Geometric Topology 3

Knots

Given a knot $K$ in $\mathbb{R}^3$ we can reflect it in any plane. The resulting knot $mK$ is well-defined up to ambient isotopy.

Proposition: If $D$ is a diagram for $K$ then $mK$ can be represented by a diagram $D'$ with the same shadow, but in which all the crossings of $D$ have been reversed, so $w(D')=-w(D)$.

To see this, choose Cartesian coordinates so that $K$ lies in the slice $0<z<1$ and $D$ is defined by projection to $z=0$. If we reflect in this plane, the projection of $mK$ to $z=-1$ will have the same shadow $\{(x,y): (x,y,z) \in K\}$ but with the crossings reversed.

Definition: If $K$ is ambient isotopic to $mK$ then $K$ is called achiral or amphichiral, otherwise chiral.

Example: The trefoil knot is chiral, so there really are two distinct knots $L3_1$ and $R3_1 = m(L3_1)$, though this is not easy to prove without some additional machinery.

The figure of eight knot $4_1$ has zero writhe, and is in fact achiral. Under certain conditions, zero writhe is a necessary condition for a diagram to represent an achiral knot, but the writhe is not invariant under ambient isotopy.

Reidemeister Moves

The Reidemeister moves are performed by unplugging a portion of a diagram $D$ of a knot $K$ and replacing it with one of three possible moves:

1. $R_1$ is a "twist": adding a loop to a strand.
2. $R_2$ is a "tuck": tucking one strand under another strand so that it pokes out the other side, adding 2 new crossings to the diagram.
3. $R_3$ is a "tuck under/over a crossing": tucking a strand under/over two crossed strands.

Throughout this process we are allowed to "stretch" strands in the plane (We can denote this by $R_0$). Crossings in a knot-diagram are based on the notion of an ordinary double point. The $R$-moves represent transitions that result from deformations of other types of singularities that are not allowed in knot diagrams:

1. Cusps
2. Tangents
3. Triple-points

Kurt Reidemeister's Theorem Suppose a link diagram $D'$ is obtained from $D$ by carrying out a finite sequence of $R$-moves. Then we say that $D,D'$ are isotopic diagrams and write $D \sim D'$. Assume that two knots or links $K,K'$ project to respective diagrams $D,D'$. It is clear that $D \sim D'$ implies that $K,K'$ are ambient isotopic. It is of enormous theoretical importance to know that the converse is true:

Theorem [Reidemeister, 1932]: If $K$ and $K'$ are ambient isotopic then $D \sim D'$. One can understand this by noticing what happens to a diagram as one carries out an ambient isotopy in space. Reidemeister's proof consisted of studying "triangle moves" on piecewise linear knots.

Effect of $R$-moves on the writhe It is obvious that applying $R_1$ to twist a single arc introduces a crossing and changes the writhe of the whole knot diagram by $\pm 1$. On the other hand: Lemma: $R_2$ and $R_3$ do not change the writhe of the diagram. For $R_2$ this is because the two crossings in the "new" diagram will always have opposite writhe-signs. In the case of $R_3$, the central crossing is unchanged. On the left there are NW and SW crossings and on the right NE and SE crossings. Using these compass points to represent their signs we see NW = SE and SW = NE. So the writhe of the new diagram is the same.

Later on we need a stronger equivalence relation of regular isotopy between diagrams $D, D'$. This means that $D'$ can be obtained from $D$ by a sequence of moves of type $R_0,R_2,R_3$, and is indicated by $D \approx D'$.

Corollary: If $D \approx D'$ then $w(D) = w(D')$.

Corollary: The absolute value of the linking number $|\ell|$ of a link $L$ is invariant by isotopy, and therefore an ambient isotopy invariant of $L$. If $\ell \neq 0$ then the two knots cannot be separated in space. The converse to this is false.

Numerical Invariants Definitions: Let $K$ be a knot or a link with more than one component. The crossing number $cr(K)$ is the least number of crossings needed in any diagram $D$ of $K$. It is a basis of traditional knot tables. If $cr(K) <3$ then $cr(K) =0$ and $K$ is trivial, i.e. ambient isotopic to an unknot like the circle.

A minimal diagram is one with exactly $cr(K)$ crossings.

The unknotting number $u(K)$ is the least number of crossing-reversals in any diagram $D$ of $K$ needed to convert $D$ to the projection of an unknot. If $K$ is a knot with $cr(K) \leq 7$ then $u(K) \leq 2$ except that $u(7_1) = 3$. The fact that $u(8_10) = 2$ was only verified in 2005 and $u$ of some knots with $cr(K) = 10$ is still unknown.

The bridge number $br(K)$ is the least number of bridges in any diagram of $K$. If $K$ is a knot with $0 < cr(K) \leq 7$ then $br(K) = 2$ but $br(8_10)=3$.

Warning: Many minimal diagrams have more than $br(K)$ bridges (for example if they are alternating). Moreover $u(K)$ is not necessarily realized by a minimal diagram of $K$ which makes it hard to determine the unknotting number.

Some Known Results

Lemma: If $D$ is a diagram of a knot with $c$ crossings then we need to reverse at most $\frac{c}{2}$ to obtain the unknot, so $u(K) \leq \lfloor{\frac{c}{2}}\rfloor$.

Lemma: If $br(K)=1$ then $K$ is trivial. (This can be shown using the DT code of a knot, shown later.)

Theorem [Scharlemann, 1985]: If $u(K) = 1$ then $K$ is prime. Equivalently the composite $K_1 \# K_2$ of two non-trivial knots must have $u \geq 2$.

Theorem [Schubert, 1954]: $br(K_1 \# K_2) = br(K_1) + br(K_2) - 1.$

Maxima and Minima Treat a portion of the image of a knot projection $f: [0,2\pi) \rightarrow S^1 \rightarrow \mathbb{R}^3 \rightarrow \mathbb{R}^2$ with $f(t) = (x(t),y(t))$ as a graph. It has a turning point where $\frac{dy}{dx} = 0$ (or $\frac{dy}{dt} = 0$). We can deduce a theorem:

Theorem: $br(K) = n$ if and only if $K$ has a diagram with $n$ (and no fewer) local maxima.

Colouring

Let $L$ be a knot or link, and let $n \geq 2$ be a positive integer.

Definition: $L$ is $n$-colourable if it possesses a diagram whose arcs can be labelled with at least two distinct integers from $\{0,1,...,n-1\}$ such that at each crossing:

No knot is 2-colourable because both underpassing arcs would necessarily have equal labels at each crossing. But any link with at least two components is 2-colourable.

It is easy to see that a link is 3-colourable if at each crossing either all three colours (i.e. 0,1 or 2) are distinct or they are all the same. By checking the effect of Reidemeister moves on coloured crossings one deduces:

Theorem: If $D$ is $n$-colourable and $D \sim D'$ then $D'$ is $n$-colourable.

We can therefore say that a knot or link in space is $n$-colourable if any one of its diagrams is. In other words, this concept is an invariant of ambient isotopy, even though colouring a knot in space makes no sense, at least without our perception of it!

Invariance by $R$-moves

The colouring equation is set up to be invariant by $R$-moves which we can consider one at a time. Bear in mind that in applying an $R$-move from left to right or vice-versa, the colours of the strands entering the boxes are fixed because they must agree with those on the rest of the link diagram outside the boxes.

The coefficient "2" deals with $R_1$: A twist will automatically satisfy the colouring equation as all the colours (i.e. integers modulo $n$) are equal.

$R_2$ (from left to right) adds an arc inside the box whose colour $x$ can be chosen freely to satisfy $a+x=b$.

$R_3$ is the only tricky case. We consider the strands inside the box before and after the $R_3$ move. The colouring equations are initially:

and after $R_3$:

If the first set of equations are satisfied then we must prove that we can choose $y$ to satisfy the second set of equations.

In this case we define $y = 2g - a$ and note that:

as required. $\blacksquare$

A $5$-colouring of $4_1$

Lemma: If a knot or link is $n$-colourable it is also $m$-colourable if $n \vert m$.

If $m = nk$ then a colouring equation $n \vert (a+b-2d)$ implies that $m \vert (ka+kb-2kd)$, which is the equation in which all the colours have been multiplied by $k$. Another observation is that for fixed $n$ we can add any constant integer to all labels and still satisfy the colouring equations. This means that we are free to choose any arc and colour it $0$.

An example of this is that we can colour the figure-eight knot $4_1$ with $n=5$, and since there are $4$ arcs we cannot use all $5$ colours. The trefoil knot is $3$-colourable but it is easy to check that $3_1$ is not $5$-colourable, and similarly $4_1$ cannot be $3$-coloured.

Corollary: $3_1$ is not ambient isotopic to $4_1$.

Exercises and Applications

• The unknot is not $n$-colourable for any $n$.
• $K$ is $n$-colourable $\iff$ its mirror image $mK$ is.
• The trefoil knot $3_1$ is $n$-colourable $\iff \ 3 \vert n$.
• The figure-eight knot $4_1$ is $n$-colourable $\iff \ 5 \vert n$.
• The knot $7_7$ is $n$-colourable $\iff \ 3 \vert n$ or $7 \vert n$.
• The Borromean rings are $n$-colourable $\iff \ 2 \vert n$.
• The composite 'Granny knot' $L3_1 \# L3_1$ is $3$-colourable.

This gives us a good way to check ambient isotopy classes of many such knots and links!