Research Article | Open Access
An Algorithm for Generating a Family of Alternating Knots
An algorithm for generating a family of alternating knots (which are described by means of a chain code) is presented. The family of alternating knots is represented on the cubic lattice, that is, each alternating knot is composed of constant orthogonal straight-line segments and is described by means of a chain code. This chain code is represented by a numerical string of finite length over a finite alphabet, allowing the usage of formal-language techniques for alternating-knot representation. When an alternating knot is described by a chain, it is possible to obtain its mirroring image in an easy way. Also, we have a compression efficiency for representing alternating knots, because chain codes preserve information and allow a considerable data reduction.
In 1993, Livingston  stated that “knot theory remains a lively topic today. Many of the basic questions, some dating to Tailt’s first paper in the subject, remain open. At the other extreme, the results of recent years promise to provide many new insights’’. Knot theory is a branch of algebraic topology. The three main techniques of knot theory are: geometric techniques, combinatorial methods, and algebraic tools. A knot is a simple closed polygonal curve in three-dimensional Euclidean space . Knots are cataloged in order of increasing complexity. One measure of complexity that is often used is the crossing number, that is, the number of double points in the simplest planar projection of the knot. There is only one knot with crossing number of three (ignoring mirror reflections) the trefoil. The figure-eight knot is the only knot with a crossing number of four. There are two knots with a crossing number of five, three knots with a crossing number of six, and seven knots with a crossing number of seven. From there on the numbers increase dramatically. There are 12,965 knots with a crossing number of 13 or fewer crossings in a minimal projection. A complex knot is a knot with a huge crossing numbers. Imagine, how many knots are there in an alternating knot which is represented by a complex knot with a crossing number of 1200? This paper deals with complex knots of this order of crossing numbers. is a more computer-friendly place for generating knots. In , every point in space has coordinates drawn from the set of real numbers. Thus, the generation of complex knots in produces a very complex computation. On the other hand, in all points have integer coordinates. Thus, we represent complex knots which are embedded on the cubic lattice. Complex knots are composed of constant orthogonal straight-line segments and are represented by means of a chain code  which produces a numerical string of finite length over a finite alphabet, allowing the usage of grammatical techniques for alternating-knot representation. The above mentioned generates a very simple computation.
Several authors have analyzed knots on the cubic lattice. In 1993, Diao presented the minimal knotted polygons on the cubic lattice  and, in 1994, the number of smallest knots on the cubic lattice . van Rensburg and Promislow define the minimal knots in the cubic lattice . Hayes  presents the “Square Knots" on the cubic lattice, which are simple closed polygonal curves embedded in . Other interesting theories: Kauffman  introduces the “Virtual Knot Theory”, which includes general Gauss codes. Nakamura and Rosenfeld  define the “Digital Knots” which represent an initial effort at the study of the concepts of knottedness and linkedness for digital objects. Complex knots are embedded on the cubic lattice. Thus, these knots are composed of constant orthogonal straight-line segments and are represented by chain coding . In order to have a self-contained paper, Appendix A describes the used chain code. The left-hand side of Figure 1 shows an example of a complex knot in a continuous representation. The right-hand side of Figure 1 illustrates the same knot embedded on the cubic lattice. In the content of this work, knots are represented as ropes. This improves the understanding of the figures.
This paper is organized as follows: Section 2 describes the proposed family of rectangular alternating links. In Section 3, we present some results. Section 4 gives some conclusions. Finally, we present Appendix A which describes the used chain code and Appendix B which presents the program of the proposed algorithm.
2. A Family of Rectangular Alternating Links
The alternating link diagrams of the type illustrated in Figure 2 are used here to construct links like that shown in Figure 1. Each diagram is a set of rectilinear polygons and can be characterized in terms of two positive integers and . In what follows we present such a characterization, from which we obtain some properties of the family of polygons, useful to construct an algorithm for generating a chain code representation of the corresponding link.
Let and be integers such that . Let be a square on the 2-dimensional integer lattice, with one corner at the origin and another at the point . Let be the rectangle with sides parallel to the diagonals of , symmetric with respect to the central point of , and with sides through the points and .
The grid nodes lying on the edges of serve as vertices of a set of rectilinear polygons, each of whose sides is the vertical or horizontal line segment that joins a corresponding pair of nodes. In Figure 2 those vertices appear counterclockwise enumerated from 0 to , beginning with the node at coordinates . Accordingly, let be the set of vertices of .
Let be the function such that is the vertex of vertically opposed to vertex . As a consequence of the way the enumeration of the vertices was done, we have that . Then, from this equation and defining , we arrive to
Now, let be the function such that is the vertex horizontally opposed to vertex . There are two cases: , if , and , otherwise. Taking into account that , in both cases we obtain
Remark 2.1. The functions and are bijections, they are self-inverse, and each one maps odd numbers to even numbers and vice versa.
By the above remark, also the function composition is bijective, but preserves the parity of its argument.
The functions , , and , being bijections, are elements of the group of symmetries of . In particular, we call attention to —the cyclic subgroup generated by on —and to its (induced) action over .
Along any polygon , the horizontal sides alternate with the vertical ones. Thus, the consecutive vertices of can be obtained by iterating alternate applications of and . For example, with and (which is the particular case presented in Figure 2), for the polygon through vertex 0, the corresponding iteration produces the (periodic) sequence which has as a shortest period.
From the first remark, it follows that in sequences like those just presented the elements alternate in parity. So, if is the set of even (odd) vertices of , then and both are equal to the set of odd (even) vertices of .
Now, to count the number of polygons in , it is useful to interpret this set as a graph, in the obvious way, and then proceed to count the number of cycles in this graph.
Definition 2.2. Let , , , , and , be as above. is the graph with vertex-set and edge-set .
We can see Figure 2 also as a drawing of . Since each vertex is an end point of exactly two edges, is a regular graph of degree 2. Therefore, every component of is a cycle . Now we will see how many components have and how long they are.
Lemma 2.3. The graph defined above is composed of cycles of length , and all the vertices belong to distinct cycles.
Proof. From (2.1) and (2.2) we have that , that is Now, to analyze the action of over (see Remark 2.1(3)), it is useful the embedding of into the additive group . From (2.5) it follows that , for . Hence, the orbit of 0 under the action of is the (finite) set which is the (cyclic) subgroup of generated by . Then, this subgroup has as its least generator, it is of order [10, Section 6.5-6], and the set of its cosets coincide with the set of orbits of on . But the set of vertices of a cycle of is equal to the union of two of these orbits (see Remark 2.1). Hence, there are cycles in , each one of length . Now, in terms of cosets, the set of vertices of a cycle of is equal to a union of the form , with . By (2.1), and because and are in the same coset, we obtain that Therefore, the cycles have, respectively, the following sets of vertices:
Remark 2.4. There are and vertices of in the upper left side and in the lower left side of , respectively. Let
Since , the vertices lie on the lower left side of .
Let be an edge of a polygon . If the end points of lie on parallel sides of , then is of even length , otherwise has odd length. Furthermore, has odd length if and only if is at distance from a side of parallel to , and in this case is of length . Thus, the lengths of the vertical and horizontal edges through the vertex are, respectively, given by the functions such that For example, the alternate application of and to the vertices obtained for the polygon in Remark 2.1 gives the following sequence of edge lengths:
If has even length, then its adjacent edges lie on distinct semiplanes with respect to the line containing , otherwise they are to the same side of such a line.
The link is constructed in such a way that is the projection of on the plane , under the mapping . Let be the component knot whose projection is the polygon trough vertex , with as defined in the preceding lemma. We orient by departing from to the right, that is, following the cycle . The knot is an axis-aligned polygon whose vertices lie on the planes and , with as the starting vertex. In turn, this polygon is a concatenation of “sections,’’ which are planar polygonal curves corresponding to the edges of , similar to that shown in Figure 3(b).
The section related to a directed edge is a concatenation of “pieces’’ like the three shown in Figure 3(c); begins with a horizontal piece, which belongs either to the plane or to the plane and is followed by alternate occurrences of the other two pieces. If we split the first piece at its midpoint, can be seen as a directed polygonal composed of line segments of constant length (), as delineated in Figure 4(a) which also shows the code obtained by means of (A.1) for each vertex apart from the first and fourth ones, whose codes and remain unknown until we have the placement of along the knot (but since labels a right angle). Thus, in formal language terminology, a section is a chain over of the form either or , where and , with the interpretation given in A. Figures 4(b)–4(f) show some configurations that correspond to five of the 20 possible values for the pair . Next we will prove that only the first two illustrated cases occur in .
Lemma 2.5. Let be an edge of and let be the section of corresponding to . Then, has or 4, and, if , .
Proof. The proof goes by induction on the position of along . If is the first section of , then its initial point is vertex which, by Remark 2.4(2.1), lies on the lower left side of . Thus, the right angle previous to along must belong to one of the two configurations shown in Figures 4(b)-4(c), depending on the length of the edge of previous to ; if , then , and if , then . Furthermore, if (thus there is a point for in ), in both cases.
Now, suppose by induction that a section satisfies the lemma. We want to show that the section next to also satisfies the lemma. Let be the section previous to . The induction hypothesis implies that the configuration for and must be one of those in Figures 4(b)-4(c), up to a rotation. Suppose is odd; then, as a consequence of Remark 2.4, and are to the same side of the plane containing and, since jumps times between the planes and , the end points of lie both in the same of these planes; therefore, the configuration determined by and must be also, up to a rotation, one of those shown in Figures 4(b)-4(c). On the other hand, if is even, from the same remark it follows that and are in opposite sides of the plane of and, since now is odd, the end points of lie one in the plane and the other in the plane , facts that lead to the same conclusion obtained in the first case.
Corollary 2.6. Let and be two successive edges of , of lengths and , respectively. Then the chain code of the section in corresponding to is given by the following function:
For example, section(1,8) evaluates to “40012042042042042042042”. The next function calculates the chain code for : where gives the list of pairs needed to compute the successive sections of . The definitions of these functions, as well as those for , , , , , , and , are local to the main function, which computes the list of strings of code for the cycles of the link : The complete definition of this program is presented in Appendix B.
In order to probe our proposed method, we present some examples of alternating-knot generation. Figure 5 illustrates an example of an alternating knot of (, ).
The chain of the alternating knot shown in Figure 5 is as follows:
Figure 6 illustrates another example of an alternating knot of using the proposed method.
We have presented a modest attempt for generating alternating knots which are represented by means of chain coding. The chain-code representation of alternating knots preserves information and allows a considerable data reduction. Also, the mirror images of alternating knots are obtained in an easy way.
A. The Chain Code
In order to have a self-contained paper, we summarize the main concepts and definitions of the used chain code. In the content of this paper, we use this chain code to represent complex knots.
Definition A.1. A discrete knot is the digitalized representation of a knot and is composed of constant orthogonal straight-line segments, whose direction changes are described as a chain.
Definition A.2. A chain is a finite sequence of elements and is represented by , where indicates the length of the chain.
The chain elements for a discrete knot are obtained by calculating the relative orthogonal direction changes of the contiguous constant straight-line segments along the knot. There are only five possible orthogonal direction changes [13, 14] for representing any discrete knot. Thus, each discrete knot is represented by a chain of elements. Figure 7(b) illustrates an example of a discrete complex knot.
An element of a chain, taken from the set , labels a vertex of the discrete knot and indicates the orthogonal direction change of the polygonal path in such a vertex. Figure 7(a) summarizes the rules for labeling the vertices: to a straight-angle vertex, a 0 is attached; to a right-angle vertex corresponds one of the other labels, depending on the position of such an angle with respect to the preceding right angle in the polygonal path.
In order to improve the understanding of the chain elements, we have colored the straight-line segments which are defined by their corresponding chain elements. Thus, the straight-line segment defined by the chain element 0 in green, 1 in cyan, 2 in yellow, 3 in magenta, and 4 in red, respectively. This is valid for the web version of the paper; however, the gray-level version of the paper also allows us to understand the above-mentioned notation. Formally, if the consecutive sides of the reference angle have respective directions and (see Figure 7(a)), and the side from the vertex to be labeled has direction (here, by direction we understand a unit vector), then the label or chain element is given by the following function, where × denotes the vector product in :
Thus, the procedure to find the chain of a discrete knot is as follows.(i)Select an arbitrary vertex of the discrete knot as the origin. Also, select a direction for traveling around the discrete knot. Figure 7(b) illustrates the selected origin which is represented by a sphere. Also, the selected direction is to the right.(ii)Compute the chain elements of the discrete knot. Figure 7(b) shows the first element of the chain which corresponds to the element 0. The second element corresponds to the chain element 0, too. The third element corresponds to the chain element 1. Note that when we are traveling around a discrete knot, in order to obtain its chain elements and find zero elements, we need to know what nonzero element (labeling a right angle) was the last one in order to define the next element. Finally, we obtain the following chain:
The main characteristics of this chain code are as follows.(i)It is invariant under translation and rotation. This is due to the fact that only relative orthogonal direction changes are used.(ii) In this code, there are only five possible orthogonal direction changes for representing any discrete knot, this produces a numerical string of finite length over a finite alphabet, allowing the usage of grammatical techniques for complex-knot generation.(iii)Using this code, it is possible to obtain the mirror image of a discrete knot with ease. The chain of the mirror image of a discrete knot is another chain (termed the reflected chain) whose elements 1 are replaced by elements 3 and vice versa. This replacement does not depend on the orthogonal reflecting plane used, it is valid for the three possible orthogonal mirroring planes. We do not prove this, only we illustrate it .
B. The Program
Here is the function that computes the links presented in Section 2. Let ; if , returns the list of chains of code for the knots of , otherwise the returned list is empty.
|—See Remark 2.1|
|—See Remark 2.4|
The functions and have definitions similar to those of the standard functions and map provided by Haskell. The invocation returns the infinite list of repeated and alternate applications of and to : and returns the list of alternate applications of and to the successive elements of the list :
The authors wish to express their gratitude to Guillermo Rojas for his help in generating patterns of alternating knots.
- C. Livingston, Knot Theory, vol. 24, Mathematical Association of America, Washington, DC, USA, 1993.
- E. Bribiesca, “A method for computing families of discrete knots using knot numbers,” Journal of Knot Theory and Its Ramifications, vol. 14, no. 4, pp. 405–424, 2005.
- Y. Diao, “Minimal knotted polygons on the cubic lattice,” Journal of Knot Theory and Its Ramifications, vol. 2, no. 4, pp. 413–425, 1993.
- Y. Diao, “The number of smallest knots on the cubic lattice,” Journal of Statistical Physics, vol. 74, no. 5-6, pp. 1247–1254, 1994.
- E. J. J. van Rensburg and S. D. Promislow, “Minimal knots in the cubic lattice,” Journal of Knot Theory and Its Ramifications, vol. 4, no. 1, pp. 115–130, 1995.
- B. Hayes, “Square knots,” American Scientist, vol. 85, no. 6, pp. 506–510, 1997.
- L. H. Kauffman, “Virtual knot theory,” European Journal of Combinatorics, vol. 20, no. 7, pp. 663–690, 1999.
- A. Nakamura and A. Rosenfeld, “Digital knots,” Pattern Recognition, vol. 33, no. 9, pp. 1541–1553, 2000.
- F. Harary, Graph Theory, Addison-Wesley, 1969.
- C. R. Jordan and D. A. Jordan, Groups, Arnold, London, UK, 1994.
- R. Bird, Introduction to Functional Programming Using Haskell, Prentice Hall, 1998.
- S. P. Jones, Ed., Haskell 98 Language and Libraries. The Revised Report, 2002, http://haskell.org.
- E. Bribiesca, “A chain code for representing 3D curves,” Pattern Recognition, vol. 33, no. 5, pp. 755–765, 2000.
- A. Guzmán, “Canonical shape description for 3-d stick bodies,” Tech. Rep. ACA-254-87, MCC, Austin, Tex, USA, 1987.
Copyright © 2012 Carlos Velarde et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.