Norwood showed that two-generator knots are prime and provided an example of a composite two-generator link of two components. Here a generalization of a theorem of Norwood shows that certain free products with amalgamation are not generated by any two group elements. This result is used to show that any composite two-generator link has Hopf link summand.

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The knot summmand is a two-generator knot in which one generator may be chosen to represent the class of a meridian. We use cookies to help provide and enhance our service and tailor content and ads. By continuing you agree to the use of cookies. Download full text in PDF Download.

These aforementioned invariants are only the tip of the iceberg of modern knot theory. 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 The Alexander-Conway polynomial is actually defined in terms of links , which consist of one or more knots entangled with each other.

The concepts explained above for knots, e. Suppose there is a link diagram which is oriented, i. Then the Alexander-Conway polynomial, C z , is recursively defined according to the rules:.

The second rule is what is often referred to as a skein relation. To check that these rules give an invariant of an oriented link, 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 the relation is applied. Applying the relation to the Hopf link where indicated,.

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The unlink takes a bit of sneakiness:. Since the Alexander-Conway polynomial is a knot invariant, this shows that the trefoil is not equivalent to the unknot. So the trefoil really 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 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 William Thurston proved many knots are hyperbolic knots , meaning that the knot complement , i. The hyperbolic structure depends only on the knot so any quantity computed from the hyperbolic structure is then a knot invariant Adams 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. 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.

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By thickening the link in a standard way, the horoball neighborhoods of the link components are obtained. Even though the boundary of a neighborhood is a torus, when viewed from inside the link complement, it looks like a sphere. Each link component shows up as infinitely many spheres 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 and shows how to extend the pattern of spheres infinitely. This pattern, the horoball pattern, 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. In four dimensions, any closed loop of one-dimensional string is equivalent to an unknot. This necessary deformation can be achieved 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 fact, there are smoothly knotted 4k-1 -spheres in 6k -space, e. The notion of a knot has further generalisations in mathematics, see: knot mathematics. Two knots can be added by cutting both knots and joining the pairs of ends.

The operation is called the knot sum , or sometimes the connected sum or composition of two knots. This can be formally defined as follows Adams : 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 a sum of the original knots.

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Depending on how this is done, two different knots but no more may result. This ambiguity in the sum can be eliminated regarding the knots as oriented , i. The knot sum of oriented knots is commutative and associative. There is also a prime decomposition for a knot which allows a prime or composite knot to be defined, analogous to prime and composite numbers Schubert For oriented knots, this decomposition is also unique.

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Higher dimensional knots can also be added but there are some differences. 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.

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 Hoste , p. Tabulation efforts have succeeded in enumerating over 6 billion knots and links Hoste , p. 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, , , , , , , While exponential upper and lower bounds for this sequence are known, it has not been proven that this sequence is strictly increasing Adams 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 Hoste 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 s. 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 one duplication in the Tait—Little tables; however he missed the duplicates called the Perko pair , which would only be noticed in by Kenneth Perko Perko This famous error would propagate when Dale Rolfsen added a knot table in his influential text, based on Conway's work.

In Rankin, Flint, and Schermann, tabulated the alternating knots through 22 crossings Hoste This is the most traditional notation, due to the paper of J.

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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. 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.

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An appropriate sign is given to indicate over and undercrossing. 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. The Conway notation for knots and links, named after John Horton Conway , is based on the theory of tangles Conway 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.