Grid2Mosaic2Grid: A Complete Pair of Polynomial Knot Algorithms

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Tame Knot Diagrams can be represented by two different discrete structures, namely, Grid Diagrams and Knot Mosaics. This report proposes two polynomial time algorithms for translations between Grid Diagrams and Knot Mosaics. It is shown that that the time complexity of both algorithms is O(\\ensuremath{n^{3}}). These results prove that Grid Diagrams and Knot Mosaics are topologically equivalent. This equivalence is efficiently computable. We also conjecture that the two Cromwell moves of Grid Diagrams, i.e. Castling and Stabilization, are equivalent to sequences of planar moves defined for Knot Mosaics. These equivalences are also conjectured to be polynomially computable.

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Grid2Mosaic2Grid: A Complete Pair of Polynomial Knot Algorithms

  1. 1. Grid2Mosaic2Grid: A Complete Pair of Polynomial Knot Algorithms Omar Shehab Department of Computer Science and Electrical Engineering University of Maryland, Baltimore County Baltimore, Maryland 21250 shehab1@umbc.edu December 4, 2011
  2. 2. OutlineDefinitionsDiscrete StructuresRationale and Related WorksThe AlgorithmsSummary of ResultsFuture WorkOmar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 2 / 60
  3. 3. My First Knots Figure: Standard jilapi Figure: Making shahi jilapi Figure: Jilapi for sale Figure: Selling shahi jilapiOmar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 3 / 60
  4. 4. Big Picture Develop discrete structures for Knot Diagrams Define a Quantum Information System using the scheme Example: Express Quantum Money protocol using knot thoery The protocol is defined in Grid Diagram. To express this using Knot Mosaic we may use Grid2Mosaic2Grid.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 4 / 60
  5. 5. Knot and It’s DiagramA Knot is an embedding of a circle in 3-dimensional Euclideanspace, R3 . Figure: Trefoil knotA Knot Diagram is a planar representation of a knot with over andunderpasses.Omar Shehab (UMBC) Figure: Trefoil knot diagram Grid2Mosaic2Grid Algorithms December 4, 2011 5 / 60
  6. 6. Use of Knot Theory Knotting of physical manifolds DNA folding Quantum field theory Spin networks Quantum cryptography ...Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 6 / 60
  7. 7. Discrete Structures for Knot DiagramDiscrete structures are necessary to process information encoded inthe physical system represented by a knot. Knot Mosaic Grid Diagram Arc presentation Cube diagram Minesweeper matrix Mirror curveWe propose a pair of algorithms to translate between Knot Mosaicand Grid Diagram.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 7 / 60
  8. 8. Grid DiagramA knot Grid Diagram, D, is an n × n arrangement of horizontaland vertical cells representing a knot diagram. Cromwell P. R. Embedding knots and links in an open book I: Basic properties. Topology Appl. 64 (1995), 3758., 1995. Each cell can have any of the following symbols - blank cell, horizontal bar, vertical bar, X or O. In each column there is only one X and one O. In each row there is only one X and one O. O and X are connected with horizontal and vertical lines in rows and columns respectively. Horizontal lines always pass under the vertical lines. n is called the complexity of D.Let’s draw the Grid Diagram of a Trefoil Knot Diagram.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 8 / 60
  9. 9. Grid Diagram Drawing a Grid Diagram from a Knot Diagram Figure: Trefoil knot diagram with sharp turnsOmar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 9 / 60
  10. 10. Grid Diagram Drawing a Grid Diagram from a Knot Diagram Figure: Trefoil knot Grid Diagram with symbols and connectorsOmar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 10 / 60
  11. 11. Grid Diagram Drawing a Grid Diagram from a Knot Diagram Figure: Trefoil knot Grid Diagram (final version)Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 11 / 60
  12. 12. Knot MosaicA Knot Mosaic, M, is an n × n arrangement of horizontal andvertical tiles representing a knot diagram. Samuel J. Lomonaco and Louis H. Kauffman. Quantum Knots and Mosaics. Journal of Quantum Information Processing, Vol. 7, Nos. 2-3, (2008), pp. 85 - 115., 2008. Mosaic symbols - T0 , T1 , T2 , T3 , T4 , T5 , T6 , T7 , T8 , T9 and T10 . n is called the complexity of M. Table: Knot Mosaic symbols Symbol Label T0 T1 T2 T3 T4 T5 Symbol Label T6 T7 T8 T9 T10Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 12 / 60
  13. 13. Knot Mosaic The Trefoil Knot Figure: Trefoil Knot MosaicOmar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 13 / 60
  14. 14. Rationale Knot Mosaic is more intuitive and has better encoding capacity given the same complexity (conjectured). Systems already modeled in Grid Diagram may be studied better using Knot Mosaic.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 14 / 60
  15. 15. Related Works Takahito Kuriya. On a Lomonaco-Kauffman conjecture. November 2008. A recently withdrawn Arxiv paper which proves the conjecture that Knot Mosaic is equivalent to Tame Knot Theory. This presentation takes hints from the paper to translate Grid Diagram into Knot Mosaic. Slavik V. Jablan, Ljiljana Radovic, Radmila Sazdanovic, Ana Zekovic. Mirror-Curves and Knot Mosaics. Topology Appl. 64 (1995), 3758. This paper converts both representations into Mirror-curves to prove the equivalence.The complexity of the translations are not known. The equivalencerelation between Knot Mosaic moves and Cromwell moves are stillunknown.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 15 / 60
  16. 16. The Proposed AlgorithmsGrid2Mosaic2Grid = Grid to Mosaic and Mosaic to Grid Grid2Mosaic (G2M): Takes a Grid Diagram as input and outputs the equivalent Knot Mosaic in polynomial time. Mosaic2Grid (M2G): Takes a Knot Mosaic as input and outputs the equivalent Grid Diagram in polynomial time.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 16 / 60
  17. 17. Grid Diagram to Knot Mosaic Issues in translation The turns and crossings of a Grid Diagram is very similar to those of a Knot Mosaic. There are only two turns in a Grid Diagram per column or per row. A Grid Diagram does not have any horizontal overpass. Eight grid scenarios are identified which have equivalent mosaic symbol com positions. Replace each scenario with corresponding mosaic symbols.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 17 / 60
  18. 18. Grid Diagram to Knot Mosaic Grid scenarios in Trefoil KnotFigure: Non trivial grid scenarios in a Trefoil knot and their mosaicreplacements.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 18 / 60
  19. 19. Grid Diagram to Knot Mosaic List of Grid Scenarios Table: Grid Diagram scenarios and equivalent Knot Mosaic symbols Label Grid Scenario Mosaic symbol Label GS0 T0 GS1 T5 , T1 , T0 , T6 GS2 T2 , T5 , T6 , T0Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 19 / 60
  20. 20. Grid Diagram to Knot Mosaic List of Grid Scenarios (contd...) Table: Grid Diagram scenarios and equivalent Knot Mosaic symbols Label Grid Scenario Mosaic symbol Label GS3 T6 , T0 , T3 , T5 GS4 T0 , T6 , T5 , T4 GS5 T5Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 20 / 60
  21. 21. Grid Diagram to Knot Mosaic List of Grid Scenarios (contd...) Table: Grid Diagram scenarios and equivalent Knot Mosaic symbols Label Grid Scenario Mosaic symbol Label GS6 T6 GS7 T0 , T0 , T0 , T5 , T10 , T5 , T0 , T0 , T0Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 21 / 60
  22. 22. Grid Diagram to Knot Mosaic Translating symbols After identifying the turn scenarios, we translate all the grid symbols into mosaic symbols. Trivial Grid scenarios are easy to replace.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 22 / 60
  23. 23. Grid Diagram to Knot Mosaic The D2M algorithm We define the algorithm G2M(G) which takes a Grid Diagram as input and outputs a Knot Mosaic. It uses TurnSymbol2MosaicSymbol(G, x, y) first to replace all the turns of the Grid Diagram with Knot Mosaic symbols. TurnSymbol2MosaicSymbol(G, x, y) uses DetermineTurnScenario(G, x, y) to determine the type of non-trivial Grid Diagram turns. Then it replaces the trivial grid scenarios. Finally it connects the turns along the columns and rows.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 23 / 60
  24. 24. Grid Diagram to Knot Mosaic Complexity Analysis Complexity of DetermineTurnScenario(G, x, y) is 3n + 16 i.e. O(n). Complexity of TurnSymbol2MosaicSymbol(G, x, y) is 3n + 27 i.e. O(n). Complexity of G2M is 3n3 + 38n2 + 3n + 2 i.e. O(n3 ).Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 24 / 60
  25. 25. Knot Mosaic to Grid Diagram Issues in translation Translating Knot Mosaic to Grid Diagram requires more considerations. We have to define local translations between mosaic symbols and Grid Diagram symbol compositions. Then we resolve the complexity issues raised by the translation.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 25 / 60
  26. 26. Knot Mosaic to Grid Diagram Issues in translation (contd...)In Knot Mosaic a column or a row may have more than two turns.Before translating the symbols, we have to factor those rows ofcolumns. Figure: Knot Mosaic column with more than two turns.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 26 / 60
  27. 27. Knot Mosaic to Grid Diagram Translating symbols Knot Mosaic symbols are larger in number. A Grid Diagram turn requires more than one symbol. No single Grid Diagram symbol represents even one turn let alone more than one turn. But, a Knot Mosaic symbol may contain more than one turn (please refer to T7 or T8 ). Knot Mosaics allow horizontal over pass which is not allowed in Grid Diagrams. So, the translation is not always complexity preserving.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 27 / 60
  28. 28. Knot Mosaic to Grid Diagram Translating symbols (contd...) An important decision to take is, of X or O, which symbol should be used to replace the cornering cell while translating a knot turn. We propose a standard that if it is the first symbol of a column it will always be O otherwise X. If the first symbol of the column of the Grid Diagram under translation process is the second symbol of a row, it will always be the symbol other than the one already in that row.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 28 / 60
  29. 29. Knot Mosaic to Grid Diagram Complexity 1:1 translations Table: Complexity 1:1 mosaic symbol translations ⇒ ⇒ ⇒Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 29 / 60
  30. 30. Knot Mosaic to Grid Diagram Complexity 1:2 translations Table: Complexity 1:2 mosaic symbol translations ⇒ or ⇒ or ⇒ or ⇒ orOmar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 30 / 60
  31. 31. Knot Mosaic to Grid Diagram Complexity 1:3 translation Table: Complexity 1:3 mosaic symbol translation ⇒Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 31 / 60
  32. 32. Knot Mosaic to Grid Diagram Complexity 1:4 translations Table: Complexity 1:4 mosaic symbol translations ⇒ or or or ⇒ or or orOmar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 32 / 60
  33. 33. Knot Mosaic to Grid Diagram Complexity 1:11 translation Table: Complexity 1:11 mosaic symbol translation ⇒ ⇒Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 33 / 60
  34. 34. Knot Mosaic to Grid Diagram Complexity issues in symbol translation Translation of Knot Mosaic symbols does not produce Grid matrix of same complexity all the time. If we replace the Knot Mosaic symbols right away, different symbols will be replaced by Grid matrices of different sizes. So, the output will no longer be a square matrix. Figure: Complexity mismatch in translation of Knot symbolsOmar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 34 / 60
  35. 35. Knot Mosaic to Grid Diagram Zooming Knot Mosaic We propose to zoom in a Knot Mosaic. The translation of the highest complexity can be done tightly while the lower complexity translation will be padded around with enough grid symbols. In this way the lower complexity translations can gracefully handle the change of the complexity of global Grid Diagram and keep themselves in comfortable positions. It helps them to preserve the connections and topological relations.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 35 / 60
  36. 36. Knot Mosaic to Grid Diagram Zoom operation and zoom factor Zooming a Knot Mosaic of complexity n by zoom factor F generates a topologically equivalent Knot Mosaic of complexity n × F. So, each of the mosaic symbols will be replaced by an n × F array of mosaic symbols. The original symbol will be at the center of the array. The connecting points will be extended to the border of the segment by adding mosaic symbols T5 or T6 . Every other cells will be T0 . We propose a convention for positioning the original Knot symbol. If k is odd, the symbol will be placed at the intersecting cell of the central row and central column. If k is even, the position of the original symbol will be ( (n × F)/2 , (n × F)/2 ).Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 36 / 60
  37. 37. Knot Mosaic to Grid Diagram Zooming the Knot Mosaic symbols Table: Zoom T0 by factor 3 ⇓Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 37 / 60
  38. 38. Knot Mosaic to Grid Diagram Zooming the Knot Mosaic symbols (contd...) Table: Zoom T1 by factor 3 ⇓Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 38 / 60
  39. 39. Knot Mosaic to Grid Diagram Zooming the Knot Mosaic symbols (contd...) Table: Zoom T2 by factor 3 ⇓Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 39 / 60
  40. 40. Knot Mosaic to Grid Diagram Zooming the Knot Mosaic symbols (contd...) Table: Zoom T3 by factor 3 ⇓Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 40 / 60
  41. 41. Knot Mosaic to Grid Diagram Zooming the Knot Mosaic symbols (contd...) Table: Zoom T4 by factor 3 ⇓Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 41 / 60
  42. 42. Knot Mosaic to Grid Diagram Zooming the Knot Mosaic symbols (contd...) Table: Zoom T5 by factor 3 ⇓Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 42 / 60
  43. 43. Knot Mosaic to Grid Diagram Zooming the Knot Mosaic symbols (contd...) Table: Zoom T6 by factor 3 ⇓Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 43 / 60
  44. 44. Knot Mosaic to Grid Diagram Zooming the Knot Mosaic symbols (contd...) Table: Zoom T7 by factor 3 ⇓Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 44 / 60
  45. 45. Knot Mosaic to Grid Diagram Zooming the Knot Mosaic symbols (contd...) Table: Zoom T8 by factor 3 ⇓Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 45 / 60
  46. 46. Knot Mosaic to Grid Diagram Zooming the Knot Mosaic symbols (contd...) Table: Zoom T9 by factor 3 ⇓Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 46 / 60
  47. 47. Knot Mosaic to Grid Diagram Zooming the Knot Mosaic symbols (contd...) Table: Zoom T10 by factor 3 ⇓Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 47 / 60
  48. 48. Knot Mosaic to Grid Diagram Issues in translating symbol (contd...) Symbol translation implies local complexity change. A solution to this issue is to zoom in the Knot Mosaic by the zoom factor equivalent to the largest complexity of the translations required. Then the translations with smaller complexities will have enough symbols padded around them to survive the higher complexity translations. After Zooming in, the Knot Mosaic is ready for further processing.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 48 / 60
  49. 49. Knot Mosaic to Grid Diagram Issues in translating symbol (contd...) A Knot Mosaic can have arbitrary number of turns in a column or row. We have to distribute the turns among columns or rows so that each column or row contains exactly one pair of connecting turns. To decouple the curves of a column or row we have to insert new columns or rows respectively.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 49 / 60
  50. 50. Knot Mosaic to Grid Diagram Issues in translating symbol (contd...) Table: Valid Bracket curves Label Curve B1 ... B2 ... B3 ... B4 ...Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 50 / 60
  51. 51. Knot Mosaic to Grid Diagram Issues in translating symbol (contd...) Table: Valid Snake curves Label Curve S1 ... S2 ... S3 ... S4 ...Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 51 / 60
  52. 52. Knot Mosaic to Grid Diagram Issues in translating symbol (contd...) A row or a column of a Knot Mosaic may contain T7 or T8 . It means that this symbol is shared by two curves. Before decoupling the curves, we need to decouple the shared symbols.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 52 / 60
  53. 53. Knot Mosaic to Grid Diagram Issues in translating symbol (contd...)Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 53 / 60
  54. 54. Knot Mosaic to Grid Diagram Issues in translating symbol (contd...) Now the Knot Mosaic is ready to be translated. This processed mosaic will not have T7 or T8 anymore. We replace the rest of the symbols according to the table.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 54 / 60
  55. 55. Knot Mosaic to Grid Diagram The M2G algorithm We define the algorithm M2G(M) which takes a Knot Mosaic as input and outputs a Grid Diagram. It determines the minimum zoom factor, F, with GetMinZoomFactor(M) first. Then it zooms in the Mosaic with ZoomMosaic(M, F). It uses DecoupleSharedSymbol(M, x, y) to decouple if the Mosaic has any T7 or T8 . Then it uses CompartmentalizeCurve(M) to factor the columns or rows which contain more than one Bracket or Snake curves. Then it uses TranslateKnotSymbols(M, F) to replace the symbols with Grid Diagram symbols.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 55 / 60
  56. 56. Knot Mosaic to Grid Diagram Complexity Analysis Complexity of GetMinZoomFactor(M) is 21n2 + n + 1 i.e. O(n2 ). Complexity of ZoomMosaic(M, F) is 1168n2 + n i.e. O(n2 ). Complexity of DecoupleSharedSymbol(M, x, y) is 34n + 5 i.e. O(n). Complexity of CompartmentalizeCurve(M) is 9n3 + 8n2 + n i.e. O(n3 ). Complexity of TranslateKnotSymbols(M, F) is 6n3 + 1358n2 + 2n i.e. O(n3 ). Complexity of M2G is 59n3 + 2575n2 + 8n + 1 i.e. O(n3 ).Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 56 / 60
  57. 57. Summary of Results Knot Mosaic can be converted into Grid Diagram and vice versa. So, these two discrete structures are equivalent. This equivalence is efficiently computable.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 57 / 60
  58. 58. Future Work Compute the complexity of translation of moves. Initial study indicates that two Cromwell moves (Castling and Stabilization) are equivalent to combinations of Knot Mosaic moves. ’Cycling’ may not be a planar move. It can be planar only if the knot is embedded on the surface of a torus. Hence it may be impossible to translate it into Knot Mosaic moves. Implement Markov process using Knot moves.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 58 / 60
  59. 59. Acknowledgement Dr. Samuel J Lomonaco Jr. Sumeetkumar Bagde.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 59 / 60
  60. 60. Questions?Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 60 / 60

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