Lecture 0: Introduction to Knot Theory

Notes by Jonier Amaral Antunes

January 12, 2016

Introduction

Knots have been elements of human culture since the beginning of History. Thefirst uses of knotting were arguably made by primitive societies still in the PaleolithicEra and knots have been represented in artistic depictions of everyday life sinceancient civilizations, like the Egiptian [3].

Even in current times, knots appear in the most diverse forms and contexts, fromelaborate artisanal braiding to magic tricks and puzzles. In climbing, for instance,stopper knots might be used to prevent the sliding of a rope. The two most elementaryof those being the overhand knot and the figure-eight, depicted below:

Figure 1: Overhand knot. Figure 2: Figure-eight knot.

Somewhat less elementary but still very common examples of knots are the Reefknot, the Bowline and the Trucker’s hitch, that might be used for uniting two linesegments, as a rescuing loop and as a pulley system to gain mechanical advantage,respectively [4].

Figure 3: Reef knot.Figure 4: Bowline Figure 5: Trucker’s hitch.

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All that diversity of applications, even though lying completely beyond the pur-poses of these lectures, might already serve as an indicator of both how elementaryand how complex the subject of knots can be. These characteristics tend to appeartogether when dealing with a subject whose consequences and ubiquity largely exceedits apparent simplicity.

The Mathematical study of knots started in the 19th century when Gauss devel-oped a way of calculating the linking number of two knots, a quantity that indicatesintuitively the amount of times the knots wind around each other. In that century,connections between the description of fluid mechanics and electromagnetism werestarting to be made, thanks to the description of the latter in terms of field lines.During that period knot theory started to take shape, mainly by the work of PeterGuthrie Tait and Lord Kelvin. Tait’s study on vortices inspired Lord Kelvin to lookfor a description of atoms in terms of knots in the aether and this, in turn, motivatedTait to start classifying knots in tables and trying to obtain their properties, sincethat could result in the attainment of a table of elements [5].

As it usually happens, the area acquired its own life inside Mathematics, pushedby the developments in topology, and it became independent from its original physicalmotivations, since the experimental and theoretical developments of the beginning ofthe 20th century gave a different direction for the study and understanding of thestructure of matter. Interestingly, by the end of that century, Knot Theory startedbeing once again relevant in Physics, through the discovery of connections to QuantumField Theory and Statistical Mechanics ( [5] and [6]). Currently, knotting phenomenaare being found in more examples of complex systems, coming from other fields suchas Chemistry and Molecular Biology [7].

Preliminaries about Mathematical Knots

As an object of topology, the mathematical description of a knot should not beconcerned with rigid aspects of localization in space, like metrics, but only with thedeformation properties of the knot. That way, we will be interested in studying knotsup to continuous deformations that in some sense correspond to the intuitive physicalnotion of knotting, that is, deforming the knot in space without cutting it.

If we were to define knots as entangled segments of line, like the examples given bythe figures in the introduction, every knot would be deformable into any other by firstunknotting it continuosly and then tying it continuously in the pattern of the secondone. To avoid that, our definition of knot should connect the loose ends, so that theknots cannot be untied. Applying that idea to the two first examples of physical

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knots given in the introduction we obtain two of the most elementary mathematicalknots, the trefoil and the figure-eight:

Figure 6: Trefoil. Figure 7: Figure-eight knot.

In both cases the resulting object is clearly homeomorphic to a circle, so we candefine a knot as a homeomorphic image of a circle in space (more regularity might berequired as we will discuss later). The trivial example is the unkotted circle:

Figure 8: The unknot.

Much of the theory developed to analyze knots can be easily extended to the studyof collections of knots. Thus we have the notion of a link, as any collection of knotswhich might be entangled in space. In Figure 9 we have some examples.

Each of the examples of knots and links in Figure 9 are distinct, in the sense thatthey cannot be deformed to become any other, and they actually present differenttopological features. For instance, the Borromean rings above form an example of aBrunnian link, which is a link with the property that by removing any of the knotsthe remaining link becomes trivial. In other words, no two circles in the Borromeanrings form a Hopf link.

Verifying that the examples we have seen so far are not equivalent might beintuitively feasible, but as the knots get more complicated, this task becomes anincreasingly difficult challenge. For example the Perko pair below, was classified astwo distinct knots for more than half a century until Kenneth Perko showed theywere representing equivalent knots in 1973.

An important part of the history of classification of knots was the development oftables, representing the simplest possible diagrams for each knot with a notation to

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Figure 9: Empty Link. Figure 10: Trivial link Figure 11: Hopf link.

Figure 12: Whitehead Link. Figure 13: Boromean rings

Figure 14: The Perko Pair.

identify them. The most traditional one is the Alexander-Briggs notation which wasmade by J. W. Alexander and G. Briggs and later included in Dale Rolfsen’s table,who also extended its use.

In the notation in Figure 15 the main number represents the minimal number ofcrossings that appear in a diagram of the knot and the index is just accounting foran arbitrary order chosen for the knots with same number of crossings.

Diagrams, as the ones being used so far, are important not only for making knottables but also, as we will see, for developing the theory. To calculate knot andlink invariants, for instance, we will usually make use of some diagram. For thatreason, we will need a good definition of a diagram so that it contains all topologicalinformation about a link without introducing new phenomena.

A Regular Plane Projection of a link is a projection of the link onto a (2-dimensional) plane retaining the information of which direction is an undercrossingand which is an overcrossing for each intersection appearing on the resulting curve.

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Figure 15: Knot table.

The projection is required to be generic, or regular, in the sense that we make thefollowing conditions:

(1) Tangent lines of the link in space are mapped onto lines (and not points) inthe projection.

(2) No more than two distinct points on the link are projected to the same pointon the plane.

(3) The projection contains only a finite number of crossings and at each one ofthese the projection of the corresponding tangent lines are not the same.

Condition (1) ensures that we are preserving regularity of the knot, by not allowingcusps to be formed. To see that, consider any segment of a knot where there is aturn, like the one indicated in the figure below with a tangent line represented. If weallowed that tangent to be mapped onto a point we could create a cusp, like in theirregular projection in the picture.

Figure 16: Cusp

Condition (2) does not allow multiple crossings, like the ones below.

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Figure 17: Multiple crossings

Condition (3) implies that the intersections appearing in the projection are notself-tangencies, because in that case the tangent lines would be projected onto thesame image.

Figure 18: Self-tangencies

To see how some properties of knots might be easily stated in terms of their dia-grams let us define the concept of alternating knot. A knot diagram is alternatingif its under and overcrossings alternate as we move along the knot. A knot is al-ternating if it has an alternating diagram. As it turns out, alternating knots haveuseful geometric properties. In fact, the definition of an alternating knot can be madeintrinsically, without refering to the concept of a knot diagram [8].

Some examples of those interesting properties will be discussed below. But thefact that they have been observed very early in the history of knot tabulating, andalso the fact that elementary knots, with fewer minimal number of crossings, tend tobe alternating made some knot diagrams to be wrongly ommited or mistaken in earlytables. Of all knots presented here so far, the only one which is not alternating is theone that appeared with a non-alternating diagram, the Perko Pair knot, 10161. Themost elementary knots that are non-alternating have 8 crossings, 819, 820 and 821.

To discuss more properties of alternating knots let us make another definitioninvolving plane projections. A knot diagram is reducible if there is a circle in theplane projection which intersects the diagram transversely at a crossing point and

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nowhere else. Intuitively, that means the diagram can be untwisted at that crossing,since it takes the schematic form:

Figure 19: Reducible knot

For example, if we take the following knot made by a combination of the trefoiland the figure-eight knot we can reduce it by untwisting:

Figure 20: Untwisting a reducible knot

Combinations of knots as the one described above are called connected sumsand they will be formally defined later. A knot that is not reducible will be calledreduced.

Now we have enough definitions to state a result that was conjectured by Taittogether with other propositions in the 19th century, all of which ended up beingproved more than a hundred years later. Since Tait was mainly concerned withtabulating knots in a time when there was not enough formalism to make his ideasprecise, it is not clear whether he intended his conjectures to apply to every knot,but now they are known to hold for alternating knots [9].Proposition 2.1 (Tait Conjecture). A reduced, alternating diagram has the minimalnumber of crossings among all diagrams for the given knot.

The proof of that conjecture was obtained in 1986 by Louis Kauffman, K. Murasugiand Morwen Thistlethwaite in 1987 using the Jones Polynomial, a powerful invariant

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of knots.

Orientation, Inverses and Mirrors

Now we can start to turn to the question of which extra structures can be addedto our definition of knots and whether they will actually give us more information,or if the resulting theory is equivalent.

As curves in space, there is a very natural notion of orientation that can beassociated with a knot as we have at every point two choices of direction for a tangentvector and we can assume this choice to be made continuously. In a plane diagram,the two orientations naturally correspond to moving along the knot projection in aclockwise or counterclockwise fashion and can be just indicated with an arrow.

For oriented knots, that is, a knot with a chosen orientation, we can define twonatural operations. The orientation reversal, σ, simply alternates between thetwo possible orientations of the knot. In the diagram it corresponds to inverting thecorresponding arrow. The mirror-reflection, τ is a reflection through the spaceorthogonal to a fixed vector. Since reflections are involutions and the compositionof two reflections is a rotation, which is one deformation we certainly allow knots tohave, it is clear that any mirror-reflection of a given knot will give the same resultingknot. In a plane diagram, we can do the reflection through the plane of the projectionan that will correspond to just exchanging over and under crossings. Given a knot Kwe will denote by σ(K) =: −K its inverse and by τ(K) =: K the mirror-reflection.In the image below we exemplify those involutions to the trefoil, K = 31.

Figure 21: Orientation-reversal and Mirror-reflection

From the first picture it is not hard to see that the trefoil is preserved underorientation-reversal, since if we rotate the diagram for −31 by 180o along the verticalaxis we obtain the same original diagram. On the other hand, it is suggested by thesecond picture that 31 6= 31. This is in fact the case, but we still have not developed

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the tools to easily prove it here. However, the existence of oriented knots that are notequivalent to their inverse or reflection has as a consequence the fact that the theoryof oriented knots is not equivalent to that of unoriented knots, since in the latter wehave more equivalences.

We can make definitions about the behavior of a knot under those operations. Aknot K is called:

(i) invertible if −K = K;(ii) plus-amphichiral if K = K;(iii) minus-amphichiral if K = −K;(iv) fully symmetric if K = −K = K = −K;(iv) totally anti-symmetric if K,−K,K and −K are all distinct.We also say that a knot is amphichiral if it is plus or minus-amphichiral and that

it is chiral if it is not amphichiral.We can see that those definitions are natural by analyzing the action of the group

generated by σ and τ over the space of knots. These involutions clearly commute,since reversing order and exchanging under and over crossings can be done in anyorder resulting in the same knot. So, the group generated by these operations isabelian generated by two elements of order 2. That is, the group of symmetries is{1, σ, τ, στ} which is isomorphic to Z2⊕Z2. Thus if we want to classify oriented knotsby considering which ones are preserved under those operations we might consider allthe subgroups of Z2 ⊕ Z2 which are five:

(1) If the stabilizer of a knotK is the subgroup {1}, that means it is not equivalentto any of its images under the symmetry operations, which means the knotsK,−K,Kand −K are all distinct. So we are in the totally anti-symmetric case. The simplestexamples happen for nine crossings, 932 and 933.

(2) If the stabilizer of K is the subgroup {1, σ}, we have K = −K and K = −K(notice that σ(K) = K implies σ(K) = στ(K) = τσ(K) = K). So this time K isinvertible and chiral. The simplest example is the trefoil, already discussed.

(3) If the stabilizer of K is {1, τ}, then K = K and −K = −K. The symmetryof K is plus-amphichiral and non-invertible. The minimal example happens only attwelve crossings.

(4) If the stabilizer of K is {1, στ}, then K = −K and −K = K. The symmetryof K is minus-amphichiral and non-invertible. The minimal example happens only ateight crossings, 817.

(5) If the stabilizer ofK is {1, σ, τ, στ}, thenK = −K = K = −K. The symmetryof K is fully-symmetric. The minimal example being the figure-eight knot, 41. To

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see that, first consider the action of σ:

It is easy to see that if we rotate the result 180o along the vertical axis we obtainthe original knot, so 41 is invertible. To see that it is also plus-amphichiral (and,consequently, minus-amphichiral) first let us deform it by moving the segment on theleft under the diagram to the other side and than rearranging it in a symmetric way:

If we rotate the last figure we clearly get 41.

As discussed above some of the symmetries only appear in higher number ofcrossings, when performing continuous deformations on knots gets increasingly morecomplicated. This type of ocurrence is an example of the general situation in knottheory, where conjectures can stand for many years until counterexamples or definitiveproofs appear.

Ambient Space

So far we have not discussed what exactly is the space where we are consideringthe knots to be. Naturally, we want its topological structure to allow our experience

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with knotting to correspond to the continuous movements we will perform on knots,so we should require our ambient space to be either euclidean or locally euclidean.

In a two dimensional space all homeomorphic images of circles are unknotted(otherwise intersections would happen). In four dimensions or more every (one di-mensional) knot is deformable into the unkot. It is possible to visualize this intuitivelyif we consider the knot projected in a 2-dimensional subspace except by a finite num-ber of crossings where it goes into the third dimension. By still having an extrafourth dimension to explore we can move the segments enough in that direction toavoid crossings and then changing their places in the third dimension. That process isanalogous of removing a point from inside of a circle, which is not possible in two di-mensions but easily done in three. Since we can exchange any crossing, consequentlywe can untie any knot.

Thus, we see not only that the three dimensional space is the most natural todefine a (one dimensional) knot, but also that the topology of the ambient space isstrongly connected with the tying itself. This fact became more pronounced in therecent development of knot theory as as connections between knots and the threedimensional manifold defined by their complements were obtained.

We could define a knot as a homeomorphic image of the circle S1 in the 3-dimensional euclidean space R3, but as it turns out it is better to use another 3-dimensional space, the 3-sphere:

S3 = {x = (x1, . . . , x4) ∈ R4 : ||x||2 = x21 + · · ·+ x2

4 = 1}

The 3-sphere can be obtained as a one-point compactification of S3. This can beseen by the stereographic projection, represented below in the case of S2:

Through this map, by removing one point, {∞}, from the sphere we can obtain ahomeomorphism from S3\{∞} to R3 by mapping any point p to the point q obtainedfrom the intersection of the plane with the line passing by ∞ and p.

This way we have that S3 is homeomorphic to R3 ∪ {∞} which means most ofthe discussions we can make about knots in S3 might be done in R3, since we cansimply move each knot to avoid {∞} if necessary. The general topological propertiesof curves will be the same because the compactification implies, in particular thatboth S3 and R3 have trivial fundamental group, since any loop in S3 is homotopicto one missing a point in S3. A difference that might be conveniently used, however,is to discuss knots passing through {∞} in R3, where they correspond to unboundedcurves.

There are many other consequences of the compacity of S3 that makes it conve-

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Figure 22: Stereographic projection

nient to use it instead of R3. For example for some classes of knots it makes sense todefine a natural metric in the complement and, in that situation, the volume of thecomplement becomes an invariant [10].

Knot Equivalences

Now let us finally turn to the question of what kind of regularity should we assumeand what notion we should give to the deformation equivalence of knots.

As it is well known in mathematics, the category of topological spaces allows manyextremely strange behaviors to appear. That is also the case with knots. If we onlydefine a knot as a subset of S3 homeomorphic to S1 we allow wild phenomena tohappen. For example, the knot depicted below contains a continuous succession ofever-shrinking overhand knots that might be made in a way so that they converge toa point. That point can be connected to the first overhand knot and the resultingfigure becomes homeomorphic to S1.

Figure 23: A wild knot

A simple way to avoid that type of object is to require smoothness of the knot,

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for instance, defining it to be diffeomorphic to S1. Alternatively, we could define aknot to be picewise-linear, that is a closed, not self-intersecting polygonal line in S3.

Figure 24: A picewise-linear trefoil

It turns out that the theories obtained by defining knots as either smooth orpicewise-linear are the same, because polygonal lines can be approximated by smoothcurves and vice-versa. But both of those definitions are more strict than the purelytopological one, since they do not allow wild knots. Working in the picewise-linearcategory allows the use of combinatorial techniques while working in the smoothcategory allows the use of differential geometry constructions.

The same way the regularity of a knot can be defined in more than one way,the notion of knot equivalence might be defined in different ways. Given two knotsX0 : S1 → S3 and X1 : S1 → S3, we say they are:

A) isotopic if there is a one parameter family of knots interpolating them. Thatis if we have F : S1 × [0, 1] → S3 where F (., t) : S1 → S3 is a knot for every t,F (., 0) = X0 and F (., 1) = X1. The map is assumed to have the correspondingregularity of the category we are working in (continuous, smooth or picewise-linear).

B) ambient isotopic if there is a one parameter family of morphisms of requiredregularity ψ : S3 × [0, 1]→ S3 such that ψ(., 0) = id and ψ(., 1) ◦X0 = X1

C) ambient equivalent if there is a commuting diagram

S1 S3

S1 S3

X0

ϕ ψ

X1

where ψ and φ are orientation preserving bijective maps of the required regularity(homeomorphisms, diffeomorphisms or picewise-linear isomorphism).

Definition (A) differs from the other two because it makes no requirement ofthe transformation extending to the ambient space. If we work in the smooth (or

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piecewise-linear) category, the three definitions are equivalent, because an embeddedsmooth manifold admits a tubular neighborhood, so the regularity in that case impliesthat deformation of the knot will correspond to a deformation of the space around it.

On the other hand, if we are working in the topological category, equivalence (A)ends up being too weak, since every tame knot (that is, a knot that does not presentwild phenomena as discussed above) is isotopic to the trivial knot. To obtain theequivalence we just have to continuosly shrink the part of the knot where the tangleis happening until it becomes a point.

For that reason, the notion of equivalence we will use is the ambient isotopy, thatexplicitly mentions the continuous ambient deformation that must happen togetherwith the knot deformation.

Now that we finished motivating the basic notions related to the study of knotswe can proceed to formally defining them and then constructing some importantinvariants through which we will start obtaining some of the fascinating topologicalproperties of these simple yet extremely interesting mathematical objects.

References[1] W. Lickorish. “An Introduction to Knot Theory,” Springer (1997).

[2] D. Rolfsen. “Knots and Links,” AMS Publishing (1976).

[3] J. C. Turner, P. Van De Griend. “History and Science of Knots,” World Scientific(1996).

[4] C. W. Ashley. “The Ashley Book of Knots,” Doubleday (1944).

[5] J. C. Baez, J. P. Muniain. “Gauge Fields, Knots and Gravity,” World Scientific(1994).

[6] L. H. Kauffman. “Knots and Physics,” World Scientific (2001).

[7] K. Murasugi. “Knot Theory and its Applications,” Birkhäuser (1996).

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[8] J. E. Greene. “Alternating links and definite surfaces,” ArXiv preprint (2015),math.GT/1511.06329.

[9] A. Stoimenow. “Tait’s conjectures and odd crossing number amphicheiralknots,”Bull. Amer. Math. Soc. 45 (2008), 285-291, math.GT/0704.1941.

[10] C. Adams, M. Hildebrand, J. Weeks. “Hyperbolic invariants of knots and links,”Transactions of the American Mathematical Society 326 (1991), 1-56.

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