We will now assume th a t the knot of interest is given in our m atrix notation w ith pki E €. A few basic properties of the notation need to be enum erated before the notation becomes useful in dealing w ith knots. These properties include th e behavior under knot addition, calculating the num ber of com ponents in a knot and calculating th e num ber of regions in the projection.
Two knots may be combined into a single knot by addition. In adding, each knot is cut a t a single point and the ends are spliced together. We find th a t given two knots A in P { i ,j ) and B in
K n o ta tio n an d B ra id in g a K n o t 100
P { i ',j ') such th a t j ' > j , the sum C = is given by
C =
oo 0 oo
a i l U i2 Û13 0 - l i 6 i i b \ 2 b i 3 bii >
U21 Û22 (^23 • 0>2i 621 622 623 &2i'
Uj i a j 2 tt j3 • CLji 6.1 b j 2 b j 3 bji> 0 0 0 • • 0 1 b j + i2 b j + 1 3 • • b j j ^ i i 0 0 0 • • 0 bj>i b j ' 2 b j>3 bj>i> 0 oo 0 • (5.13)
h
(a)Figure 5.7; A ddition of two knots in our notation.
This addition formula is the straightforw ard consequence of the geometrical procedure of splic ing two polyhedra together and then m aking them both fit into another larger one. We illustrate this procedure for two P ( 2 ,2) polyhedra in figure 5.7. As can be seen in figure 5.7, this m ethod chooses a particular cutting point for each polyhedron. It can be shown th a t th e operation of knot addition ^ is independent of the cutting point. This is tru e only w ithin a com ponent of a knot. If a knot has more th a n one com ponent, th e m ethod of adding described by equation 5.13 is not general b u t makes a specific choice. Because such additions rely on the particular stru ctu re of the specific knots to be added, such a form ulation can not be m ade in general.
This new notation can be readily used in calculating some invariants of th e knot. For example, to calculate a polynomial invariant for which we have a state model, we simply replace each ±1 tangle by the 0 or oo tangles in all possible ways to yield all possible states of the knot. If a knot has n double points, this means 2” states. We associate an algebraic factor w ith the way in which this replacem ent is m ade and then m ultiply it by an algebraic factor depending on th e num ber of unknots left (since there are no double points left, this is equivalent to the num ber of regions in the resultant projection). All these contributions are added and yield a polynom ial invariant of the knot. The key is to be able to calculate th e num ber of regions and com ponents of a knot given
5.2 K n o t N o ta tio n 101
its m atrix.
The num ber of regions into which the knot projection partitions the plane is im p o rtan t in a few applications such as the calculation of polynomial invariants, as stated above, and also in the braiding algorithm which follows. Each vertex has exactly one region lying to its right in the polyhedron and thus we may label all these regions by th e row and column indices of th e associated vertex. Only two regions are not indexed by this m ethod, these are the two regions w ith j vertices directly on th e top and bottom of the m atrix construction. These will be labeled by the pairs (0,1) and (jf + 1,1). Thus the regions m ay also be represented by a m atrix. If the entries rki are m ade to take integer values we may count the num ber of regions by the following algorithm .
A lg o r ith m 5 .2 .3 Input: A m atrix describing a knot in our notation. Output: A m atrix describing the regions o f the knot. Each element o f the m atrix receives a label from 1 to R , the number o f regions. This gives complete inform ation about which regions o f the polyhedron are connected and how m any there are.
1. Begin a t the top left of vertex (1,1) and follow th e boundary downwards, as for counting regions, the orientation of th e knot does not m atter. M ark the region (0,1) w ith a 1, the current m arker, in the region m atrix.
2. In following the boundary, one will come to vertex (1,1); we assess its value and continue. If we stay in the same region of the polyhedron we continue, if we enter a new region of the polyhedron, then this new region of the polyhedron belongs to the same region of the knot as th e previous one and thus we m ark it w ith the current m arker in the region m atrix. The whole issue a t hand is th a t the regions of the polyhedron are known while we wish to gain knowledge of the regions of the knot.
3. We continue to follow the boundary until we reach th e point of origin.
4. We search the m atrix for an unm arked region. If there exist unm arked regions, we increm ent our current m arker and choose one of th e regions as our new startin g region and choose a point upon its boundary as our new startin g point. Then, we repeat the algorithm from step 1, m arking the region w ith the current marker.
5. Once no unm arked region of the polyhedron exists, the algorithm is finished. T he largest m arker used in the m atrix which we have obtained is clearly the num ber of regions of the knot. Furtherm ore, since all connected regions are labeled w ith the same m arker, we have a complete knowledge of which regions of the polyhedron belong to th e same region of the knot.
The algorithm considers each vertex exactly twice and moves and m arks accordingly. Therefore the complexity is 0 (n ) . An algorithm to find th e num ber of com ponents in a knot is sim ilar bu t differs in a few details.
K n o ta tio n and B ra id in g a K n o t 102
A lg o r ith m 5 .2 .4 Input: A m atrix describing a knot in our notation. Output: The number o f components in this knot.
1. Each vertex has four points in which the two polygonal curves intersect B ^. These are shown in figure 5.1. S tart at point N W of the vertex (1,1) and follow the orientation of the knot. 2. We follow th e orientation and not th e boundary, as in algorithm 5.2.3, m arking each point
as we pass it.
3. W hen we reach the point of origin again, we increment th e com ponent counter and look for an unm arked point.
4. If there is an unm arked point, we begin w ith step 1, if there is not, we are finished.
T his m ethod calculates the num ber of com ponents considering each point on each vertex once, therefore the complexity is also 0 ( n ) . Note th a t a m atrix of only 0 tangles contains i + 1 unknots and a m atrix composed of only oo tangles contains j unknots.
Clearly smaller polyhedra P { i,j) m ay be embedded in larger ones by filling in th e rest with 0 and oo tangles. Conversely, if the configuration of the tangles is right, we m ay delete rows and columns accordingly. For example, we may create an extra row a t the bottom or top of the m atrix containing
(0 0 . . . 0 oo) (5.14)
and we m ay add an ex tra column a t the left or right of the m atrix containing only 0 tangles. Likewise, such columns or rows m ay be removed w ithout changing the knot type. Thus if a given knot can be expressed in the polyhedron P { i ,j ) it can also be expressed in any polyhedron P { i ',j') for which i' > i and j ' > j . An internal row of 0 tangles splits th e polyhedron into two p arts each described by the m atrix above and below th e row of zeros. T hus if two knots should be described in a single diagram w ithout touching, this is a way in which this may be done.
5.3
B raiding a K not
Having constructed a new notation for knots, we wish to solve the problem of how to extract a closed b raid from the m atrix which is isotopic to the knot described by the m atrix. A few algorithm s have been constructed in the past, which convert a knot into a closed braid b u t th ey are difficult to im plement because they depend upon topological deform ation of the knot projection [97] [22]. The best known algorithms have been im plemented [151] [162] and have com plexity O(n^). We shall present an algorithm which achieves th e conversion w ith complexity 0 ( n ) , increases the num ber of crossings only in a few cases (and then only by a few crossings) and uses a linearly bounded num ber of strings. There exists no algorithm to calculate the num ber of strings which are a t least necessary to describe a specific knot — th e braid index of the knot. Because of this, it is not