Introduction to Knot Theory 18/02/2018

1. Knots and knot diagrams

1.1. Embeddings and isotopies. LetX andY be Hausdorﬀ topo- logical spaces. Deﬁnition 1.1. A continuous mapf:Y→X is an embedding if it is injective and closed.

Recall that every functorf:Y→X is closed ifY is compact. Deﬁnition 1.2. An embedding Ψ:Y×I→X×I is an isotopy if there is a continuous mapψ:Y×I→X such that Ψ(y, t)=(ψ(y, t), t). We say that embeddingsf,g:Y→X are isotopic if there exists an isotopy Ψ:Y×I→X×I, such thatf(y)=Ψ(y, 0) andg(y)=Ψ(y, 1).

We will often write Ψt for the mapy 7→Ψ(y, t) with ﬁxedt. Deﬁnition 1.3. Embeddingsf,g:Y→X are ambient isotopic if there is an isotopy Ψ:X×I→X×I, such that Φ 0 = idX andg=Ψ 1 ◦f. Such Φ is called an ambient isotopy.

Not every isotopy can be extended to an ambient isotopy. In particular, there are embeddings that are isotopic, but not ambient isotopic.

1.2. Knots and links. Deﬁnition 1.4. A knot is an embeddingK:S 1 →R 3. We consider oriented and non-oriented knots. Two knotsK 1 andK 2 are equivalent, writtenK 1 ∼K 2, if they are ambient isotopic. An embedding ofn disjoint circlesL:S 1 ⊔...⊔S 1 →R 3 is called a link. Example 1.5.

unknot right trefoil left trefoil ﬁgure-eight 1 2

two unlink Hopf link Borromean rings Whitehead link Remark 1.6. The word ,,ambient” in Deﬁnition 1.4 is important: all knots are isotopic as embeddings. Indeed, one can take a ball that contains almost the whole knot, and shink it to a point:

−→ −→

There are pathological knots, such that the following one:

We want to avoid such knots. Deﬁnition 1.7. A knot/link is tame if it is equivalent to a union of ﬁnitely many intervals (such knots/links are called polyhedral or PL). Otherwise, a knot/link is wild. Example 1.8. Knots and links from Example 1.5 are tame. For example:

∼ ∼

In order to avoid wild knots, one can also restrict to smooth embeddings and smooth isotopies. In such case, there is no need for ambient isotopies: it is known that smooth embeddings are smoothly isotopic if and only if they are smoothly ambient isotopic.

1.3. Knot diagrams. LetK be a knot of a link. 3

Deﬁnition 1.9. A projectionπ:R 3 →H onto a planeH is regular if every point fromπ(K) is covered by at most two points ofK, there are only ﬁnitely many double points, and at each of them the two arcs ofK crosses themselves:

regular not regular Theorem 1.10. Every tame knot/link admits a regular projection.

Sketch of Proof. LetK be a tame knot/link. Assume ﬁrst that it is polygonal, i.e.K=I 1 ∪ · · · ∪I r, where eachI i is a straight segment. Consider the following sets:

•V 1 is the set of directions of segmentsI i, •V 2 is the set of directions of vectors with origin at a vertex of K and pointing to a point ofK, •V 3 is the set of directions of lines that intersectK three times. where by a direction we understand a line going through the origin of 3 R .V 1 is a finite union of lines,V 2 is contained in a finite union of planes, whereasV 3 is a union of surfaces. Hence, the complement of V :=V 1 ∪V 2 ∪V 3 is dense, in particular not empty, and every point from it is a direction of a regular projection. In the general case, letK ′ be a polygonal knot/link equivalent toK, and which is contained in anǫ-neighborhood ofK for some smallǫ> 0. Then regular projection ofK are given by points from the complement ofV that are in distance at leastǫ fromV. � Definition 1.11. A diagram of a knot/linkK is the imageπ(K) ofK under a regular projection, where at each double point the part ofK closer toH is broken.

It follows from Theorem 1.10 that tame knots and links have diagrams. Example 1.12. The pictures from Example 1.5 are knot/link diagrams. 4

The main problem of knot theory is to determine whether two diagrams represent the same knot/link. For instance, all the following are diagrams of the unknot:

There are certain transformations, called the Reidemeister moves, that change the diagram but preserves the represented knot/link:

←−−−→ ←−−−→ R1 R2

←−−−→ R3

Theorem 1.13 (Reidemeister). Link diagramsD 1 andD 2 represent equivalent links if and only ifD 2 can be obtained fromD 1 through ﬁnitely many Reidemeister moves.

Sketch of Proof. LetL 1 andL 2 be links with diagramsD 1 andD 2 projected on the same planeH. We may assume thatL 1 andL 2 are polygonal: every tame linkL is equivalent to a polygonal oneL ′, such that their diagrams are ambient isotopic. We now use the fact that polygonal links inR 3 are ambient isotopic if and only if they are related by a sequence of Δ-move:

When the moves are suﬃciently small, their projections correspond to Reidemeister moves. For instance: 5

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Example 1.14. It is not always an easy task to check if two diagrams represent equivalent links. One of the most famous examples is the Perko pair:

It was conjectured for a long time that these diagrams represent diﬀer- ent knots, until a sequence of Reidemeister moves have been found by Keneth Perko in 1973.