**Knot
theory**

**
III. Fundamental concepts of knot theory (continued)**

Recall our primary goal in knot theory:

*How can we distinguish two knots?*

We have seen several examples in the previous two lessons.
Before looking at our goal directly, we need to have some necessary mathematical
tools in our hands. This is exactly the content of the lesson here. It
turns out that it is useful, first, to *project* the knot onto a *
two-dimensional plane. *This is the content of section 1.

**Section1. Regular
diagram**

As usual, we always think of a knot sitting in space. To be
more precise, *a knot is in a 3-dimensional space* that we are living in.
So, we are able to assign coordinates P(x,y,z) to every point on the knot, as an
indication of the position of the point in our 3-dimensional space. Let us
denote by p, the map that projects the point P(x,y,z) onto the point P'(x,y,0)
in the *xy*-plane, see fig. 24.

**fig. 24.**

If **K** is a knot (or link), we shall say that p(**K**)=**K' **is
the projection of **K**. Further, if **K** has an orientation assigned,
then in a natural way **K'** inherits its orientation from the orientation of
**K**. However, **K'** is not a simple closed curve lying on the plane,
since **K'** possesses everal points of intersections. But by performing
several elementary knot moves on **K**, intuitively this is akin to slightly
shifting **K** in space, we can impose the following conditions on these
projections :

**(1) ** **K'** has at most a finite
number of points of intersection.

**(2) ** If **Q** is a point of
intersection of **K'**, then the inverse image of **Q** back in **K**
has exactly two point. That is, **Q** is a double point of **K'**, see
fig. 25(a), it cannot be a multiple point of the kind shown in fig. 25(b).

**(3) ** A vertex of **K** (the knot
considered now as polygon) is never mapped onto a double point of **K'**. In
the two examples in fig. 25(c), 25(d), a polygonal line projected from K comes
into contact with a vertex point(s) of **K'**, so both of these cases are not
permissible.

**fig. 25**

A projection that satisfies the above conditions is said to be a **
regular projection**.

Throughout these lessons, we will work exclusively with *regular
projections*, and to simplify matters, we shall refer to them just as
projections, we will draw a distinction only if some confusion might otherwise
arise. However, even if we restrict ourselves to (regular) projections, there
are still a considerable number of them. Secondly, and at this juncture of quite
some importance, is the ambiguity of the double points. At a double points of
projection, it is not clear whether the knot passes over or under itself. To
remove this ambiguity, we slightly change the projection close to the double
points, drawing the projection so that it *appears* to have been cut.
Hopefully, this will give a *trompe l'oeil * effect of a continuous
knot passing over and under itself. Such an altered projection is called a **
regular diagram. **See fig. 26(a), fig26(b).

A regular diagram gives us a sense of how the knot may in fact lie in 3-dimensions, that is, it allows us to depict the knot as a spatial diagram on the plane. Further, we can use the regular diagram to recover information lost in the projection, for example, in fig. 26(c), it is the projection of the two non-equivalent knots in fig. 26(a) and fig. 26(b).

**fig. 26**

Therefore, we need to be a bit more precise with regard to the exact nature
of a regular diagram and its crossing (double) points, since from the above
description a regular diagram has no double points. The crossing points of a
regular diagram are exactly the double points of its projection, p(**K**),
with an over- and under- crossing segment assigned to them. Henceforth, we shall
think of knots in terms of this diagrammatic interpretation, since, as we shall
see shortly, this approach gives us one of the easiest ways of obtaining insight
and hence results into the nature of knots.

For a particular knot (or link), **K**, the number of regular diagrams is
innumerable. To be more exact, there is *only one* regular diagram of a
knot, **K**, in our 3-dimensional space. However, from our discussion in the
previous lessons, the knot **K** and a knot **K'** obtained from **K**
by applying the elementary knot moves are thought of as being the same knot. So,
we can think of the regular diagram of **K'** as being a regular diagram for
**K**. Hence, it follows that for **K,** the number of regular diagrams is
innumerable.

It is possible that a regular diagram may have crossing points of the type shown in fig. 27(a), fig.27(b).

**fig. 27**

More generally, suppose two regular diagrams of two knots (or links) are
connected by a single twisted band. We can, in fact, remove this 'central'
crossing point by applying a twist, either to the left or right, to the knot. A
regular diagram that does not possess any crossing points of this type is called
a * reduced regular diagram*.

**Exercise 3.1** Show that a
regular diagram for **K1*K2** can be obtained by placing the regular diagrams
of the oriented knots **K1** and **K2** side by side, and connecting them
by means of two parallel segments, see fig. 28 for an example.

**fig. 28**

**Section2. What is a
knot invariant?**

As a way of determining whether two knots are equivalent,
the concept of the *knot invariant *plays a very important role. The
types of knot invariants are not just limited to, say, numerical quantities.
These knot invariants can also depend on commonly used mathematical tools, like
polynomials.

Suppose that to each knot, **K, **we can assign a
specific quantity q(**K**). if for two equivalent knots the assigned
quantities are always equal, then we call such a quantity, q(**K**), a knot
invariant. This concept of assigning some mathematical quantity to an object
under investigation is not limited just to knot theory, it can be found in many
branches of mathematics.

We knot that if a knot **K** and another knot **K'**
are equivalent, then it is possible to change **K** into **K'** by
applying elementary knot moves to **K** a finite number of times. Therefore,
for a quantity q(**K**) to be a knot invariant, q(**K**) should not change
as we apply the finite number of elementary knot moves to the knot **K**. It
follows from this, for example, that the number of edges of a knot is not a knot
invariant. The reason is that elementary knot moves (recall: the definition of
elementary knot moves is in lesson 2, section1) either increase or decrease the
number of edges. Similarly, if we consider the size of a knot, it is not an
invariant as well since the size of a knot changes when we apply the elementary
knot moves.

A ** knot invariant**, in general, is
unidirectional, that is:

*if two knots are equivalent, then their
invariants are equal.*

For many cases the reverse of ' if...then... ' does not hold. In
contraposition, if two knot invariants are different then the knots themselves
cannot be equivalent, and so a knot invariant gives us an extremely effective
way to show whether two knots are *non*-equivalent. The history of knot
theory may be said to be an account of how the various knot invariants were
discovered and their subsequent application to various problems. To find such
knot invariants is by definition a 'global' problem. On the other hand, to
actually calculate many of these knot invariants, which we shall discuss a bit
later, is quite difficult. Further, to find a method to calculate these
invariants is also a 'global' problem.

Notice that it is still an open problem in knot theory to find a *complete*
knot invariant. There is no known complete knot invariants at this stage. By
'completeness', what we mean here is that *the reverse of the above statement*
'*if two knots are equivalent, then their invariants are
equal. ' holds. *More concretely speaking, for any two given knots, if the
knot invariants of these two knots are equal, then these two knots are equal. If
this condition is satisfied, we call such a knot invariant *complete*. Put
it in the other way, a complete knot invariant means a complete classification
of the set of all knots in our space, up to equivalence.

**Section3.
Reidemeister moves**

A knot (or link) invariant, by its very definition, as
discussed in the previous section, does not change its value if we apply one of
the elementary knot moves. It is often useful to project the knot onto the
plane, and then study the knot via its regular diagram. If we wish to pursue
this line of thought, we must now ask ourselves what happens to, what is the
effect on, the regular diagram if we perform a single elementary knot move on
it? This question was studied by **K. Reidemeister** in the 1920s. In the
course of time, many knot invariants were defined on Reidemeister's seminal
work. In this section, we are going to study the moves defined by him, called
the *Reidemeister moves*.

A solitary elementary knot move, as might be expected,
gives rise to various changes in the regular diagram. However, it is possible to
restrict ourselves to just the four *moves* (strictly speaking, changes)
shown in fig. 29 and their inverse moves, also in fig. 29:

**fig. 29**

That these moves may, in fact, be made is reasonably straightforward to
understand. For example, **M2** may be thought of as the move that
corresponds to an elementary knot move on a regular diagram, which replaces, **
AB** by (**AC**)U(**CB**), as shown in figure 30.

**fig. 30**

**Exercise 3.2** Verify that, in
fact, **M3, M4** are possible, that is, they are a consequence of some finite
sequence of elementary knot moves.

**Definition 3.1** If we can
change a regular diagram, **D**, to another **D'** by performing, a finite
number of times, the operations **M2, M3, M4,** and/or their inverses, then
**D** and **D'** are said to be equivalent. We shall denote the
equivalence between **D** and **D' **by **D=D'**.

These three moves **M2, M3, M4** and/or their inverses are called the **
Reidemeister moves. **Notice that after making any elementary knot
moves to the original knot, the regular diagram remains essentially unchanged by
making suitable (sequences of) Reidemeister moves corresponding to the
elementary knot moves we have made. That is, we may state the following theorem:

**Theorem 3.1 **Suppose **D**
and **D'** are regular diagrams of two knots (or links) **K** and **K'**,
respectively. Then,

*K=K' if and only if D=D'*

We may conclude, from the above theorem, that the problem of equivalence of
knots, in essence is just a problem of the equivalence of regular diagrams.
Therefore, a knot (or link) invariant may be thought of as a quantity that
remains unchanged when we apply any one of the above Reidemeister moves to a
regular diagram. In the following, we shall often need to perform locally a
finite number of times a composition of Reidemeister moves, for simplicity we
shall call such a composition an *R-moves.*

**fig. 31**

**Theorem 3.2** The moves shown in
fig. 31 are R-moves.

__ Proof__: For the upper picture :

For the lower picture :

**Exercise 3.3** Using similar
methods as in theorem 4.2, show that the following moves are R-moves.

In the next lesson, we shall explain several knot invariants that have played substantial role in research into knots.