Lecture 2: Planar diagrams and Reidemeister moves
Notes by Nate Bade
January 19, 2016
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Connected Sums and Prime Knots
In this section we will discuss tame and wild knots, connected sums of knots,
prime knots and the Reidemeister moves. First, let us give some useful definitions.
Definition 1.1. A tame knot K is a subset of R3 homeomorphic to the circle S 1
and locally flat at each point p ∈ K. Pictorially, for each p ∈ K, there exists a ball
U ⊂ R3 around p such that the “ball-arc pair” (U, U ∩ K) is homeomorphic to the
ball with a bisecting line through the origin, see Figure 1.
Figure 1: A knot K and a ball-arc pair (U, U ∩ K) around a point p ∈ K.
Definition 1.2. A knot K0 is said to be ambient isotopic to a knot K1 if there exists
an isotopy H : R3 × [0, 1] → R3 such that H(·, 0) = id and H(K0 , 1) = K1
1.1
Connected sums of knots
In this sections, we will define the operation of taking the connected sum of two
knots. Consider two oriented knots K1 and K2 such that K1 and K2 can be separated
by some sphere S 2 ⊂ R3 . We can then construct the connected sum of K1 and K2 by
the following steps, illustrated in Figure 2.
1. Pick small, locally flat intervals U1 ⊂ K1 and U2 ⊂ K2 .
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2. Excise these intervals, leaving a pair of tangles T1 and T2 .
3. Join the ends, smoothing if necessary, along the boundary of a rectangular strip
that meets the separating S 2 in a single arc. The join must maintain consistency
between the orientation of the tangles T1 and T2 .
We denote the connected sum of K1 and K2 by K1 # K2 .
Figure 2: The connected sum of the knots K1 and K2 , connected after excising the
intervals U1 and U2 .
Lemma 1.3. The connected sum of two oriented knots is well defined on knot types,
commutative and associative. Furthermore, the connected sum of any knot with the
unknot is simply the original knot K # = K.
Proof. To prove that the connected sum is well defined, we note that the only
ambiguity in the construction is the choice of the intervals Ui .
Figure 3: By shrinking a tangle, we can slide it through another tangle by ambient
isotopy.
Consider a different choice of flat interval V1 on K1 , and write the oriented segment
from U1 to V1 as the tangle T1U V , and the segment from V1 to U1 as T1V U . If we attach
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K1 and K2 at U1 , and then shrink the tangle T2 by isotopy we can pull the tangle T2
through to V1 as in Figure 3. Therefore we can transform the sum connected at U1
to the sum connected at V1 by ambient isotopy.
Note that the same argument also proves commutativity K1 # K2 = K2 # K1 .
Associate and K # = K are immediate.
Remark 1.4. Connected sums of links are not well defined a priori, there must be
something akin to a labeling on the components of the link.
Remark 1.5. Connected sums of unoriented knots are also not well defined. Indeed,
if K1 6= −K1 and K2 6= −K2 then while K1 # K2 = −K1 # − K2 , neither is equal to
−K1 # K2 = K1 # − K2 .
We have seen that the operation of connected sum turns the set of knot types into
a semigroup. It is natural to ask whether this semigroup is in fact a group. If that
were the case, every knot would have to have an inverse, that is to say for each knot
K there must exists a knot K −1 such that K # K −1 = . It turns out not only is
this statement untrue in general, it fails to hold for any nontrivial knot.
Proposition 1.6. If K 6= is nontrivial, then so is the connected sum K1 # K2 for
any other knot K2 .
Proof. The Mazur Swindle.
By contradiction, assume K1 # K2 is trivial. Then there exists an isotopy ft :
[0, 1] × R3 → R3 with f1 (K1 #K2 ) = . Without loss of generality, we can take ft to
act as the identity outside of the ball B3 containing tangles Ti .
Remark 1.7. To see that we may take ft to act trivially outside of B3 in the topological category requires the Isotopoy Extension Theorem of Edwards and Kirby proved
in [1]. However, since each pair of tangles is tame, we can consider them in the
PL-category.
In the PL category, the isotopy ft corresponds to a finite series ∆-moves taking
T1 − T2 to T0 . While there may be ∆ moves on the segment outside of B3 , it is
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clear that the isotopy will be unchanged if we ignore them. Using ambient isotopy
that acts trivally outside of B3 , we can shrink the tangles small enough that we may
assume that all ∆ moves are contained in a compact set C ⊂ B3 \S 2 . However, each
∆ move corresponds to an ambient isotopoy which is the identity outside of a small
ball around the triangle ∆. This means that the total isotopoy constructed from any
series of piecewise linear moves is entirely local, and can be taken to be the identity
outside of B3 .
Since connected sum is a commutative operation, there must be a similar isotopy
gt deforming the tangles T2 − T1 into the trivial tangle T0 . Again gt preserves the
segment common to both knots.
Now, consider the wild knot made by alternating sums of the knots K1 and K2 ,
connected such that each pair of tangles lies in an ball of radius rj = 2−j . Consider
the isotopy Ft that acts as the identity outside of the triangular region, but inside
acts as ft on each pair of tangles. Such an isotopy takes W → , and so W is trivial.
Similarly, consider the isotopy Gt that acts simultaneously by gt on each pair
of tangles T2 − T1 and acts as the identity elsewhere. Such an isotopy clearly
takes W → K1 . But then K1 must be isotopic to both W and , contradicting our
assumption that K1 was nontrivial.
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Definition 1.8. A nontrivial knot K is prime if K = K1 #K2 always implies that
either K1 or K2 is trivial.
We already know a couple of examples of primes knots: as we will prove later on
both the Trefoil and the Figure-Eight are prime.
Theorem 1 (Unique Prime Decomposition). Every nontrivial tame knot K is a finite
connected sum of prime knots K = K1 #K2 # . . . #Kn . The prime knots Ki are unique
up to order.
Proof. We will prove this in the lectures on Seifert Surfaces.
Remark 1.9. The Unique Prime Decomposition theorem shows that the semigroup
of knot types is isomorphic to the semigroup of natural numbers. However, it is
important to note that this isomorphism is not canonical.
1.2
Plane Diagrams and Reidemeister Moves
Recall that a regular plane diagram is a smoothly immersed curve in R2 with no
cusps, no self-tangencies and only transverse double point singularities.
To construct a knot diagram, we imagine projecting a knot in general position
down onto a plane. We want to understand how a general ambient isotopy of the
knot acts on a particular choice of knot diagram. Obviously, some isotopes in space
simple reduce to isotopes of the entire knot diagram. These include rotation of the
knot around any normal vector to the plane, uniform dilation of the knot and any
perturbations small enough not to change any of the crossing data.
To understand the action of a general isotopy we need to add three local transitions
which cannot be obtained through isotopies of the knot diagram. These transitions
are known as the three Reidemeister Moves. The Reidemeister Moves describe the
transition of a plane diagram through a cusp, a self-tangency and a triple point:
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Move Type
Singularity
Ω1
←−−−−
I:
Cusp
−−−−1
−→
Ω1
Ω2
←−−−−
II:
Self-Tangency
−−−−1
−→
Ω2
Ω3
←−−−−
III:
Triple-Point
−−−−1
−→
Ω3
Two diagrams represent the same knot if one can be transformed into the other by a
series diagram isotopies and Reidemeister Moves.
Some equivalent transitions. The Reidemeister Moves imply the following diagram transitions:
Ω−1
1
Ω2
←−−−−
←−−−−
−−−−1
−→
−−−−→
Ω2
Ω1
Similarly we can derive versions of Ω3 with other crossing orientations
Ω−1
3
Ω2
←−−−−
←−−−−
−−−−1
−→
−−−−→
Ω2
Ω3
Ω−1
3
Ω2
←−−−−
←−−−−
−−−−→
−−−−1
−→
Ω2
Ω3
Finally, a rotation of Reidemeister Move III gives
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Ω3
←−−−−
Diagram Rotation
−−−−→
−−−−1
−→
Ω3
Ω3
←−−−−
Diagram Rotation
−−−−→
−−−−1
−→
Ω3
Theorem 2 (Reidemeister). Two planar diagrams correspond to isotopic links if and
only if one can be obtained from the other by a finite sequence of Reidemeister Moves
and plane isotopies.
Before beginning the proof we give a quick definition:
Definition 1.10. A ∆-move on a piecewise linear knot K consists of (1) replacing an
edge E ⊂ K with two edges of F and G of the triangle EF G, provided Int(EF G) ∩
K = E or (2) replacing adjacent edges F G with an edge E of a triangle EF G,
provided Int(EF G) ∩ K = E. We illustrate the ∆ move below.
∆+
←−−−−
−−−−−→
∆
Proof. We will write out this proof-sketch in the piecewise linear category, with
combinatorial equivalence given by the ∆-moves. It suffices to show that any ∆-move
can be reduced to a sequence of Reidemeister moves and plane isotopies.
Without loss of generality we can assume that the link L is in general position
with respect to the plane projection π : R3 → R2 . By general position we mean
1. No edge is parallel to the projection π.
2. Each singular point x ∈ π(L) is a double point, and the preimage π −1 (x) does
not contain any vertices.
3. The triangles ∆ in any ∆-moves project to triangles in R2 .
There are two main cases we must check. First, assume that Int(∆) ∩ π(L) = ∅.
Then the ∆-move is a simply a plane isotopy.
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Plane Isotopy
−−−−→
If the interior of the triangle Int(∆) intersects K there are three basic possibility,
summarized in the following table
Ω01
−−−−→
Ω02
−−−−→
Ω03
−−−−→
By subdividing ∆ if necessary, we can always assume Int(∆) contains at most one
crossing or vertex of the form listed above. This concludes the proof of Theorem 2.
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Diagrammatic Invariants
The Reidemeister Moves allow us to promote certain invariants of planar diagrams
to invariants of knot types. If a planar diagram is invariant under the Reidemeister
Moves, it is a isotopy invariant of the knot. Furthermore, is often more straight forward to check invariance under Reidemeister Moves than under all ambient isotopies.
Define the following convention for the crossing of in a digram
Left handed crossing, w = −1
Right handed crossing, w = 1
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Definition 2.1. The writhe of a planar diagram D is the signed sum
n
X
w(D) =
wi .
crossings
i=1
Remark 2.2. On its own, writhe is not a knot invariant. Although it is invariant under Reidemeister II and III, it is not invariant under Reidemeister I since RI increases
the number of crossings in a planar diagram.
Definition 2.3. Let L = K1 t K2 be an oriented two component link. The linking
number lk(K1 , K2 ) is the sum over crossings in any diagrams D1 and D2 for K1 and
K2 respectively
n
X
1
wi .
lk(K1 , K2 ) =
2 crossings D1 ∩D2
i=1
Example 2.4. For the positive/negative Hopf link shown below the linking number is
±1 respectively.
+
-
lk = 1,
lk = −1.
The linking number is a topological invariant. Since we only sum over the crossing
between K1 and K2 , Reidemeister I preserves linking number, as do Reidemeister II
and III.
Example 2.5. The linking number does not detect the unlink. That is to say, there
are many nontrivial links with linking number 0. For example the Whitehead link
pictured below
Definition 2.6. A knot diagram is three-colorable if each segment the the diagram
from each underpass to the next can be colored red, green or blue in such a way that
1. All three colors are used.
2. At each crossing, either a single color or all three appear.
We need to check that three colorability is preserved under the Reidemeister
moves. Reidemeister I and II are obvious. For Reidemeister III, we can fix the
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Figure 4: For the Whitehead link, lk = 0.
color on the over string to be green and simply iterate through all the other permissible colorings. It turns out that up to an overall reordering of the colors there are
four cases we must check separately. The case in which all strands carry the same
color is obvious, so we will check the last three:
Ω3
←−−−−
−−−−1
−→
Ω3
Ω3
←−−−−
−−−−1
−→
Ω3
Ω3
←−−−−
−−−−1
−→
Ω3
Thus we see that if a diagram was three colored before a Reidemeister III move, it
will be three colored after. Therefore three colorability may be promoted from a
diagrammatic invariant to a knot invariant.
Example 2.7. While the unknot is obviously not three colorable the trefoil is. This
proves that that trefoil is a nontrivial knot.
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References
[1] R. Edwards, R. Kirby, ‘Deformations of Spaces of Imbeddings,” The Annals of
Mathematics, 2nd Ser., Vol. 93, No. 1 (Jan. 1971), 63-88.
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