By Victoria Celetti, Cameron Martin, Kirsten Pasterczyk, Slavik Yanyuk, and Cindy Zheng.
Mirror-curve codes for knots and links Ljiljana Radovic, Slavik Jablan 3.-8.9. 2012, Zlatibor.
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Transcript of Mirror-curve codes for knots and links Ljiljana Radovic, Slavik Jablan 3.-8.9. 2012, Zlatibor.
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Mirror-curve codes for knots and linksMirror-curve codes for knots and links
Ljiljana Radovic, Slavik JablanLjiljana Radovic, Slavik Jablan
3.-8.9. 2012 , Zlatibor3.-8.9. 2012 , Zlatibor
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Snake with interlacing coil, Cylinder seal, Ur, Mesopotamia, 2600-2500 B.C.
9x5
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Tchokwe sand drawings
``The Tchokwe people of northeast Angola are wellknown for their beautiful decorative art. When they meet, they illustrate their conversations by drawings on the ground. Most ofthese drawings belong to a long tradition. They refer to proverbs, fables, games, riddles, etc. and play an important role in thetransmission of knowledge from one generation to the other.'‘
(Gerdes, 1990)
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Tchokwe sand drawings
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"During the harvest month, Tamil women in South India draw designs in front of the thresholds of their houses. In order to prepare their drawings, they set out a rectangular reference frame of equidistant points. Then curves are drawn in such a way that they surround the dots without touching them. The (culturally) ideal design is composed of a single closed line."
P.P.J.Gerdes: On ethnomathematical research and symmetry, Symmetry: Culture and Science 1, 2 (1990), 154-170.
Tamil drawings
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Tamil knots (“Pavitram” or “Brahma Mudi”)
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Tamil Knots
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``Leonardo spent much time in making a regular design of a series ofknots so that the cord may be traced from one end to the other, thewhole filling a round space...''
Bain, G.: Celtic Art - the Methods of Construction, Dover, New York, 1973.
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Plates
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Number of curves(components) cin regular squaregrid RG[a,b]
c = GCD(a,b)
10×3
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Celtic knots
Celtictangles
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Celtic monolinear design withbroken symmetry
Celtic mirror curves
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Mirror curvesMirror curves
Reflection in a mirror.
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Square grid with a two-sided mirror between cells.
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Mirror curve in a square grid 3x3 with two internal mirrors.
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“Over-under” alternating.
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Mirror-curves can be constructed any polygonal edge-to-edge tilingT of a part of an arbitrary surface. We propose the followingconstruction:
First, construct all different curves in T containing lines that connect different cell-edge midpoints until T is uniformly covered by k components. Then, in order to obtain a single curve, place internal mirrors according tothe following rules:
1) any mirror placed in a crossing point of two distinctcurves connects them in one curve;
2) depending on the position of a mirror, a mirror placedinto a self-crossing point of an (oriented) curve either does notchange the number of curves, or breaks the curve in two closedcurves.
Construction rules
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Construction rules
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Construction of a single mirror curve from the tiling (a)by connecting edge mid-points (b), tracing components (c) andintroducing a mirror (d).
(a) (b)
(c) (d)
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Lomonaco, S. J. and Kauffman,L.H.: Quantum knots and mosaics, Quantum Information Processing 7, 2-3 (2008),85--115 (arXiv:quant-ph/0805.0339v1).
212=2 =
RII
412=4
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Consider a rectangular grid RG[p,q] with edges of integer lengths p and q. Every internal edge will carry labels 2, -2, 1, and -1, where 2 denotes a two-sided mirror incident with the edge, -2 two-sided mirror perpendicular to the edge in its middle point, 1 the positive crossing +1, and -1 the negative crossing in the middle point of the edge (Fig. a). Mirror curves will be denoted by a list of lists (i.e., a matrix), containing labels of internal edges corresponding to rows and columns of the RG[p,q]. For example, to the labeled RG[3,2] (Fig. b) corresponds the code Ul={{-2,-1,-1,2},{1,2,-1,1},{2,1,-1},{1,-2,-1},{1,-2,-1}}.
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Ul={{-2,-1,-1,2},{1,2,-1,1},{2,1,-1},{1,-2,-1},{1,-2,-1}}
2 = a mirror incident to an internal edge-2 = a mirror perpendicular to an internal edge-1 = negative crossing 1 = positive crossing
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Reidemeister moves (knots and links are Reidemeister moves (knots and links are equivalence classes with regard to equivalence classes with regard to ambient isotopies)ambient isotopies)
I II III
I loopsII two standsIII three stands
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I II
II III
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Every unknot or unlink can be reduced to the code containing only labels 2 and -2.
For example, unknot with three crossings given by the code Ul = {{-2, -1}, {1, 1}} (Fig. a) after RII applied on the upper right crossing gives the code Ul={{-2, -2},{1, -2}} (Fig. b), which after RI applied on the remaining crossing becomes Ul = {{-2, -2}, {2, -2}}. The obtained code contains only labels 2 and -2, so it is unknot.
Minimal diagrams of mirror curves correspond to the codes with the minimal number of labels ±1.
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In order to make the reduction (or ambient isotopy), we use themirror-moves.
Mirror moves
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Reducing the code Ul={{-2,-1,-1,2},{1,2,-1,1},{2,1,-1},{1,-2,-1},{1,-2,-1}} of the 2-component link shown on Fig. a) to the 2-component unlink (f) is more complicated.
First we apply RI to the right lower crossing, and three moves RII, in order to obtain the code (Fig. b){{-2, -2, -1, 2}, {-2, 2, 2, -2}, {2, 1, -2}, {-2, -2, -1}, {-2, -2, -2}}, then the mirror-move to the first mirror in the upper row and obtain the code (Fig. c) {{-2, -2, -1, 2}, {-2, 2, -1,-2}, {2, 1,-1}, {-2, -2, -2}, {-2, -2, -2}}, RI and obtain the code (Fig. d) {{-2, -2, -1, 2}, {-2, 2, 2, -2}, {2, 1, -1}, {-2, -2, -2}, {-2, -2, -2}}, RI and obtain the code (Fig. e) {{-2, -2, 2, 2},{-2, 2, 2, -2}, {2, 1, -1}, {-2, -2, -2}, {-2, -2, -2}}, and finally, by RII we eliminate the remaining two crossings and obtain the code (Fig. f) {{-2,-2,2,2},{-2,2,2,-2},{2,2,2},{-2,-2,-2},{-2,-2,-2}} of the two-component unlink.
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• Same as about mosaics and quantum knots and links, we can pose many similar open questions about mirror-curves, e.g., the question about mosaic number.
• For knot mosaics this will be the minimal squarefrom which certain knot or link can be obtained. For mirror curves this will be p+q.
• In RG[p,q] every alternating knot or link is given by a code which contains either 1 or -1, but not the both of them.
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Which knots and links can be obtained from a RG[p,q]?
From RG[1,1] we can derive only unknot. From RG[2,1] we can also derive two-component unlink. It is clear that from every RG[p,1] we can obtain p-component unlink.
In order to simplify, for every RG[p,q] we will be interested only for new knots or links,i.e., knots or links which cannot be derived from some smaller rectangular grid.
4=412 2=21
2 2=212 2=21
2 31
RG[2,2]
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RG[3,2]
(a) 3 1 3 = 74 {{1,1,1},{1,1},{1,1}}(b) 4 2 = 61 {{1,1,-1},{1, 1},{-1,-1}} (c) 3 1 2 = 62 {{1,1,1},{1,1},{-2,1}} (d) 6 = 61
2 {{1,2,1},{1,1},{1,1}}(e) 5 = 51 {{1,2,1},{-2,1},{1,1}} (f) 3 2 = 52 {{1,1,1},{1,1},{-2,-2}} (g) 2 1 2 = 51
2 {{1,1,1},{-2,1},{1,-2}}(h) 2 2 = 41 {{-2,1,1},{1,1},{-2,-2}} (i) 3#3 {{1,-2,1},{1,1},{1,1}}(j) 3#2 {{1,-2,1},{-2,1},{1,1}}(k) 2#2 {{1,-2,1},{1,-2},{-2,1}}.
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(a) 3 1 2 1 3 = L10a101 (b) 5 1 3 = 95 (c) 3 1 2 1 2 = 920
(d) 4 1 1 3 = 952
(e) 3 1 3 2 = 982
(f) 3 1 1 1 3 = 992
(g) 5 1 2 = 82
(h) 4 1 3 = 84
(i) 3 1 1 1 2 = 813
(j) 8 = 812
(k) 4 2 2 = 832
(l) 3 2 3 = 842
(m) 3 1 2 2 = 852(n) 2 4 2 = 86
2
(o) 2 12 1 2 = 872
(p) 7 = 71
(q) 5 2 = 72
(r) 2 2 1 2 = 76
(s) 2 1 1 1 2 = 77
(t) 4 1 2 = 712
(u) 3 1 1 2 = 722
(v) 2 3 2 = 732
(x) 2 1 1 2 = 63
(y) 3 3 = 622
(z) 2 2 2 = 632
RG[4,2], prime KLs
Theorem: All rational knots and links can be derived as mirror-curves fromrectangular grids RG[p,2] (p>1).
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The main problem is that some KLs created as mirror-curves will be placed in non-minimal rectangle grids. This problem can be solved by grid reduction, where by "all-over move" froma KL placed in a RG[p,q] we obtain the same KL placed in the grid RG[p-1,q].
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Reduction of the knot 3 -1 3 placed in the RG[3,2] tothe trefoil placed in its minimal RG[2,2].
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The next question is: how to construct a mirror-curve correspondingto some KL given in Conway notation.
Construction of a mirror-curve diagram of figure-eight knot2 2 from its Conway symbol.
Remark: Probably the simplest method of construction is to use the programs KnotAtlas or gridlink to construct grid diagram (arc presentation) of a given link, then transform it into a mirror-curve and reduce the obtained mirror-curve.
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From RG[3,3] with its corresponding alternating 3-component link 8*2:2:2:2 (a) with n=12 we derive a lot of new KLs, among them the smallest basic polyhedron- Borromean rings 6* = 62
3 {{-1,-1,-2},{-2,-1,-2},{-2,2,-2},{-1,-1,-1}} and the first non-alternating 3-component link 2,2,-2 =63
3 {{-1,-1,1},{-1,-1,1},{-2,2,-2},{-2,2,-2}}.
(a) RG[3,3] with 3-component link 8*2:2:2:2 (b) non-alternating 3-component link 2,2,-2 =63
3.
(a) Figure-eight knot and (b) Borromean rings mosaic transformed into mirror-curves.
The approach based on mirror-curves is equivalent to the approach based on link mosaics: every link mosaic can be easily transformed into a mirror-curve and vice versa.
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Even more illustrative are knot mosaics from T. Kyria’s paper: On a Lomonaco-Kauffman Conjecture,arXiv:math.GT/0811.0710v3 (2008), page 15: take any of them, rotate it by 45o, cut the emptyparts and add the two-sided mirrors in appropriate places.
4151 52
62 74
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Since Takahito Kuriya proved Lomonaco-Kauffman Conjecture, showing that the knot mosaic theory and the tame knot theory are equivalent, the same holds for the mirror-curve theory and the tame knot theory. According to the Proposition 8.4 from the same paper, there is also a correspondence between knot mosaics and grid diagrams, enabling the connection between grid diagrams and mirror-curves.
Definition: The mosaic number m(L) of a link L is the smallest integer n for which L is representable as a knot n-mosaic.
For every link L, the mosaic number m(L) equals p+q, where p and q are dimensions of the minimal RG[p,q] in which L can be placed, and the dimension of the grid (arc) representation matrix equals m(L)+1=p+q+1.
Using this we can prove Kyria’s Conjecture 10.4, claiming that the mosaic number of the knot 2 1 1 2 = 63 is 6, since its minimal rectangular grid is RG[3,3].
63
Conjecture: For direct product of two links and their mosaic numbers holds the relationship:m(L1#L2)= m(L1)+m(L2)-3.
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Product of mirror-curves
In order to define a product of mirror-curves derived from the same RG[p,q] we can substitute in their codes 2, -2, 1, and -1 by the elements of some semigroup of order 4. For example, if S is the semigroup of order 4, with elements A={a,aba}, B={b,bab}, C={ab}, and D={ba}, given by the Cayley table
A B C D A A C C A B D B B D C A C C A D D B B D
after substitutions 2 → a, -2 → b, 1→ ab, -1→ ba, the product of mirror-curvesM1={{-2,-2,1,1},{1,2},{-1,1},{-1,-2}} andM2={{-2,-2,1,1},{-1,-2},{1,-1},{2,-1}} will beM1*M2={{-2,-2,1,1},{2,1},{-2,2},{-1,-1}}.
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(a) 2*2 → 2; (b) 2*-2 → 1; (c) 2*1 → 1; (d) 2*-1 → 2; (e) -2*2 → -1; (f) -2*-2 → -2; (g) -2*1 → -2; (h) -2*-1 → -1; (i) 1*2 → 2; (j) 1*-2 → 1; (k) 1*1 → 1; (l) 1*-1 → 2; (m) -1*2 → -1; (n) -1*-2 → -2; (o) -1*1 → -2; (p) -1*-1 → -1.
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Since the elements a, b, ab and ba are idempotents, for every mirror-curve M holds the relationship M*M=M2=M. If M[p,q] is the set of all mirror-curves derived from RG[p,q],
the basis (minimal set of mirror-curves from which M[p,q] can be obtained by the operation of product) is the subset of all mirror-curves with codes consisting only from 2 and -2, i.e. the set of all unlinks belonging to RG[p,q]. The product of two different mirror-curves belonging to the basis is a mirror-curve not belonging to the basis, i.e., a mirror-curve with at least onecrossing. In particular, as the product of mirror-curves containing only vertical and horizontal mirrors we obtain alternating knot or link corresponding to RG[p,q].
Alternating link 3 1 2 1 3 = L10a101 corresponding to RG[4,2] obtained as the product M1*M2={{1,1,1,1},{1,1},{1,1},{1,1}} of mirror-curves M1={{2,2,2,2},{2,2},{2,2},{2,2}} andM2={{-2,-2,-2,-2},{-2,-2},{-2,-2},{-2,-2}}.
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Minimal representation of mirror-curves
If a link L is given by a mirror-curve M=M1*M2, its mirror-image is M'=M2*M1.
Moreover, every mirror curve M can be decomposed a unique way in a product of two mirror-curves representing two Kauffman states, meaning that for every mirror-curve we can obtain exactly one pair of mirror-curves (M1,M2) such that M=M1*M2, and M1 and M2 represent Kauffman states (i.e., mirror-curves containing only 2 and -2 in their codes).