional Euclidean space. This was investigated most actively in the period 1960-1980, when a number of breakthroughs were made. In recent years, low dimensional phenomena has garnered the most interest.
Research in knot theory began with the creation of knot tables and the systematic tabulation of knots. While tabulation remains an important task, today's researchers have a wide variety of backgrounds and goals. Classical knot theory, as initiated by Max Dehn, J. W. Alexander, and others, concerns primarily knot invariants such as the knot group or those coming from homology theory, such as the Alexander polynomial.
Vaughan Jones' discovery of the Jones polynomial in 1984 and subsequent contributions from Edward Witten, Maxim Kontsevich, and others, revealed deep connections between knot theory and mathematical methods in statistical mechanics and quantum field theory. A plethora of knot invariants have been invented since, including the quantum invariants and finite type invariants. These have been shown to be connected to more general invariants of 3-manifolds.
In the last 30 years, knot theory has also become a tool in applied mathematics. Chemists and biologists use knot theory to understand, for example, chirality of molecules and the actions of enzymes on DNA.
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[edit] History
Knots were studied by Carl Friedrich Gauss, who developed the Gauss linking integral for computing the linking number of two knots. His student Johann Benedict Listing, after whom Listing's knot is named, furthered their study. The early, significant stimulus in knot theory would arrive later with Sir William Thomson (Lord Kelvin) and his theory of vortex atoms.
In 1867 after observing Scottish physicist Peter Tait's experiments involving smoke rings, Thomson came to the idea that atoms were knots of swirling vortices in the æther. Chemical elements would thus correspond to knots and links. Tait's experiments were inspired by a paper of Helmholtz's on vortex-rings in incompressible fluids. Thomson and Tait believed that an understanding and classification of all possible knots would explain why atoms absorb and emit light at only the discrete wavelengths that they do. For example, Thomson thought that sodium could be the Hopf link due to its two lines of spectra.
Tait subsequently began listing unique knots in the belief that he was creating a table of elements. He formulated what are now known as the Tait conjectures on alternating knots. The conjectures spurred research in knot theory, and were finally resolved in the 1990s. Tait's knot tables were subsequently improved upon by C. N. Little and T. P. Kirkman.
James Clerk Maxwell, a colleague and friend of Thomson's and Tait's, also gained a strong interest of knots. Maxwell studied Listing's work on knots. He re-interpreted Gauss' linking integral in terms of electromagnetic theory. In his formulation, the integral represented the work done by a charged particle moving along one component of the link under the influence of the magnetic field generated an electric current along the other component. Maxwell also continued the study of smoke rings by considering three interacting rings.
When the luminiferous æther was not detected in the Michelson-Morley experiment, vortex theory became completely obsolete, and knot theory ceased to be of great scientific interest. Modern physics demonstrates that the discrete wavelengths depend on quantum energy levels.
Following the development of topology in the early 20th century spearheaded by Henri Poincare, topologists such as Max Dehn, J. W. Alexander, and Kurt Reidemeister, investigated knots. Out of this sprang the Reidemeister moves and the Alexander polynomial. Dehn also developed Dehn surgery, which related knots to the general theory of 3-manifolds and formulated the Dehn problems in group theory, such as the word problem. Early pioneers in the first half of the 20th century include Ralph Fox, who popularized the subject. In this early period, knot theory primarily consisted of study of the knot group and homological invariants of the knot complement.
A few major discoveries in the late 20th century greatly revived knot theory. The first was Thurston's hyperbolization theorem which introduced the theory of hyperbolic 3-manifolds into knot theory and made it of prime importance. Thurston's work also led, after much expansion by others, to the effective use of tools from representation theory and algebraic geometry. Important results followed, including the Gordon-Luecke theorem, which showed that knots were determined (up to mirror-reflection) by their complements, and the Smith conjecture.
Interest in knot theory grew significantly after Vaughan Jones' discovery of the Jones polynomial. This led to other knot polynomials such as the bracket polynomial, HOMFLY polynomial, and Kauffman polynomial. Jones was awarded the the highest honor in mathematics, the Fields medal, in 1990 for this work. In 1988 Edward Witten proposed a new framework for the Jones polynomial, utilizing existing ideas from mathematical physics, such as Feynman path integrals, and introducing new notions such as topological quantum field theory. Witten also received the Field medal in 1994 partly for this work. Witten's description of the Jones polynomial implied related invariants for 3-manifolds. Different attempts to put Witten's work on rigorous mathematical foundations created a number of interesting developments, such as the Witten-Reshetikhin-Turaev invariants and various so-called "quantum invariants".
The last several decades of the 20th century, scientists and mathematicians began finding applications of knot theory to problems in biology and chemistry. Knot theory can be used to determine if a molecule is chiral (has a "handedness") or not. Chemical compounds of different handedness can have drastically differing properties. More generally, knot theoretic methods have been used in studying topoisomers, topologically different arrangements of the same chemical formula. The closely related theory of tangles have been effectively used in studying the action of certain enzymes on DNA. (Flapan 2000)
[edit] Knot equivalence
A knot is created by beginning with a one-dimensional line segment, wrap it around itself arbitrarily, and then fuse its two free ends together to form a closed loop. Some movements, such as small perturbations of the knot embeds the loop differently in three dimensional space, but intuitively, this embedding should really be considered the "same" as the first. The idea of knot equivalence is to make precise when two embeddings should be considered the same.
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equivalent to it |
When mathematical topologists consider knots and other entanglements such as links and braids, they describe how the knot is positioned in the space around it, called the ambient space. If the knot is moved smoothly, without cutting or passing a segment through another, to coincide with another knot, the two knots are considered equivalent.
The basic problem of knot theory, the recognition problem, can thus be stated as: given two knots, determine whether or not they are equivalent or not. Algorithms exist to solve this problem, with the first given by Wolfgang Haken. Nonetheless, these algorithms use significantly many steps, and a major issue in the theory is to understand how hard this problem really is.[1] The special case of recognizing the unknot, called the unknotting problem, is of particular interest.
[edit] Knot diagrams
A useful way to visualise and manipulate knots is to project the knot onto a plane - think of the knot casting a shadow on the wall. A small perturbation in the choice of projection place will ensure that the projection is one to one except at the double points, called crossings, where the "shadow" of the knot crosses itself once transversely (Rolfsen 1976). At each crossing we must indicate which section is "over" and which is "under", so as to be able to recreate the original knot. This is often done by creating a break in the understrand.
[edit] Reidemeister moves
In 1927, working with this diagrammatic form of knots, J.W. Alexander and G. B. Briggs, and independently Kurt Reidemeister, demonstrated that two knot diagrams belonging to the same knot can be related by a sequence of three kinds of moves on the diagram, shown right. These operations, now called the Reidemeister moves, are:
- Twist and untwist in either direction.
- Move one loop completely over another.
- Move a string completely over or under a crossing.
[edit] Knot invariants
A knot invariant is a "quantity" that is the same for equivalent knots (Adams 2001, Lickorish 1997, Rolfsen 1976). An invariant may take the same value on two different knots, so by itself may be incapable of distinguishing all knots. An elementary invariant is tricolorability.
"Classical" knot invariants include the knot group, which is the fundamental group of the knot complement, and the Alexander polynomial, which can be computed from the Alexander invariant, a module constructed from the infinite cyclic cover of the knot complement (Lickorish 1997, Rolfsen 1976). In the late 20th century, invariants such as "quantum" knot polynomials and hyperbolic invariants were discovered. These aforementioned invariants are only the tip of the iceberg of modern knot theory.
[edit] Knot polynomials
A knot polynomial is a knot invariant that is a polynomial. Well-known examples include the Jones and Alexander polynomials. A variant of the Alexander polynomial, the Alexander-Conway polynomial, is a polynomial in the variable z with integer coefficients (Lickorish 1997).
Suppose we are given a link diagram which is oriented, i.e. every component of the link has a preferred direction indicated by an arrow. Also suppose L + ,L − ,L0 are oriented link diagrams resulting from changing the diagram at a specified crossing of the diagram, as indicated in the figure:
Then the Alexander-Conway polynomial, C(z), is recursively defined according to the rules:
- C(O) = 1 (where O is any diagram of the unknot)
- C(L + ) = C(L − ) + zC(L0)
The second rule is what is often referred to as a skein relation. To check that these rules give an invariant, one should determine that the polynomial does not change under the three Reidemeister moves. Many important knot polynomials can be defined in this way.
The following is an example of a typical computation using a skein relation. It computes the Alexander-Conway polynomial of the trefoil knot. The yellow patches indicate where we applied the relation.
- C(
)=C(
) + z C(
)
gives the unknot and the Hopf link. Applying the relation to the Hopf link where indicated,
- C(
) = C(
) + z C(
)
gives a link deformable to one with 0 crossings (it is actually the unlink of two components) and an unknot. The unlink takes a bit of sneakiness:
- C(
) = C(
)+ z C(
)
which implies that C(unlink of two components) = 0, since the first two polynomials are of the unknot and thus equal.
Putting all this together will show:
- C(trefoil) = 1 + z (0 + z) = 1 + z2
Note that if we believe that the Alexander-Conway polynomial is actually a knot invariant, this shows that the trefoil is not equivalent to the unknot. So there is really a knot that is "knotted".
Actually, there are two trefoil knots, called the right and left-handed trefoils, which are mirror images of each other (take a diagram of the trefoil given above and change each crossing to the other way to get the mirror image). These are not equivalent to each other! This was shown by Max Dehn, before the invention of knot polynomials, using group theoretical methods (Dehn, 1914). But the Alexander-Conway polynomial of each kind of trefoil will be the same, as can be seen by going through the computation above with the mirror image. The Jones polynomial can in fact distinguish between the left and right handed trefoil knots (Lickorish 1997).
[edit] Hyperbolic invariants
 The Borromean rings are a link with the property that removing one ring unlinks the others. |
 SnapPea's cusp view: the Borromean rings complement from the perspective of an inhabitant living near the red component. |
William Thurston proved many knots are hyperbolic knots, meaning that the knot complement, i.e. the points of 3-space not on the knot, admit a geometric structure, in particular that of hyperbolic geometry. The hyperbolic structure depends only on the knot so any quantity computed from the hyperbolic structure is then a knot invariant. (Adams 2001)
Geometry lets us visualize what the inside of a knot or link complement looks like by imagining light rays as traveling along the geodesics of the geometry. An example is provided by the picture of the complement of the Borromean rings. The inhabitant of this link complement is viewing the space from near the red component. The balls in the picture are views of horoball neighborhoods of the link. By thickening the link in a standard way, we obtain what are called horoball neighborhoods of the link components. Even though the boundary of a neighborhoods is a torus, when viewed by inside the link complement, it looks like a sphere, called a horoball. Each link component shows up as infinitely many horoballs (of one color) as there are infinitely many light rays from the observer to the link component. The fundamental parallelogram (which is indicated in the picture), tiles both vertically and horizontally.
The pattern of horoballs is itself a useful invariant. Other hyperbolic invariants include the shape of the fundamental paralleogram, length of shortest geodesic, and volume. Modern knot and link tabulation efforts have utilized these invariants effectively. Fast computers and clever methods of obtaining these invariants make calculating these invariants, in practice, a simple task. (Adams, Hildebrand, & Weeks, 1991)
[edit] Higher dimensions
In four dimensions, any closed loop of one-dimensional string is equivalent to an unknot. We can achieve the necessary deformation in two steps. The first step is to "push" the loop into a three-dimensional subspace, which is always possible, though technical to explain. The second step is changing crossings. Suppose one strand is behind another as seen from a chosen point. Lift it into the fourth dimension, so there is no obstacle (the front strand having no component there); then slide it forward, and drop it back, now in front. An analogy for the plane would be lifting a string up off the surface.
Since a knot can be considered topologically a 1-dimensional sphere, the next generalization is to consider a two dimensional sphere embedded in a four dimensional sphere. Such an embedding is unknotted if there is a homeomorphism of the 4-sphere onto itself taking the 2-sphere to a standard "round" 2-sphere. Suspended knots and spun knots are two typical families of such 2-sphere knots.
The mathematical technique called "general position" implies that for a given n-sphere in the m-sphere, if m is large enough (depending on n), the sphere should be unknotted. In general, piecewise-linear n-spheres form knots only in (n+2)-space (Zeeman 1963), although this is no longer a requirement for smoothly knotted spheres. In fact, there are smoothly knotted 4k-1-spheres in 6k-space, e.g. there is a smoothly knotted 3-sphere in the 6-sphere (Haefliger 1962, Levine 1965). Thus the codimension of a smooth knot can be arbitrarily large when not fixing the dimension of the knotted sphere; however, any smooth 4k-1-sphere in an n-sphere with n > 6k is unknotted.
[edit] Adding knots
Two knots can be added by cutting both knots and joining the pairs of ends. This can be formally defined as follows (Adams 2001): consider a planar projection of each knot and suppose these projections are disjoint. Find a rectangle in the plane where one pair of opposite sides are arcs along each knot while the rest of the rectangle is disjoint from the knots. Form a new knot by deleting the first pair of opposite sides and adjoining the other pair of opposite sides. The resulting knot is the sum of the original knots.
This operation is called the knot sum, or sometimes the connected sum or composition of two knots. The knot sum is commutative and associative. There is also a prime decomposition for a knot which allows us to define a prime or composite knot, analogous to prime and composite numbers. The trefoil knot is the simplest prime knot. Higher dimensional knots can be added by splicing the n-spheres. While you cannot form the unknot in three dimensions by adding two non-trivial knots, you can in higher dimensions, at least when one considers smooth knots in codimension at least 3.
[edit] Tabulating knots
Traditionally, knots have been catalogued in terms of crossing number. The number of nontrivial knots of a given crossing number increases rapidly, making tabulation computationally difficult. Knot tables generally include only prime knots and only one entry for a knot and its mirror image (even if they are different). The sequence of the number of prime knots of a given crossing number, up to crossing number 16, is 0, 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, 46972, 253293, 1388705... .[2] While exponential upper and lower bounds for this sequence are known, it has not been proven that this sequence is strictly increasing. (Adams 2001)
The first knot tables by Tait, Little, and Kirkman used knot diagrams, although Tait also used a precursor to the Dowker notation. Different notations have been invented for knots which allow more efficient tabulation.
The early tables attempted to list all knots of at most 10 crossings, and all alternating knots of 11 crossings. The development of knot theory due to Alexander, Reidemeister, Seifert, and others eased the task of verification and tables of knots up to and including 9 crossings were published by Alexander-Briggs and Reidemeister in the late 1920s.
The first major verification of this work was done in the 1960s by John Horton Conway, who not only developed a new notation but also the Alexander-Conway polynomial. (Conway 1970, Doll-Hoste 1991) This verified the list of knots of at most 11 crossings and a new list of links up to 10 crossings. Conway found a number of omissions but only duplication in the Tait-Little tables; however he missed the duplicates called the Perko pair, which would only be noticed in 1974 by Kenneth Perko. (Perko 1974) This famous error would propogate when Dale Rolfsen added a knot table in his influential text, based on Conway's work.
[edit] Alexander-Briggs notation
This is the most traditional notation, due to the 1927 paper of J. W. Alexander and G. Briggs and later extended by Dale Rolfsen in his knot table. The notation simply organizes knots by their crossing number. One writes the crossing number with a subscript to denote its order amongst all knots with that crossing number. This order is arbitrary and so has no special significance.
[edit] The Dowker notation
The Dowker notation, also called the Dowker-Thistlethwaite notation or code, for a knot is a finite sequence of even integers. The numbers are generated by following the knot and marking the crossings with consecutive integers. Since each crossing is visited twice, this creates a pairing of even integers with odd integers. An appropriate sign is given to indicate over and undercrossing. For example, in the figure the knot diagram has crossings labelled with the pairs (1,6) (3,-12) (5,2) (7,8) (9,-4) and (11,-10). The Dowker notation for this labelling is the sequence: 6 -12 2 8 -4 -10. A knot diagram has more than one possible Dowker notation, and there is a well-understood ambiguity when reconstructing a knot from a Dowker notation.
[edit] Conway notation
The Conway notation for knots and links, named after John Horton Conway, is based on the the theory of tangles. (Conway, 1970) The advantage of this notation is that it reflects some properties of the knot or link.
The notation describes how to construct a particular link diagram of the link. Start with a basic polyhedron, a 4-valent connected planar graph with no digon regions. Such a polyhedron is denoted first by the number of vertices then a number of asterisks which determine the polyhedron's position on a list of basic polyhedron. For example, 10** denotes the second 10-vertex polyhedron on Conway's list.
Each vertex then has an algebraic tangle substituted into it (each vertex is oriented so there is no arbitrary choice in substitution). Each such tangle has a notation consisting of numbers and + or - signs.
An example is 1*2 -3 2. The 1* denotes the only 1-vertex basic polyhedron. The 2 -3 2 is a sequence describing the continued fraction associated to a rational tangle. One inserts this tangle at the vertex of the basic polyhedron 1*.
A more complicated example is 8*3.1.2 0.1.1.1.1.1 Here again 8* refers to a basic polyhedron with 8 vertices. The periods separate the notation for each tangle.
Any link admits such a description, and it is clear this is a very compact notation even for very large crossing number. There are some further shorthands usually used. The last example is usually written 8*3:2 0, where we omitted the ones and kept the number of dots excepting the dots at the end. For an algebraic knot such as in the first example, 1* is often omitted.
Conway's pioneering paper on the subject lists up to 10-vertex basic polyhedra of which he uses to tabulate links, which have become standard for those links. For a further listing of higher vertex polyhedra, there are nonstandard choices available.
[edit] See also
[edit] References
- Colin Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, 2001, ISBN 0-7167-4219-5
- Adams, Colin; Hildebrand, Martin; Weeks, Jeffrey; Hyperbolic invariants of knots and links. Trans. Amer. Math. Soc. 326 (1991), no. 1, 1--56.
- John Horton Conway, An enumeration of knots and links, and some of their algebraic properties. 1970 Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) pp. 329--358 Pergamon, Oxford
- Max Dehn, Die beiden Kleeblattschlingen, Math. Ann. 75 (1914), 402-413.
- Helmut Doll and Jim Hoste, A tabulation of oriented links. With microfiche supplement. Math. Comp. 57 (1991), no. 196, 747--761.
- Erica Flapan, When topology meets chemistry: A topological look at molecular chirality. Outlooks. Cambridge University Press, Cambridge; Mathematical Association of America, Washington, DC, 2000. xiv+241 pp. ISBN 0-521-66254-0; ISBN 0-521-66482-9
- André Haefliger, Knotted (4k-1)-spheres in 6k-space. Ann. of Math. (2) 75 1962 452--466.
- Jerome Levine, A classification of differentiable knots. Ann. of Math. (2) 82 1965 15--50.
- W.B. Raymond Lickorish, An Introduction to Knot Theory, Graduate Texts in Mathematics, Springer, 1997, ISBN 0-387-98254-X
- Kenneth A. Perko Jr., On the classification of knots. Proc. Amer. Math. Soc. 45 (1974), 262--266.
- Dale Rolfsen, Knots and Links, 1976, ISBN 0-914098-16-0
- Silver, Dan, Scottish physics and knot theory's odd origins (expanded version of Silver, "Knot theory's odd origins," American Scientist, 94, No. 2, 158-165)
- E. C. Zeeman, Unknotting combinatorial balls. Ann. of Math. (2) 78 1963 501--526.
[edit] Further reading
There are a number of introductions to knot theory. A classical introduction for graduate students or advanced undergraduates is Rolfsen (1976), given in the references. Other good texts from the references are Adams (2001) and Lickorish (1997). Adams is informal and accessible for the most part to high schoolers. Lickorish is a rigorous introduction for graduate students, covering a nice mix of classical and modern topics.
- Richard H. Crowell and Ralph Fox,Introduction to Knot Theory, 1977, ISBN 0-387-90272-4
- Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, 1985, Walter de Gruyter, ISBN 3-11-008675-1
- Louis H. Kauffman, On Knots, 1987, ISBN 0-691-08435-1
[edit] External links
[edit] History
- Thomson, Sir William (Lord Kelvin), On Vertex Atoms, Proceedings of the Royal Society of Edinburgh, Vol. VI, 1867, pp. 94-105.
- Silliman, Robert H., William Thomson: Smoke Rings and Nineteenth-Century Atomism, Isis, Vol. 54, No. 4. (Dec., 1963), pp. 461-474. JSTOR link
- Movie of a modern recreation of Tait's smoke ring experiment
[edit] Knot tables and software
- KnotInfo: Table of Knot Invariants and Knot Theory Resources
- The wiki Knot Atlas - detailed info on individual knots in knot tables
- KnotPlot - software to investigate geometric properties of knots
