Grid2Mosaic2Grid: A Complete Pair of Polynomial Knot Algorithms

Grid2Mosaic2Grid: A Complete Pair ofPolynomial Knot Algorithms

Omar Shehab

Department of Computer Science and Electrical EngineeringUniversity of Maryland, Baltimore County

Baltimore, Maryland 21250

[email protected]

December 4, 2011

Outline

Definitions

Discrete Structures

Rationale and Related Works

The Algorithms

Summary of Results

Future Work

Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 2 / 60

My First Knots

Figure: Standard jilapi

Figure: Jilapi for sale

Figure: Making shahi jilapi

Figure: Selling shahi jilapi


Big Picture

I Develop discrete structures for Knot Diagrams

I Define a Quantum Information System using the scheme

I Example: Express Quantum Money protocol using knot thoery

I The protocol is defined in Grid Diagram. To express this usingKnot Mosaic we may use Grid2Mosaic2Grid.


Knot and It’s DiagramA Knot is an embedding of a circle in 3-dimensional Euclideanspace, R3.

Figure: Trefoil knot

A Knot Diagram is a planar representation of a knot with over andunderpasses.

Figure: Trefoil knot diagramOmar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 5 / 60

Use of Knot Theory

I Knotting of physical manifolds

I DNA folding

I Quantum field theory

I Spin networks

I Quantum cryptography

I ...


Discrete Structures for Knot Diagram

Discrete structures are necessary to process information encoded inthe physical system represented by a knot.

I Knot Mosaic

I Grid Diagram

I Arc presentation

I Cube diagram

I Minesweeper matrix

I Mirror curve

We propose a pair of algorithms to translate between Knot Mosaicand Grid Diagram.


Grid Diagram

A knot Grid Diagram, D, is an n × n arrangement of horizontaland vertical cells representing a knot diagram.

I Cromwell P. R. Embedding knots and links in an open book I:Basic properties. Topology Appl. 64 (1995), 3758., 1995.

I Each cell can have any of the following symbols - blank cell,horizontal bar, vertical bar, X or O.

I In each column there is only one X and one O.

I In each row there is only one X and one O.

I O and X are connected with horizontal and vertical lines inrows and columns respectively.

I Horizontal lines always pass under the vertical lines.

I n is called the complexity of D.

Let’s draw the Grid Diagram of a Trefoil Knot Diagram.


Grid DiagramDrawing a Grid Diagram from a Knot Diagram

Figure: Trefoil knot diagram with sharp turns



Figure: Trefoil knot Grid Diagram with symbols and connectorsOmar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 10 / 60


Figure: Trefoil knot Grid Diagram (final version)Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 11 / 60

Knot MosaicA Knot Mosaic, M, is an n × n arrangement of horizontal andvertical tiles representing a knot diagram.

I Samuel J. Lomonaco and Louis H. Kauffman. QuantumKnots and Mosaics. Journal of Quantum InformationProcessing, Vol. 7, Nos. 2-3, (2008), pp. 85 - 115., 2008.

I Mosaic symbols - T0, T1, T2, T3, T4, T5, T6, T7, T8, T9 andT10.

I 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 T10


Knot MosaicThe Trefoil Knot

Figure: Trefoil Knot MosaicOmar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 13 / 60

Rationale

I Knot Mosaic is more intuitive and has better encodingcapacity given the same complexity (conjectured).

I Systems already modeled in Grid Diagram may be studiedbetter using Knot Mosaic.


Related Works

I Takahito Kuriya. On a Lomonaco-Kauffman conjecture.November 2008. A recently withdrawn Arxiv paper whichproves the conjecture that Knot Mosaic is equivalent to TameKnot Theory. This presentation takes hints from the paper totranslate Grid Diagram into Knot Mosaic.

I Slavik V. Jablan, Ljiljana Radovic, Radmila Sazdanovic, AnaZekovic. Mirror-Curves and Knot Mosaics. Topology Appl. 64(1995), 3758. This paper converts both representations intoMirror-curves to prove the equivalence.

The complexity of the translations are not known. The equivalencerelation between Knot Mosaic moves and Cromwell moves are stillunknown.


The Proposed Algorithms

Grid2Mosaic2Grid = Grid to Mosaic and Mosaic to Grid

I Grid2Mosaic (G2M): Takes a Grid Diagram as input andoutputs the equivalent Knot Mosaic in polynomial time.

I Mosaic2Grid (M2G): Takes a Knot Mosaic as input andoutputs the equivalent Grid Diagram in polynomial time.


Grid Diagram to Knot MosaicIssues in translation

I The turns and crossings of a Grid Diagram is very similar tothose of a Knot Mosaic.

I There are only two turns in a Grid Diagram per column or perrow.

I A Grid Diagram does not have any horizontal overpass.

I Eight grid scenarios are identified which have equivalentmosaic symbol com positions.

I Replace each scenario with corresponding mosaic symbols.


Grid Diagram to Knot MosaicGrid scenarios in Trefoil Knot

Figure: Non trivial grid scenarios in a Trefoil knot and their mosaicreplacements.Omar Shehab (UMBC) Grid2Mosaic2Grid Algorithms December 4, 2011 18 / 60

Grid Diagram to Knot MosaicList 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, T0


Grid Diagram to Knot MosaicList of Grid Scenarios (contd...)



GS3 T6, T0, T3, T5

GS4 T0, T6, T5, T4

GS5 T5


Grid Diagram to Knot MosaicList of Grid Scenarios (contd...)



GS6 T6

GS7 T0, T0, T0,T5, T10, T5,T0, T0, T0


Grid Diagram to Knot MosaicTranslating symbols

I After identifying the turn scenarios, we translate all the gridsymbols into mosaic symbols.

I Trivial Grid scenarios are easy to replace.


Grid Diagram to Knot MosaicThe D2M algorithm

I We define the algorithm G2M(G) which takes a Grid Diagramas input and outputs a Knot Mosaic.

I It uses TurnSymbol2MosaicSymbol(G, x, y) first to replace allthe turns of the Grid Diagram with Knot Mosaic symbols.

I TurnSymbol2MosaicSymbol(G, x, y) usesDetermineTurnScenario(G, x, y) to determine the type ofnon-trivial Grid Diagram turns.

I Then it replaces the trivial grid scenarios.

I Finally it connects the turns along the columns and rows.


Grid Diagram to Knot MosaicComplexity Analysis

I Complexity of DetermineTurnScenario(G, x, y) is 3n + 16 i.e.O(n).

I Complexity of TurnSymbol2MosaicSymbol(G, x, y) is 3n + 27i.e. O(n).

I Complexity of G2M is 3n3 + 38n2 + 3n + 2 i.e. O(n3).


Knot Mosaic to Grid DiagramIssues in translation

I Translating Knot Mosaic to Grid Diagram requires moreconsiderations.

I We have to define local translations between mosaic symbolsand Grid Diagram symbol compositions.

I Then we resolve the complexity issues raised by thetranslation.


Knot Mosaic to Grid DiagramIssues 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.


Knot Mosaic to Grid DiagramTranslating symbols

I Knot Mosaic symbols are larger in number.

I A Grid Diagram turn requires more than one symbol.

I No single Grid Diagram symbol represents even one turn letalone more than one turn. But, a Knot Mosaic symbol maycontain more than one turn (please refer to T7 or T8).

I Knot Mosaics allow horizontal over pass which is not allowedin Grid Diagrams. So, the translation is not always complexitypreserving.


Knot Mosaic to Grid DiagramTranslating symbols (contd...)

I An important decision to take is, of X or O, which symbolshould be used to replace the cornering cell while translating aknot turn.

I We propose a standard that if it is the first symbol of acolumn it will always be O otherwise X. If the first symbol ofthe column of the Grid Diagram under translation process isthe second symbol of a row, it will always be the symbol otherthan the one already in that row.


Knot Mosaic to Grid DiagramComplexity 1:1 translations

Table: Complexity 1:1 mosaic symbol translations

⇒

⇒

⇒




⇒ or

⇒ or

⇒ or

⇒ or


Knot Mosaic to Grid DiagramComplexity 1:3 translation

Table: Complexity 1:3 mosaic symbol translation

⇒




⇒ or or or

⇒ or or or


Knot Mosaic to Grid DiagramComplexity 1:11 translation

Table: Complexity 1:11 mosaic symbol translation

⇒ ⇒


Knot Mosaic to Grid DiagramComplexity issues in symbol translation

I Translation of Knot Mosaic symbols does not produce Gridmatrix of same complexity all the time.

I If we replace the Knot Mosaic symbols right away, differentsymbols will be replaced by Grid matrices of different sizes.

I So, the output will no longer be a square matrix.

Figure: Complexity mismatch in translation of Knot symbols


Knot Mosaic to Grid DiagramZooming Knot Mosaic

I We propose to zoom in a Knot Mosaic.

I The translation of the highest complexity can be done tightlywhile the lower complexity translation will be padded aroundwith enough grid symbols.

I In this way the lower complexity translations can gracefullyhandle the change of the complexity of global Grid Diagramand keep themselves in comfortable positions.

I It helps them to preserve the connections and topologicalrelations.


Knot Mosaic to Grid DiagramZoom operation and zoom factor

I Zooming a Knot Mosaic of complexity n by zoom factor Fgenerates a topologically equivalent Knot Mosaic ofcomplexity n × F . So, each of the mosaic symbols will bereplaced by an n × F array of mosaic symbols.

I The original symbol will be at the center of the array. Theconnecting points will be extended to the border of thesegment by adding mosaic symbols T5 or T6.

I Every other cells will be T0. We propose a convention forpositioning the original Knot symbol. If k is odd, the symbolwill be placed at the intersecting cell of the central row andcentral column. If k is even, the position of the originalsymbol will be (b(n × F)/2c, b(n × F)/2c).


Knot Mosaic to Grid DiagramZooming the Knot Mosaic symbols

Table: Zoom T0 by factor 3

⇓


Knot Mosaic to Grid DiagramZooming the Knot Mosaic symbols (contd...)


⇓




⇓




⇓




⇓




⇓




⇓




⇓




⇓




⇓




⇓


Knot Mosaic to Grid DiagramIssues in translating symbol (contd...)

I Symbol translation implies local complexity change.

I A solution to this issue is to zoom in the Knot Mosaic by thezoom factor equivalent to the largest complexity of thetranslations required.

I Then the translations with smaller complexities will haveenough symbols padded around them to survive the highercomplexity translations.

I After Zooming in, the Knot Mosaic is ready for furtherprocessing.



I A Knot Mosaic can have arbitrary number of turns in acolumn or row.

I We have to distribute the turns among columns or rows sothat each column or row contains exactly one pair ofconnecting turns.

I To decouple the curves of a column or row we have to insertnew columns or rows respectively.



Table: Valid Bracket curves

Label Curve

B1 ...

B2

...

B3 ...

B4

...



Table: Valid Snake curves

Label Curve

S1 ...

S2

...

S3 ...

S4

...



I A row or a column of a Knot Mosaic may contain T7 or T8 .It means that this symbol is shared by two curves.

I Before decoupling the curves, we need to decouple the sharedsymbols.



Figure: Curve compartmentalization process on a row with 3 curves.



I Now the Knot Mosaic is ready to be translated.

I This processed mosaic will not have T7 or T8 anymore.

I We replace the rest of the symbols according to the table.


Knot Mosaic to Grid DiagramThe M2G algorithm

I We define the algorithm M2G(M) which takes a Knot Mosaicas input and outputs a Grid Diagram.

I It determines the minimum zoom factor, F , withGetMinZoomFactor(M) first.

I Then it zooms in the Mosaic with ZoomMosaic(M, F).

I It uses DecoupleSharedSymbol(M, x, y) to decouple if theMosaic has any T7 or T8.

I Then it uses CompartmentalizeCurve(M) to factor thecolumns or rows which contain more than one Bracket orSnake curves.

I Then it uses TranslateKnotSymbols(M, F) to replace thesymbols with Grid Diagram symbols.


Knot Mosaic to Grid DiagramComplexity Analysis

I Complexity of GetMinZoomFactor(M) is 21n2 + n + 1 i.e.O(n2).

I Complexity of ZoomMosaic(M, F) is 1168n2 + n i.e. O(n2).

I Complexity of DecoupleSharedSymbol(M, x, y) is 34n + 5 i.e.O(n).

I Complexity of CompartmentalizeCurve(M) is 9n3 + 8n2 + ni.e. O(n3).

I Complexity of TranslateKnotSymbols(M, F) is 6n3 + 1358n2

+ 2n i.e. O(n3).

I Complexity of M2G is 59n3 + 2575n2 + 8n + 1 i.e. O(n3).


Summary of Results

I Knot Mosaic can be converted into Grid Diagram and viceversa.

I So, these two discrete structures are equivalent.

I This equivalence is efficiently computable.


Future Work

I Compute the complexity of translation of moves.I Initial study indicates that two Cromwell moves (Castling and

Stabilization) are equivalent to combinations of Knot Mosaicmoves.

I ’Cycling’ may not be a planar move. It can be planar only ifthe knot is embedded on the surface of a torus. Hence it maybe impossible to translate it into Knot Mosaic moves.

I Implement Markov process using Knot moves.


Acknowledgement

I Dr. Samuel J Lomonaco Jr.

I Sumeetkumar Bagde.


Questions?


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Grid2Mosaic2Grid: A Complete Pair of Polynomial Knot Algorithms

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