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5. Knots, Links, Braids

A knot is a simple closed curve (homeomorphic image of S(1)) in
Euclidean 3-space E(3). Two knots are called equivalent when there is
an orientation-preserving homeomorphism of E(3) onto itself sending
one knot to the other.

5.1 Wild embeddings

Schoenflies proved in 1908 that any homeomorphism from a simple
closed curve in the plane E(2) onto the unit circle S(1) can be
extended to a homeomorphism of the plane onto itself. Similar things
do not hold in higher dimensions.

[wildarc]

For example, there exist wild embeddings of simple arcs into E(3):
homeomorphic images of the unit interval such that the complement is
not simply connected. Thus, one usually restricts knots to be tamely
embedded, e.g., as a simple closed polygonal curve, and we'll do so
as well.

Two more examples of wild embeddings:

[hornedsphe]

Example (Alexander's horned sphere) A homeomorphic image of the
sphere S(2) in E(3) such that the complement of the image is not
simply connected.

[antoine]

Example (Antoine's necklace) A homeomorphic image of the Cantor set
(which is compact and totally disconnected) in E(3) such that the
complement of the image is not simply connected.

(The pictures here were taken from Hocking & Young, Topology, pp.
176-177.)

5.2 Knot diagrams and Reidemeister moves

Given a knot in E(3), we project it from a point in general position
into E(2), so that the resulting curve never passes three times
through the same point, and indicate for each crossing whether it is
an over- or undercrossing. The resulting diagram suffices to retrieve
the knot up to equivalence.

Now equivalence between knots can be translated into equivalence
between diagrams. This was done by Reidemeister, who showed that two
diagrams represent the same knot if and only if one is obtained from
the other by a sequence of Reidemeister moves:

[reidemeist]

in each of the three types of move, we may replace the upper picture
by the lower, or vice versa; type I also has a mirror image, type I'.

There is a unique knot with a diagram without crossings: the unknot.

5.3 Prime knots and Seifert surfaces

There is a composition of knots: roughly: tying one after the other.
One takes two oriented knots, places two straight line segments with
opposite orientation on top of each other, making them cancel. The
resulting knot is uniquely determined by the two knots one started
with. A knot is called prime if it is not the sum (defined in this
way) of two knots, both different from the unknot. Of course this is
precisely the 1-dimensional analogue of the # glueing operation used
in the previous section.

[knotsum]

(This picture was taken from Lickorish, An Introduction to Knot
Theory, p. 6.)

Note that the unknot is a zero element for this sum operation. We
have a unique `prime factorization': each knot is a sum of a finite
number of prime knots, and the prime knots that go into a knot are
uniquely determined.

This can be proved by associating a Seifert surface to the knot. A
Seifert surface of a link is a compact oriented surface in S(3) with
the given link as boundary. Every link has a Seifert surface: give
the link some orientation, take a diagram, and replace each crossing
by a non-crossing by making the upper strand turn left. Now the
diagram has become a union of circuits and we can pick disjoint discs
with these circuits as boundaries. Now for each crossing that we
replaced, insert a half-twisted strip joining the discs. This yields
an oriented surface with the original link as boundary.

Now one can define the genus of a knot as the minimal genus of a
Seifert surface Ffor it. (Where one might define the genus in this
case by g = (1 - chi(F))/2 where chi(F) is the Euler characteristic.
The genus is additive: g(K1 + K2) = g(K1) + g(K2), and is
nonnegative, and is zero only for the unknot. It follows that the sum
of two knots can only be the unknot when both knots are the unknot
themselves.

The Seifert surface of a knot is orientable and has a single boundary
component, so g = (1 - chi(F))/2 = h is the number of handles, a
nonnegative integer, and zero precisely for the unknot.

5.4 Catalog

Pictures of the prime knots with a plane drawing with at most 8
crossings (after G. Burde, `Knoten', Jahrbuch Ueberblicke Mathematik,
B.I. Mannheim, 1978, pp. 131-147 which contains drawings of the knots
with at most 9 crossings. A table of the knots with at most 10
crossings is found in D. Rolfsen, `Knots and Links', Publish or
Perish, 1976). The unknot is omitted and orientation is disregarded.

[03-001] [04-001] [05-001] [05-002] [06-001] [06-002] [06-003]
[07-001] [07-002] [07-003] [07-004] [07-005] [07-006] [07-007]
[08-001] [08-002] [08-003] [08-004] [08-005] [08-006] [08-007]
[08-008] [08-009] [08-010] [08-011] [08-012] [08-013] [08-014]
[08-015] [08-016] [08-017] [08-018] [08-019] [08-020] [08-021]

All except the last three are alternating: they have a diagram where
along the knot overcrossings and undercrossings alternate.

5.5 Invariants - the Kauffman bracket - the Jones polynomial

One of the objects of knot theory is to distinguish inequivalent
knots. To this end many invariants have been invented. In this
context the most obvious invariant of a knot K is the fundamental
group of its complement E(3) \ K.

A complete invariant is the complement itself as a topological space:
C. Gordon and J. Luecke, `Knots are Determined by their Complements',
J. Amer. Math. Soc. 2, 371-415, 1989.

Kauffman defines a Laurent polynomial in a variable A (i.e., a
polynomial in A and A^(-1)) for each unoriented link diagram D in S
(2) as follows: If D is a simple loop, then <D> = 1. If D is the
disjoint union of D' and a simple loop, then <D> = (-A^2-A^(-2))<D'>.
Otherwise, pick a crossing in D and make diagrams D' and D'' by
replacing the crossing by two turns where the upper strand turns left
in D' and right in D''. Now define <D> = A <D'> + A^(-1)<D''>.

[kauffman]

Then (1) the Kauffman bracket is well-defined, i.e., does not depend
on the order in which crossings are eliminated, and (2) the Kauffman
bracket is invariant for Reidemeister moves of types II and III. It
is not invariant for Reidemeister moves of type I. Indeed, we find

[kauffman1]

Now given an oriented link we can do a bit more, and distinguish two
types of crossing: those where the top comes from the left (+1) and
those where the top comes from the right (-1). Adding the values for
all crossings we get the writhe of the diagram. For an unoriented
knot this is defined too: the sign of a crossing does not change when
the orientation is reversed.

[writhe]

The Jones polynomial V(L) of a link L is defined by V(L) = ((-A)^
(-3w) <D>) where D is any oriented diagram of L and w is the writhe
of D. The usual variable in V(L) is t, where t = A^(-4). Now V(L) is
a polynomial in t when the number of components of L is odd, and in
particular when L is a knot.

5.6 Links

A link is a disjoint union of simple closed curves in E(3). Thus,
each connected component is a knot, and these knots may be entangled.
The smallest few examples are shown here.

[00-02-01] [02-02-01] [04-02-01] [05-02-01] [06-02-01] [06-02-02]
[06-02-03] [07-02-01] [07-02-02] [07-02-03] [07-02-04] [07-02-05]
[07-02-06] [07-02-07] [07-02-08] [08-02-01] [08-02-02] [08-02-03]
[08-02-04] [08-02-05] [08-02-06] [08-02-07] [08-02-08] [08-02-09]
[08-02-10] [08-02-11] [08-02-12] [08-02-13] [08-02-14] [08-02-15]
[08-02-16]

(Two links can be nonequivalent but have homeomorphic complements,
see C.C. Adams, `The Knot Book: An Elementary Introduction to the
Mathematical Theory of Knots', W. H. Freeman, New York, pp. 280-286,
1994.)

A link is trivial if and only if the fundamental group of the
complement is free.

5.7 Braids

A braid on n strings is a collection of n arcs in E(3) starting at
the n points (0,j,1) and ending at the n points (0,j,0), j =
1,2,...,n, each meeting the planes z = c (with 0 < c < 1) in a unique
point.

The elementary braid s(i) is the braid interchanging the two strands
numbered i and i+1, where the former overcrosses the latter, where
all other strands go straight from top to bottom.

We have the obvious concept of composition of braids (similar to that
of composition of paths), and find that every braid is (equivalent
to) the product of elementary braids.

The braid group B(n) of braids with n strings is isomorphic to the
group with generators s(i) (i=1,...,n-1) and relations (i) s(i) s(j)
= s(j) s(i) when |i-j| > 1, and (ii) s(i) s(i+1) s(i) = s(i+1) s(i) s
(i+1) for all i. (Artin)

It is possible to view B(n) as the fundamental group of a topological
space. (Fox)

A theorem by Alexander (J.W. Alexander, `A lemma on systems of
knotted curves', Proc. Nat. Acad. Sci. USA 9 (1923) 93-95) shows that
every link is equivalent to one obtained from a braid by identifying
starting and ending points.

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