Introduction to Knot Theory 25/02/2018

1.4. Operations on knots. We will now discuss certain operations on knots and links. Definition 1.15. The mirror image L! of a knot/linkL is the result of applying toL the reflection (x,y,z) 7→(−x,y,z). The reversed ←− knot/link L isL with orientation of all components reversed. We say ←− thatL is anticheiral ifL∼L ! and cheiral otherwise. IfL∼ L , then we say thatL is reversible . Example 1.16. Figure-eight is anticheiral (exercise), whereas the mir- ror image of a right-handed trefoil is the left-handed one:

mirror

We will show later that these two knots are not equivalent, so that the trefoil is a cheiral knot. It is also reversible: to see this rotate its diagram by 180 degrees with respect to an axis in the plane of the projection:

rotate

Definition 1.17. LetL 1 andL 2 be two links and pick pointsp i ∈L i fori=1, 2. The connected sum L1#L2 ofL 1 andL 2 is the result of cutting the links at the pointsp i and regluing them in an oriented way:

p1 p2 L1 L2 L1 L2

1 2

Theorem 1.18. The connected sumL 1#L2 does not depend on the choice ofp 1 andp 2 whenL 1 andL 2 are knots.

Sketch of Proof. We will use the following facts from topology. (1) A tame knotK has a regular neighborhood , that is an embed- ding of a solid torush:S 1 ×D 2 →R 3 such thatK(t)=h(t, 0). (2) Given closed ballsO ′ ⊂ O there is an ambient isotopy that takesO intoO ′ and does not move points outside a given open neighborhoodU ofO. (3) There is an isotopy Ψ of a solid torus, such that it it (e , r) ifr=1, and Ψ(e , r)= i(t+s) 1 �(e , r) ifr< 2 for a givens. ′ Letp 1 be another point onL 1 and fix a regular neighborhoodV of the knot. Consider a ballO that intersectsL 1 in an arc aroundp 1 ′ and which contains knotL 2. Shrink it into a ballO aroundp 1 that is ′ contained inV , then move it alongV to reach the pointp 1 using (3). Then expand it back using (2). This describes an isotopy that takes the connected sum with respect to pointsp 1 andp 2 to the one with ′ respect top 1 andp 2. � Property 1.19. The connected sum is a commutative monoid opera- tion on knots:

(1) the unknotU is the neutral element:U#K∼K∼K#U, (2) it is associative:K#(K ′#K′′)∼(K#K ′)#K′′, (3) it commutes:K#K ′ ∼K ′#K.

We will see later that # has no inverses: the only invertible knot is the unknot. Example 1.20. The connected sum of two right trefoils is the granny knot, whereas the connected sum of the left and right trefoil is the square knot:

granny knot square knot 3

There is an empirical ,,proof” that these two knots are not equivalent: shoelaces tied as the square knot are more durable that those tied as the granny knot.

2. Simple invariants

It is a difficult problem to see whether two knots/links are equiva- lent or not. Such questions can be often answered by computing link invariants: certain quantities that are the same for equivalent knots. There are two sources of link invariants:

• topological invariants, which are computed directly from a link and are preserved under ambient isotopies, and • combinatorial invariants, which are computed from a link dia- grams and are preserved under Reidemeister moves.

The former can be easily defined and are usually very powerful, but very hard to compute. Contrary, combinatorial invariants are harder to define (one has to check all Reidemeister moves), but usually easier to compute. The following is an example of a full invariant, which takes different values for different knots, but completely useless in practice: there is no good way to compare such invariants algorithmically. Example 2.1. Define the knot type [K] ofK as the equivalence class of its diagram modulo Reidemeister moves.

In what follows we construct other, more tackable, invariants.

2.1. Crossing and unknotting numbers. Example 2.2. Define the crossing number c(L) of a linkL as the min- imal number of crossings among its diagrams. It has the following properties:

(1)c(L) = 0 if and only ifL is an unlink, (2)c(K)� 3 for a nontrivial knotK, (3) the crossing number of a trefoil is 3.

It is very easy to find an upper bound forc(K) (any diagram gives one), but very hard to computec(K). 4

Proposition 2.3. The crossing number is subadditive:c(L#L ′)� c(L)+c(L ′) for any linksL andL ′.

Proof. Choose minimal diagramsD andD ′ of the linksL andL ′. Then D#D′ is a diagram ofL#L ′ withc(L)+c(L ′) crossings.�

Additivity ofc(L) is an open problem: there is neither a proof nor a counter-example. Problem. Is it true thatc(L#L ′) =c(L)+c(L ′)?

Consider the following move, called the crossing change:

With this move we can reduce every link diagram to a diagram of an unlink. Proposition 2.4. LetD be a diagram of a linkL. Then there is a diagramD ′ of an unlink obtained fromD by changing some of its crossings.

Proof. Consider the regular projectionπ(L) of the linkL used to con- structD. It is a union of loops, one per component ofL. Pick one loopγ parametrized asγ(t)=(x(t), y(t)) witht∈ [0, 1] and consider a particle P moving inR 3 aboveγ in a descending way except near the endpoints, where it goes quickly upwards, i.e.P(t)=(x(t), y(t), z(t)), wherez(t) has the following graph:

z(t)

t 0 1

ThenP is an unknot (exercise: find the isotopy). Repeat the above for each component ofL, putting each of them below the previous one. This results in an unlinkL ′ with the same projection asL. Hence, 5 their diagramD andD ′ have the same shape, but they may differ at crossings. � Definition 2.5. The unknotting number u(D) of a link diagramD is the smallest number of crossing changes required to reduceD to a diagram of an unlink. The unknotting number u(L) of a linkL is the minimal unknotting number of its diagrams. Example 2.6. The unknotting numberu(K) = 0 if and only ifK is an unlink. Both a trefoil and a figure-eight knot have unknotting number 1:

−→ −→

The unknotting number of the cinquefoil is at most 2:

−→ −→

2.2. Linking number and writhe. Define the sign sgn(c) of a cross- ingc as follows:

+ −

This convention follows the right-hand rule: bend slightly the fingers of your right hand except the thumb, which you should make orthogonal to the rest. Thinking of the thumb as the underpass and the other fin- gers as the overpass, they define the orientation of the positive crossing. Definition 2.7. The writhe w(D) of a link diagramD is the sum of signs of its crossings. The linking number lk(L1,L2) of componentsL 1 andL 2 of a linkL is half the sum of signs of crossings betweenL 1 and L2. 6

Example 2.8.

w=+3 lk = +1

w=−3 lk =−1

The linking number is a link invariant, which implies that the two Hopf links are not equivalent; they are also nontrivially link. On the other hand, writhe is not a link invariant, as its value is not preserved by the first Reidemeister move.

Proposition 2.9. The linking number lk(L1,L2) is preserved by Rei- demeister moves. The writhew(D) is preserved by the second and third Reidemester move, but not by the first one:

w =w +1 andw =w −1. � � � � � � � �

Proof. The second Reidemester moves creates or destroys two cross- ings of opposite signs, whereas the third Reidemesiter moves preserves the amount of positive and negative crossings. The first Reidemeis- ter move only changes the number of self-crossings, so that the linking number is not affected. � Definition 2.10. A framed knot is a embedding of an annulusS 1×[0, 1] intoR 3. A embedding of several annuli is a framed link.

Framed knots/links also admits diagrams. Usually we require that the projection is one-to-one on each section{p}×[0, 1] of the annulus— if so we say that the diagram has a blackboard framing. There is a variant of Reidemeister Theorem for diagrams of framed links: we 7 are still allowed to perform the second and third move, but the first one is replaced with

←→

Notice that the other configuration of kinks already follows from the sec- ond and third move. Example 2.11. It follows from Proposition 2.9 that the writhe is an in- variant of framed knots. Because the two trefoils have different writhe, they are not equivalent as framed knots.

2.3. Colorings. Definition 2.12. A 3-coloring of a link diagramD is an assignment of a color—red, green, or blue—to each arc ofD, such that at each crossing either all three arcs have the same color or all three colors meet. We say thatD is 3-colorable if it admits a coloring by all three colors. Example 2.13. The standard diagrams of a trefoil and unknot are 3-colorable:

whereas the standard diagrams of the unknot and figure-eight are not. Theorem 2.14. If a linkL admits a 3-colorable diagram, then all diagrams ofL are 3-colorable.

Hence, trefoil is not trivial and not equivalent to the figure-eight knot.