Research Article Volume 10 Issue 1
^{1}Program of Mathematics and Statistics and Center of Applied Physics, Louisiana Tech University, USA
^{2}Department of Mathematics, Computer Science, and Engineering, Georgia State University, Perimeter College, USA
Correspondence: Songming Hou, Program of Mathematics and Statistics and Center of Applied Physics Louisiana Tech University Ruston, Louisiana, 71272, USA, Tel 3182573270, Fax 3182572182
Received: February 29, 2024 | Published: March 8, 2024
Citation: Songming H, Jianning S, Ramon M. Shortest paths of Rubik’s snake composite knots with 9 crossings. Int Rob Auto J. 2024;10(1):25-30. DOI: 10.15406/iratj.2024.10.00279
A Rubik’s Snake is an interesting toy that was invented over 40 years ago. It can be twisted to many interesting shapes, including non-trivial knots. In this paper we proposed a systematic way to study how many blocks are needed to form a composite knot with 9 crossings. We also improved some results from our previous papers.
Keywords: Rubik’s snake, composite knots
In the interesting terrain of recreational mathematics, the Rubik’s Snake exemplifies Erno Rubik’s creativity.^{1,2} The Rubik’s Snake, a cryptic member of the Rubik’s puzzle family, came into the hands of puzzle lovers who wanted to unravel not just the permutations but also the underlying geometry and topology. With its colorful, interlocking prismatic blocks, it provides a malleable, three-dimensional shape for exploration, captivating the interest of both casual players and dedicated mathematicians. Each twist and turn of the snake results in a variety of permutations, sometimes changing its appearance into a complicated knot. As we look deeper into the history of this engaging puzzle, we realize that it is more than just a hobby; it is also a useful tool for exploring mathematical concept.
The Rubik’s Snake has been used as a tool for the study of protein folding^{3,4} and for the construction of reconfigurable modular robots.^{5–7} More applications of robots can be found in.^{8,9} Some ideas in the study of Rubik’s Snake such as the use of rotation matrix is also used in rigid Origami folding.^{10,11} In previous papers that the first author collaborated with others, strategies have been given for the design of a Rubik’s Snake,^{12} and some mathematical problems concerning a Rubik’s Snake have been studied.^{13} Rotations that are not 0^{◦} (or 360^{◦}), 90^{◦}, 180^{◦}, or 270^{◦} are mentioned in^{12} but not much theoretical work is presented. On the other hand,^{13} has quite some theoretical work but is only concerned with integer multiple of 90 degree rotations. In, Hou S^{14 }Rubik’s snakes with general rotation angles were studied with theoretical work presented. In, Hou S^{15} theorems about palindromic, periodic and Mo¨bius Rubik’s snakes were proved. In, Hou S^{16} a general strategy for designing box shapes using a Rubik’s snake was given and a counting formula was proved.
Knot theory^{17} is an interesting research area that attracted a lot of mathematicians. It also has interesting applications in architectural design, for example, some work by Zaha.^{18} Knot classification tool is available.^{19} The Rubik’s Snake prime knots up to 6 crossings and composite knots up to 8 crossings have been studied in the past. The 3_{1} (trefoil) knot was studied in our previous paper^{20} and 4_{1}, 5_{1} and 5_{2} studied in our previous work.^{21} The prime knots with 6 crossings 6_{1}, 6_{2} and 6_{3} were studied in our recent work.^{22} The result for 4_{1} was improved from 46 blocks to 44 blocked by using symmetry in our paper on composite knots up to 8 crossing^{23} and the result for 5_{2} was improved from 56 blocks to 54 blocks by a non-local change in the same paper.
As we explore the complexities of Rubik’s Snake composite knots, our goal is not only to gain a better understanding of their mathematical properties, but also to contribute to the ongoing story at the intersection of knot theory, recreational mathematics, and the Rubik’s Snake’s ingenious design. Here we study the shortest path for a Rubik’s snake composite knot with 9 crossings. The organization is as follows: In Section 2, we explained the idea of using local structures. In Section 3, we improved our previously published result for 5_{1} knot and 3_{1}#5_{1} knot. In Section 4, we presented an innovative method to improve our previous result for 6_{1} knot. The results in Sections 3 and 4 will be used in later sections. In Section 5 to 7, we present results for Rubik’s Snake composite knots with 9 crossings. We conclude in Section 8.
Local structures
In, Hou S^{23} Rubik’s snakes’ shortest paths for composite knots up to 8 crossings were studied. However, some of the study was based on some inspirations rather than a systematic way. We take a different approach here. In, Hou S^{20} a complete list of 6 shortest palindromic trefoil knots with 34 blocks were presented. From there, we could extract the following 6 local structures for the knotted part of 3_{1}:
A_{1} = [0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 0, 0, 1, 2, 0, 1, 1, 0, 1, 2, 1]
A_{2} = [0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 0, 0, 1, 3, 0, 3, 3, 0, 3, 1, 1]
A_{3} = [0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 0, 3, 2, 3, 3, 2, 3, 0, 3, 1, 1]
A_{4} = [0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 0, 3, 3, 0, 2, 3, 0, 0, 1, 2, 1]
A_{5} = [0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 0, 3, 3, 0, 3, 1, 0, 0, 3, 1, 1]
A_{6} = [0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 0, 3, 3, 1, 1, 3, 3, 0, 3, 1, 1]
Similarly, in, Hou S^{23} the shortest 4_{1} knot was improved to 44 blocks. There are 12 different solutions of the form [t, −t, t, −t], where t is listed below:
[0, 0, 0, 0, 1, 0, 1, 1, 0, 2, 1],[0, 0, 0, 1, 0, 0, 3, 2, 0, 3, 3]
[0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 3],[0, 0, 1, 0, 1, 2, 1, 0, 0, 1, 2]
[0, 0, 1, 0, 3, 2, 3, 0, 0, 1, 2],[0, 0, 1, 0, 3, 3, 1, 0, 0, 3, 1]
[0, 0, 1, 1, 0, 0, 3, 1, 0, 1, 1],[0, 0, 1, 1, 0, 1, 1, 3, 3, 1, 1]
[0, 0, 1, 1, 0, 1, 2, 1, 1, 2, 1],[0, 1, 0, 1, 1, 3, 3, 0, 3, 1, 1]
[0, 1, 0, 1, 2, 1, 0, 3, 3, 2, 3],[0, 1, 1, 3, 0, 3, 2, 3, 3, 3, 1]
Based on above, we have the following 12 local structures for the knotted part of 4_{1}:
B_{1} = [0, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 0, 0, 3, 0, 3,3, 0, 2, 3, 0, 0, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0]
B_{2} = [0, 0, 0, 1, 0, 0, 3, 2, 0, 3, 3, 0, 0, 0, 3, 0, 0,1, 2, 0, 1, 1, 0, 0, 0, 1, 0, 0, 3, 2, 0, 3, 3, 0]
B_{3} = [0, 1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 0, 3, 3, 0, 3, 3,0, 3, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 0]
B_{4} = [0, 0, 3, 2, 0, 0, 1, 0, 1, 2, 1, 0, 0, 1, 2, 0, 0,3, 0, 3, 2, 3, 0, 0, 3, 2, 0, 0, 1, 0, 1, 2, 1, 0]
B_{5} = [1, 2, 1, 0, 0, 3, 2, 0, 0, 1, 0, 3, 2, 3, 0, 0, 1,2, 0, 0, 3, 0, 1, 2, 1, 0, 0, 3, 2, 0, 0, 1, 0, 3]
B_{6} = [0, 1, 1, 3, 0, 0, 1, 3, 0, 0, 1, 0, 3, 3, 1, 0, 0,3, 1, 0, 0, 3, 0, 1, 1, 3, 0, 0, 1, 3, 0, 0, 1, 0]
B_{7} = [0, 1, 1, 0, 0, 3, 1, 0, 1, 1, 0, 0, 3, 3, 0, 0, 1,3, 0, 3, 3, 0, 0, 1, 1, 0, 0, 3, 1, 0, 1, 1, 0, 0]
B_{8} = [0, 1, 1, 0, 1, 1, 3, 3, 1, 1, 0, 0, 3, 3, 0, 3, 3,1, 1, 3, 3, 0, 0, 1, 1, 0, 1, 1, 3, 3, 1, 1, 0, 0]
B_{9} = [0, 1, 1, 0, 1, 2, 1, 1, 2, 1, 0, 0, 3, 3, 0, 3, 2,3, 3, 2, 3, 0, 0, 1, 1, 0, 1, 2, 1, 1, 2, 1, 0, 0]
B_{10} = [0, 1, 0, 1, 1, 3, 3, 0, 3, 1, 1, 0, 3, 0, 3, 3,1, 1, 0, 1, 3, 3, 0, 1, 0, 1, 1, 3, 3, 0, 3, 1, 1, 0]
B_{11} = [1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 3, 2, 3, 0, 3,0, 3, 2, 3, 0, 1, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 3]
B_{12} = [0, 1, 1, 3, 0, 3, 2, 3, 3, 3, 1, 0, 3, 3, 1, 0,1, 2, 1, 1, 1, 3, 0, 1, 1, 3, 0, 3, 2, 3, 3, 3, 1, 0]
Other knots are more complicated and it is not realistic to make such lists. However, for 5_{1}, 5_{2}, 6_{1}, 6_{2}, 6_{3}, we could at least provide one local structure for each.
When combining local structures, we need to search for some connection blocks and also keep in mind that we have 4 choices: the original sequence, the negative sequence (left hand becomes right hand), the reverse sequence, and the negative of the reverse sequence. For a composite knot A#B, there are 4 choices as mentioned above. For a composite knot A#B#C, there are 4 ∗ 4 = 16 choices.
We will apply the above method to the construction of possible shortest paths for composite knots in the next few sections.
Improving the 5_{1} and 3_{1}#5_{1} knot paths
The shortest Rubik’s Snake 5_{1} knot we found in Hou S^{21} has 52 blocks. We rotate the sequence and write as: [1, 1, 1, 3, 3, 0, 2, 1, 1, 2, 1, 0, 1, 2, 1, 1, 3, 0, 0, 1, 0, 0, 2, 3, 0, 0,0, 3, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 0, 3, 0, 0, 1, 3, 0, 0, 0, 3, 2, 0, 0]
Now we replace the subsequence [0, 1, 0, 0, 2, 3, 0] with [3, 0, 1, 2, 1, 0, 1] without changing the relative location and orientation of the starting and end blocks of this part.
By doing so, we make room for another part of the path. We now replace the subsequence [0, 0, 3, 2, 0, 0] with [3, 3, 0, 3] and improved the 5_{1} path to only 50 blocks:
[1, 1, 1, 3, 3, 0, 2, 1, 1, 2, 1, 0, 1, 2, 1, 1, 3, 0, 3, 0, 1, 2, 1, 0, 1, 0,0, 3, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 0, 3, 0, 0, 1, 3, 0, 3, 3, 0, 3]
Figure 1 and 2 show the Rubik’s snake 5_{1} knot with 50 blocks and its line representation.
Since the 5_{1} knot path is improved, the new result can improve the composite knot path 3_{1}#5_{1} in Hou S.^{23} We use one of our 6 local structures for 3_{1} combined with a local structure of the new 5_{1}. The new path has 72 blocks instead of 74. The notation “...” separates local knot structures and connections between local knot structures:
[0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 0, 0, 1, 2, 0, 1, 1, 0, 1, 2, 1, ...2, 1, 0, ...3, 3, 1, 0, 1, 0, 3, 2, 3, 0, 3, 0, 0, 1, 0, 3, 2, 3, 0, 3, 0, 3, 2, 3, 0,0, 1, 0, 0, 3, 1, 0, 1, 1, 0, 1, 3, 3, ...2, 3, 3, 3, 1, 0, 0, 3]
Figure 3 and 4 show the Rubik’s snake 3_{1}#5_{1} knot with 72 blocks and its line representation.
Improving the 6_{1} knot path
The shortest Rubik’s Snake 6_{1} knot we found in^{22} has 64 blocks. We verified that could not be improved locally. A new idea is needed to improve it.
Based on the structure of 6_{1} knot, we could try to construct a period 2 Rubik’s snake 6_{1} knot. However, the computational cost is not acceptable for an exhaustive search. A trick is needed.
At the website,^{19} knot diagrams of prime knots with a small crossing number are available. We could see that the 6_{1} knot and 8_{3} knot differ only by a local change. Also, 8_{3} has nice knot symmetry so that we could use the pattern [t, −t, t, −t] to construct.
We first use the pattern [t, −t, t, −t] to construct a 8_{3} knot, where t = [0, 0, 0, 0, 0, 1, 0, 1, 3, 0, 3, 3, 0, 3, 3, 3, 1, 0, 1].
This is done by an exhaustive search with rotations 0,1,3 only to save computational cost. Note that 8_{3} knot has this kind of symmetry while 6_{1} knot does not. We keep periodic 2 structure and do local improvement. This can change the knot structure from 8_{3} to 6_{1}. When we shrink to 60 blocks, we get a possibly shortest 6_{1} with 60 blocks:
[3, 0, 3, 0, 0, 0, 0, 0, 1, 0, 1, 3, 0, 3, 3, 0, 3, 3, 3, 1, 0, 1, 0, 0, 0, 0, 3, 2, 0, 3] repeated twice.
We could also make a local change to remove the 2 and rotate the sequence to [0, 0, 0, 0, 1, 0, 1, 3, 0, 3, 3, 0, 3, 3, 3, 1, 0, 1, 0, 0, 0, 0, 3, 1, 0, 1, 1, 0, 1, 1] repeated twice.
Figure 5 and 6 show the Rubik’s snake 6_{1} knot with 60 blocks and its line representation.
We will use part of this sequence in the next section.
3_{1}#6_{1}, 3_{1}#6_{2}, 3_{1}#6_{3}
We exhaust the 6 choices for the local 3_{1} structure and search for the smallest number of connection blocks with a local structure of 6_{1}, 6_{2} and 6_{3}, where the local structure of 6_{1} comes from the previous section (the improved 6_{1}) and the other two come from.^{22} The following are the shortest solutions we found (only one representative for each tied solutions, again the notation “...” separates local knot structures and connections between local knot structures):
[0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 0, 0, 1, 2, 0, 1, 1, 0, 1, 2, 1, ...0, 3, 0, 0, ...0, 0, 0, 1, 0, 1, 3, 0, 3, 3, 0, 3, 3, 3, 1, 0, 1, 0, 0, 0, 0, 3,1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 3, 0, 3, 3, 0, 3, 3, 3, 1, 0, 1, 0, 0,...0, 0, 0, 1, 2, 0, 0, 3]
[0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 0, 0, 1, 2, 0, 1, 1, 0, 1, 2, 1, ...1, 3, 0, 3, 0, 1, 0, ...0, 1, 1, 3, 3, 1, 1, 0, 0, 3, 3, 1, 1, 1, 1, 1, 0, 2, 1,0, 0, 3, 0, 0, 0, 1, 2, 1, 1, 2, 1, 0, 1, 1, 0, 2, 1, 0, 0, 1, 2, 0, 0, 3, 0, 3,2, 3, 3, ...0, 1, 0, 0, 3]
[0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 0, 0, 1, 3, 0, 3, 3, 0, 3, 1, 1, ...1, 3, 0, 3, 3, 0, ...0, 0, 1, 1, 0, 2, 3, 0, 3, 0, 3, 3, 0, 0, 1, 0, 0, 3, 2, 3,3, 2, 0, 0, 1, 0, 1, 2, 1, 1, 3, 0, 0, 1, 0, 0, 2, 3, 0, 0, 0, 3, 0, 1, 2, 1, 0,1, 0, 1, 2, 1, 0, 3, 0, 0, ...3, 0, 0]
Figure 7,8,9,10,11 and 12 show the Rubik’s snake knots constructed in this section and their line representations.
4_{1}#5_{1}, 4_{1}#5_{2}
By exhausting the 12 choices for local patterns related to shortest Rubik’s snake 4_{1} knots with the pattern [t, −t, t, −t] and using local patterns for 5_{1} and 5_{2}, we constructed 4_{1}#5_{1 }with 82 blocks and 4_{1}#5_{2} with 86 blocks:
[0, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 0, 0, 3, 0, 3, 3, 0, 2, 3, 0, 0,0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, ...0, 0, 3, 0, ...1, 1, 3, 0, 3, 0, 1, 2,1, 0, 1, 0, 0, 3, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 0, 3, 0, 0, 1, 3, 0, 3,3, 0, 3, 1, 1, ...1, 3, 3, 0, 3, 0]
[0, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 0, 0, 3, 0, 3, 3, 0, 2, 3, 0, 0,0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, ...0, 1, 0, 3, 0, 0, 2, ...1, 0, 3, 0, 0,0, 3, 2, 0, 3, 3, 0, 0, 1, 2, 1, 1, 3, 0, 3, 3, 0, 3, 1, 0, 0, 3, 0, 0, 3,2, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 0, ...0, 1]
Figure 13, 14, 15 and 16 show the knots constructed in this section and their line representations.
3_{1}#3_{1}#3_{1} knot
The natural way to construct a 3_{1}#3_{1}#3_{1} knot with short path is to use one of the local structures of 3_{1} and add a few blocks then repeat three times, hopefully to form a closed loop. By a simple search, we found the shortest has 84 blocks, for example [0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 0, 0, 1, 2, 0, 1, 1, 0, 1, 2, 1, 0, 3, 3, 0, 3] repeated 3 times.
Figure 17 shows the 3_{1}#3_{1}#3_{1} knot with 84 blocks and period 3.
Figure 18 shows the line representation.
However, if we break the symmetry, it allows us more degree of freedom. As mentioned in the previous section, for a composite knot A#B#C, there are 4 ∗ 4 = 16 choices. For each part using local 3_{1} structure, there are 6 choices. 16 ∗ 6 ∗ 6 ∗ 6 is a small amount we could afford to exhaust and for each case we seek for the shortest closed loop connections.
We found the shortest path has 82 blocks. For example, let t = [0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 0, 0, 1, 2, 0, 1, 1, 0, 1, 2, 1].
Then [t, 0, 3, t, 1, 0, 0, 3, t, 1, 0, 0, 2, 0, 0, 3] is a 3_{1}#3_{1}#3_{1} that has 82 blocks.
Figure 19 shows the 3_{1}#3_{1}#3_{1} knot with 82 blocks.
Figure 20 shows the line representation.
Finding the shortest path for certain Rubik’s snake’s nontrivial knot is a challenging problem. With each joint having 4 choices, the total number of attempts is too many to have an exhaustive search. Based on our previous results for prime knots, we found possible shortest composite knots as follows: 3_{1}#6_{1} with 34 + 60 − 10 = 84 blocks, 3_{1}#6_{2} with 34 + 62 − 12 = 84 blocks, 3_{1}#6_{3} with 34 + 64 − 10 = 88 blocks, 4_{1}#5_{1} with 44 + 50 − 12 = 82 blocks, 4_{1}#5_{2} with 44 + 54 − 12 = 86 blocks, 3_{1}#3_{1}#3_{1} with 34 + 34 + 34 − 20 = 82 blocks. We summarize the results in Table 1. It is verified that no further local improvement could be made. Although there is no proof that these are optimal globally, the results look reasonable that they might be the shortest.
Shortest A |
Shortest B |
Shortest composite |
3_{1}: 34 |
6_{1}: 60 |
3_{1}#6_{1}: 34+60-10=84 |
3_{1}: 34 |
6_{2}: 62 |
3_{1}#6_{2}: 44+62-12=84 |
3_{1}: 34 |
6_{3}: 64 |
3_{1}#6_{3}: 34+64-10=88 |
4_{1}: 44 |
5_{1}: 50 |
4_{1}#5_{1}: 34+50-12=82 |
4_{1}: 44 |
5_{2}: 54 |
4_{1}#5_{2}: 34+54-12=86 |
3_{1}: 34 |
3_{1}: 34 |
3_{1}#3_{1}#3_{1}: 34+34+34-20=82 |
Table 1 A list of length of shortest paths found for composite knots with 9 crossings based on prime knots
We thank Yin Sun, Si Gao and Jin Fu for some interesting discussions. S. Hou’s research is partially supported by the Walter Koss Endowed Professorship. The title of the professorship is made available through the State of Louisiana Board of Regents Support Funds.
Authors declare that there is no conflict of interest.
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