Algebraic topology: knots links and braids
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## Contents
[Contents](algtop.html#toc5)5. Knots, Links, Braids
A knot is a simple closed curve (homeomorphic image of (S^{1})) in Euclidean 3‑space (\mathbb{E}^{3}).
Two knots are called equivalent when there is an orientation‑preserving homeomorphism of (\mathbb{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 (\mathbb{E}^{2}) onto the unit circle (S^{1}) can be extended to a homeomorphism of the plane onto itself. Similar statements fail in higher dimensions.
For example, there exist wild embeddings of simple arcs into (\mathbb{E}^{3}): homeomorphic images of the unit interval whose complement is not simply connected. Hence one usually restricts knots to be tamely embedded (e.g. as a simple closed polygonal curve); we will do the same.
Two more classic examples of wild embeddings are:
-
Alexander’s horned sphere – a homeomorphic image of the sphere (S^{2}) in (\mathbb{E}^{3}) whose complement is not simply connected.
-
Antoine’s necklace – a homeomorphic image of the Cantor set (compact and totally disconnected) in (\mathbb{E}^{3}) whose complement is not simply connected.
(The pictures here are taken from Hocking & Young, Topology, pp. 176‑177.)
5.2 Knot diagrams and Reidemeister moves
Given a knot in (\mathbb{E}^{3}), we project it from a point in general position onto (\mathbb{E}^{2}) so that the resulting curve never passes three times through the same point. At each crossing we indicate whether it is an over‑ or under‑crossing. The resulting knot diagram determines the knot up to equivalence.
Reidemeister showed that two diagrams represent the same knot iff one can be obtained from the other by a finite sequence of Reidemeister moves:
| Type I | Type II | Type III |
|---|---|---|
 |  |  |
In each case the upper picture may be replaced by the lower (or vice‑versa). Type I also has a mirror image, denoted I′.
The unique knot with a diagram containing no crossings is the unknot.
5.3 Prime knots and Seifert surfaces
There is a natural composition (connected sum) of knots: take two oriented knots, place two straight line segments with opposite orientation on top of each other so that they cancel, and join the remaining ends. The resulting knot is uniquely determined by the two summands.
A knot is prime if it is not the sum of two non‑trivial knots (i.e. knots different from the unknot). This is the 1‑dimensional analogue of the gluing operation discussed earlier.
Figure (from Lickorish, An Introduction to Knot Theory, p. 6)
The unknot serves as the zero element for this operation. Every knot admits a prime factorisation: it can be expressed as a finite sum of prime knots, and the multiset of prime factors is unique.
The proof uses Seifert surfaces.
A Seifert surface for a link (L\subset S^{3}) is a compact, oriented surface whose boundary is (L). Every link has a Seifert surface:
- Choose an orientation of the link and a diagram.
- Replace each crossing by a non‑crossing, turning the upper strand left. The diagram becomes a disjoint union of circles.
- Fill each circle with a disjoint disc.
- For each original crossing, attach a half‑twisted strip joining the appropriate discs.
The resulting surface is oriented and has the original link as its boundary.
The genus of a knot (K) is the minimal genus among all Seifert surfaces (F) for (K):
[
g(K)=\min_{F}; g(F),\qquad
g(F)=\frac{1-\chi(F)}{2},
]
where (\chi(F)) is the Euler characteristic.
Genus is additive:
[
g(K_{1}# K_{2}) = g(K_{1}) + g(K_{2}),
]
non‑negative, and equals zero iff the knot is the unknot. Consequently, a connected sum can be the unknot only when both summands are the unknot.
Since a Seifert surface of a knot is orientable with a single boundary component, its genus equals the number of handles, a non‑negative integer that is zero precisely for the unknot.
5.4 Catalog of prime knots
Below are plane drawings of the prime knots with at most eight crossings (the unknot is omitted and orientation is disregarded). The pictures are taken from:
- G. Burde, Knoten, Jahrbuch Überblicke Mathematik B.I. Mannheim, 1978, pp. 131‑147 (drawings up to nine crossings).
- D. Rolfsen, Knots and Links, Publish or Perish, 1976 (table up to ten crossings).
(Insert the series of knot diagrams here.)
All except the last three knots are alternating: they admit a diagram in which over‑ and under‑crossings alternate as one travels around the knot.
5.5 Invariants – the Kauffman bracket and the Jones polynomial
A central problem in knot theory is to distinguish inequivalent knots. Many invariants have been introduced for this purpose. The most immediate invariant of a knot (K) is the fundamental group of its complement (\mathbb{E}^{3}\setminus K). A complete invariant is the complement itself as a topological space (Gordon & Luecke, “Knots are Determined by their Complements”, J. Amer. Math. Soc. 2 (1989), 371‑415).
The Kauffman bracket
For an unoriented link diagram (D\subset S^{2}), Kauffman defines a Laurent polynomial (\langle D\rangle\in\mathbb{Z}[A,A^{-1}]) recursively:
-
Loop rule – If (D) is a simple closed curve, then (\langle D\rangle = 1).
-
Disjoint‑union rule – If (D = D’\sqcup\bigcirc) (a disjoint union of a diagram (D’) and a simple loop), then
[ \langle D\rangle = (-A^{2}-A^{-2}),\langle D’\rangle . ] -
Skein rule – For any crossing, replace it by the two smoothings:
- (D’): the smoothing where the upper strand turns left,
- (D”): the smoothing where the upper strand turns right.
Then
[ \langle D\rangle = A,\langle D’\rangle + A^{-1},\langle D”\rangle . ]
Kauffman proved:
- The bracket (\langle D\rangle) is well‑defined (independent of the order in which crossings are eliminated).
- It is invariant under Reidemeister moves of types II and III, but not under type I. In fact, a type I twist multiplies the bracket by (-A^{\pm3}).
From the Kauffman bracket one obtains the Jones polynomial (V_{K}(t)) by a suitable normalization that restores invariance under all three Reidemeister moves.
End of Section 5.
Writhe and the Jones Polynomial
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.
The Jones polynomial (V(L)) of a link (L) is defined by
[ V(L)=(-A)^{-3w},\langle D\rangle , ]
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 (\mathbb{E}^3). Thus each connected component is a knot, and these knots may be entangled. The smallest few examples are shown here.
(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 iff the fundamental group of its complement is free.
5.7 Braids
A braid on (n) strings is a collection of (n) arcs in (\mathbb{E}^3) starting at the points ((0,j,1)) and ending at the points ((0,j,0)) for (j=1,2,\dots ,n), each meeting the planes (z=c) (with (01);
2. (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.