Grid2Mosaic2Grid: A Complete Pair of Polynomial Knot Algorithms

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

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: Making shahi jilapi

Figure: Jilapi for sale Figure: Selling shahi jilapi

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.


Knot and It’s Diagram
A Knot is an embedding of a circle in 3-dimensional Euclidean
space, R3 .

Figure: Trefoil knot

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

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

Use of Knot Theory

Knotting of physical manifolds
DNA folding
Quantum ﬁeld theory
Spin networks
Quantum cryptography
...


Discrete Structures for Knot Diagram

Discrete structures are necessary to process information encoded in
the physical system represented by a knot.
Knot Mosaic
Grid Diagram
Arc presentation
Cube diagram
Minesweeper matrix
Mirror curve
We propose a pair of algorithms to translate between Knot Mosaic
and Grid Diagram.


Grid Diagram
A knot Grid Diagram, D, is an n × n arrangement of horizontal
and 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.


Grid Diagram
Drawing a Grid Diagram from a Knot Diagram

Figure: Trefoil knot diagram with sharp turns


Grid Diagram

Figure: Trefoil knot Grid Diagram with symbols and connectors

Grid Diagram

Figure: Trefoil knot Grid Diagram (ﬁnal version)

Knot Mosaic
A Knot Mosaic, M, is an n × n arrangement of horizontal and
vertical 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 T10


Knot Mosaic
The Trefoil Knot

Figure: Trefoil Knot Mosaic

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.


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 equivalence
relation between Knot Mosaic moves and Cromwell moves are still
unknown.


The Proposed Algorithms

Grid2Mosaic2Grid = 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.


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.


Grid scenarios in Trefoil Knot

Figure: Non trivial grid scenarios in a Trefoil knot and their mosaic
replacements.

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 , T0


List of Grid Scenarios (contd...)



GS3 T6 , T0 , T3 , T5

GS4 T0 , T6 , T5 , T4

GS5 T5


List of Grid Scenarios (contd...)



GS6 T6

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


Translating symbols

After identifying the turn scenarios, we translate all the grid
symbols into mosaic symbols.
Trivial Grid scenarios are easy to replace.


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.


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 ).


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.


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 of
columns.

Figure: Knot Mosaic column with more than two turns.


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.


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.


Complexity 1:1 translations

Table: Complexity 1:1 mosaic symbol translations

⇒

⇒

⇒




⇒ or

⇒ or

⇒ or

⇒ or


Complexity 1:3 translation

Table: Complexity 1:3 mosaic symbol translation

⇒




⇒ or or or

⇒ or or or


Complexity 1:11 translation

Table: Complexity 1:11 mosaic symbol translation

⇒ ⇒


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 symbols

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.


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 ).


Zooming the Knot Mosaic symbols

Table: Zoom T0 by factor 3

⇓


Zooming the Knot Mosaic symbols (contd...)


⇓




⇓


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.



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.



Table: Valid Bracket curves
Label Curve
B1 ...
B2

...

B3 ...
B4

...



Table: Valid Snake curves
Label Curve
S1 ...
S2

...

S3 ...
S4

...



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.



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.


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.


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 ).


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.


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.


Acknowledgement

Dr. Samuel J Lomonaco Jr.
Sumeetkumar Bagde.


Questions?


Grid2Mosaic2Grid: A Complete Pair of Polynomial Knot Algorithms

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