Small knot mosaics and partition matrices
aa r X i v : . [ m a t h . G T ] N ov SMALL KNOT MOSAICS AND PARTITION MATRICES
KYUNGPYO HONG, HO LEE, HWA JEONG LEE, AND SEUNGSANG OH
Abstract.
Lomonaco and Kauffman introduced knot mosaic systemto give a definition of quantum knot system. This definition is intendedto represent an actual physical quantum system. A knot ( m, n )-mosaicis an m × n matrix of mosaic tiles which are T through T depictedas below, representing a knot or a link by adjoining properly that iscalled suitably connected. An interesting question in studying mosaictheory is how many knot ( m, n )-mosaics are there. D m,n denotes thetotal number of all knot ( m, n )-mosaics. This counting is very importantbecause the total number of knot mosaics is indeed the dimension of theHilbert space of these quantum knot mosaics.In this paper, we find a table of the precise values of D m,n for4 ≤ m ≤ n ≤ D m,n n = 4 n = 5 n = 6 m = 4 2594 54 ,
226 1 , , m = 5 4 , ,
954 331 , , m = 6 101 , , , Introduction
The connection between knots and quantum physics has been of greatinterest. One of remarkable discovery in the theory of knots is the Jonespolynomial, and it turned out that the explanation of the Jones polynomialhas to do with quantum theory. The readers refer [3, 4, 5, 6, 9, 11, 15].Lomonaco and Kauffman introduced a knot mosaic system to set the foun-dation for a quantum knot system in the series of papers [10, 12, 13, 14].Their definition of quantum knots was based on the planar projections ofknots and the Reidemeister moves. They model the topological informationin a knot by a state vector in a Hilbert space that is directly constructedfrom knot mosaics. They proposed several questions in [12], and this paperaims to answer to one of them.Throughout this paper the term “knot” means either a knot or a link.We begin by introducing the basic notion of knot mosaics. Let T denote theset of the following eleven symbols which are called mosaic tiles ; PACS numbers . 02.10.Kn, 02.10.Ox, 03.67.-a.The corresponding author(Seungsang Oh) was supported by Basic Science ResearchProgram through the National Research Foundation of Korea(NRF) funded by the Min-istry of Science, ICT & Future Planning(MSIP) (No. 2011-0021795).This work was supported by the National Research Foundation of Korea(NRF) grantfunded by the Korea government(MEST) (No. 2011-0027989). T T T T T T T T T T T For positive integers m and n , we define an ( m, n ) -mosaic as an m × n matrix M = ( M ij ) of mosaic tiles. We denote the set of all ( m, n )-mosaicsby M ( m,n ) . Obviously M ( m,n ) has 11 mn elements. Indeed this rectangularversion of knot ( m, n )-mosaics is a generalization of a square version of knot n -mosaics.A connection point of a tile is defined as the midpoint of a mosaic tileedge which is also the endpoint of a curve drawn on the tile. Then each tilehas zero, two or four connection points as illustrated in the following figure;Two tiles in a mosaic are called contiguous if they lie immediately nextto each other in either the same row or the same column. A mosaic is saidto be suitably connected if any pair of contiguous mosaic tiles have or donot have connection points simultaneously on their common edge. Notethat this definition is slightly different from the original definition in [12],in which boundary edges of a mosaic do not have connection points. Thisnew definition is convenient to define a quasimosaic (in Section 2.) which issuitably connected and allows connection points on boundary edges. A knot ( m, n ) -mosaic is a suitably connected ( m, n )-mosaic whose boundary edgesdo not have connection points. Then this knot ( m, n )-mosaic represents aspecific knot. The examples of mosaics in Figure 1 are a non-knot (4 , , Figure 1.
Examples of mosaicsLet K ( m,n ) denote the subset of M ( m,n ) of all knot ( m, n )-mosaics. Oneof the problems in studying mosaic theory is how many knot ( m, n )-mosaicsare there. Let D m,n denote the total number of elements of K ( m,n ) . Indeedthe original definition of D n,n for m = n is the dimension of the Hilbertspace of quantum knot ( n, n )-mosaics. The main theme in this paper is toestablish a table of the precise values of D m,n for small m and n by usingthe partition matrix argument. Lomonaco and Kauffman [12] showed that D , = 1, D , = 2 and D , = 22, and presented a complete list of K (3 , . MALL KNOT MOSAICS AND PARTITION MATRICES 3
The authors [1] found the precise value D , = 2594, and also a lowerbound and an upper bound on D n,n for n ≥ D m,n for m, n ≥ Theorem 1. [1]
For m, n ≥ , ( m − n − ≤ · m − + 1)(9 · n − + 1) · D m,n ≤ . ( m − n − . Also we can easily get the precise values of D m,n for small m = 1 , , D ,n is obtained from Theorem 1 by applying m = 3 directly. Corollary 2.
For m = 1 , , and a positive integer n , • D ,n = 1 • D ,n = 2 n − for n ≥ • D ,n = (9 · n − + 1) for n ≥ D m,n for m, n = 4 , , D m,n = D n,m . In Section 4, we create two partition matriceswhich turn out to be remarkably efficient to count small knot mosaics. Theorem 3.
For ≤ m ≤ n ≤ , D m,n ’s are as follows; D m,n n = 4 n = 5 n = 6 m = 4 2594 54 ,
226 1 , , m = 5 4 , ,
954 331 , , m = 6 101 , , , D , , D , and D , byusing a combination of counting techniques and computer algorithms.Recently the authors [2] announced that they constructed an algorithmgiving the precise value of D m,n for m, n ≥ mosaic number m ( K )of a knot K as the smallest integer n for which K is representable as aknot ( n, n )-mosaic. For example, the mosaic number of the trefoil is 4 as isillustrated in Figure 1. They asked “Is this mosaic number related to thecrossing number of a knot?” The authors [8] established an upper bound onthe mosaic number as follows; If K be a nontrivial knot or a non-split linkexcept the Hopf link, then m ( K ) ≤ c ( K ) + 1. Moreover if K is prime andnon-alternating except the 6 link, then m ( K ) ≤ c ( K ) −
1. Note that themosaic numbers of the Hopf link and the 6 link are 4 and 6 respectively.2. Sets of quasimosaics of nine types A quasimosaic is a part of a mosaic where mosaic tiles are located at aparticular places of connected M ij ’s and these tiles are suitably connected.A quasimosaic does not need to be rectangular. Especially a rectangularquasimosaic is called ( p, q )-quasimosaic if it consists of p rows and q columns, K. HONG, H. LEE, H. J. LEE, AND S. OH and let Q ( p,q ) denote the set of all ( p, q )-quasimosaics. A ( p, q )-quasimosaicis a submosaic of a knot mosaic in Lomonaco and Kauffman’s definition.An edge e on a quasimosaic will be marked by “x” if it does not have aconnection point and “o” if it has. Sometimes we use a word of x and o tomark several edges together like e e = xo which means that edge e doesnot have a connection point but edge e has. Choice rule.
Each M ij in a suitably connected mosaic has four choices T , T , T or T of mosaic tiles if its boundary has four connection points, and itis uniquely determined if it has zero or two connection points. Furthermoreit can not have odd number of connection points on its boundary. Now we introduce useful sets of quasimosaics of nine types named P through P . As in Figure 2, let M ∗ be some M ij of a given quasimosaic,and five edges e through e are its typical edges in each type. A set ofquasimosaics of type P consists of single mosaic tiles M ∗ with the restrictionon the related edges e and e so that e e = oo (more precisely, for each fixedone among xx, xo or ox). A set of quasimosaics of type P is defined similarlywith the condition e e = oo. Sets of quasimosaics of next four types P through P consist of two contiguous mosaic tiles with the restriction e e =oo and e = x, e e = oo and e = o, e e e = oox, and e e e = ooo,respectively. Sets of quasimosaics of last three types P , P and P consistof three contiguous mosaic tiles, not on the same row or the same column,with the restriction e e = oo and e e = oo, e e = oo and e e = oo (or e e = oo and e e = oo), and e e e e = oooo, respectively. Note that thisset is exhaustive. For, there are two types for single mosaic tiles ( e e iseither oo or not), four types for two contiguous mosaic tiles ( e e is eitheroo or not, and e is x or o) and three types for three contiguous mosaictiles ( e e is either oo or not, and e e is either oo or not) where type P comprises two symmetric cases. | P i | denotes the number of elements of a set of quasimosaics of type P i for each i . Lemma 4.
Each type P i has following values; | P | = 2 , | P | = 5 , | P | = 4 , | P | = 7 , | P | = 10 , | P | = 22 , | P | = 11 , | P | = 32 and | P | = 98 .Proof. | P | and | P | can be obtained easily from the fact that the mosaictile at M ∗ has 2 choices among 11 mosaic tiles if e e = oo, and 5 choices if e e = oo.For type P or P , the mosaic tile at M ∗ has 4 or 7 choices depending on e = x or o, respectively. After this mosaic tile is settled, the contiguousmosaic tile must be uniquely determined because of e e = oo by Choicerule. The arrows in the figures indicate that the mosaic tiles at arrowheadsare “uniquely determined”. For type P , we distinguish into two cases e =x or o. In either case, the mosaic tile at M ∗ has 2 choices, but the contiguousmosaic tile is uniquely determined or has 4 choices, respectively. So a setof this type has 10 kinds of quasimosaics in total. For type P , we similarlydistinguish into two cases e = x or o. When e = x, the mosaic tile at M ∗ The same notation P i ’s are used for planar isotopy moves on knot mosaics in theoriginal Lomonaco and Kauffman’s paper [12] MALL KNOT MOSAICS AND PARTITION MATRICES 5 type P type P type P e nottype P not e e type P M * e x not M * e e type P not e e M * e e type P notnot type P type P x e Figure 2.
Sets of quasimosaics of nine types. The notation‘not oo’ with an arrowed half circle, for example, in type P means that e e is one among xx, xo or ox. The numbers incircles indicate the number of choices of mosaic tiles in M ij ’swhere the circles take possession. We draw an arrow startedat a numbered circle when the mosaic tile at the arrowheadis uniquely determined.has 2 choices and the contiguous mosaic tile is uniquely determined. When e = o, the mosaic tile at M ∗ has 5 choices and the contiguous mosaic tilehas 4 choices. So a set of this type has 22 kinds of quasimosaics.For type P , the mosaic tile at M ∗ has 11 choices, and the two contiguousmosaic tiles are uniquely determined after the first mosaic tile is settled.For type P , we distinguish into two cases e = x or o. In either case, themosaic tile at the second row is uniquely determined. But two contiguousmosaic tiles at the first row is in type P or P depending on e = x or o.So a set of this type has 32 kinds of quasimosaics. Finally for type P , wedistinguish into two cases e = x or o. When e = x, two contiguous mosaictiles at the first row are in type P , and the mosaic tile at the second row isuniquely determined. When e = o, two contiguous mosaic tiles at the firstrow are in type P , and the mosaic tile at the second row has 4 choices. Soa set of this type has 98 kinds of quasimosaics. (cid:3) K. HONG, H. LEE, H. J. LEE, AND S. OH D , = 4 , , D , which can not be handled inthe argument proving the other cases. Let Q (3 , denote the set of (3 , M ij ’s where i, j = 2 , ,
4. Wename the interior twelve edges as in Figure 3. M b M b M M b M b M M b M b M b b a a a a Figure 3. (3 , a i ’s. See Figure 4.First we consider the case of a a a a = xxxx. As the first figure, all of M , M , M and M are pieces of quasimosaics of type P . We will saythis briefly as ( M , M , M , M ) is of type ( P , P , P , P ). This meansthat each of M , M , M and M has 11 choices independently, andthen four contiguous mosaic tiles M , M , M and M are uniquely de-termined by Choice rule. These produce 11 = 14 ,
641 kinds of quasimosaicsin total.Now consider the case of a a a a = oxox or xoxo (assume the for-mer) as four figures in the second row of the figure. When b b = xx, xo,ox and oo, ( M , M , M , M ) is of type ( P , P , P , P ), ( P , P , P , P ),( P , P , P , P ) and ( P , P , P , P ) respectively. These four occasions pro-duce (11 · + 32 · · · · · · · ) × ,
648 kindsof quasimosaics. We multiplied by 2 because of the two possible choices of a a a a .Next consider the case of a a a a = xxoo, xoox, ooxx or oxxo (as-sume the first one) as four figures in the third row. When b b = xx, xo,ox and oo, ( M , M , M , M ) is of type ( P , P , P , P ), ( P , P , P , P ),( P , P , P , P ) and ( P , P , P , P ) respectively. These four occasions pro-duce (11 · · · · · · · × ,
880 kindsof quasimosaics. We multiplied by 4 because of the four possible choices of a a a a .Lastly consider the case of a a a a = oooo as in the last two rows inthe figure. When b b b b = xxxx, xxxo (and similarly for xxox, xoxxor oxxx), xxoo (and similarly for ooxx), xoox (and similarly for oxxo),xoxo (and similarly for oxox), xooo (and similarly for oxoo, ooxo or ooox)and oooo, ( M , M , M , M ) has type ( P , P , P , P ), ( P , P , P , P ),( P , P , P , P ), ( P , P , P , P ), ( P , P , P , P ), ( P , P , P , P ) and ( P , P , P , P )respectively. These sixteen occasions produce (11 · + 4 · · · + 2 · · MALL KNOT MOSAICS AND PARTITION MATRICES 7
11 1111 11 xx xx
11 44 11 xx xx
11 47 xx x x xx
32 74 11 x x
32 732711 1111 2 xx x x
11 112 xx x xx x xx
32 3211 112 2 x x x x
112 2 x x x
32 112 2 x x
98 5 2 x x
322 2 x x x
32 98 5 59898
Figure 4.
All possible (3 , a a a a where the four groups are drawn at the first, the second, thethird and the last two rows, respectively.11 · + 2 · · · · · + 4 · · · · · ) × , ,
808 kindsof quasimosaics. We multiplied by 4 because of the four possible choices ofmosaic tiles of M by Choice rule.By summing all up, we got that the total number of elements of Q (3 , is 2,091,977. The following rule is very useful to get knot mosaics from aquasimosaic. Lemma 5 ( Twofold rule ) . A ( p, q ) -quasimosaic can be extended to exactlytwo knot ( p + 2 , q + 2) -mosaics.Proof. A ( p, q )-quasimosaic can be extended to knot ( p + 2 , q + 2)-mosaics byadjoining proper mosaic tiles surrounding it, called boundary mosaic tiles. K. HONG, H. LEE, H. J. LEE, AND S. OH
Since each mosaic tile has even number of connection points, suitable con-nectedness guarantee that this ( p, q )-quasimosaic has exactly even numberof connection points on its boundary. To make a knot ( p + 2 , q + 2)-mosaic,all these connection points must be connected pairwise via mutually disjointarcs when we adjoin boundary mosaic tiles. There are exactly two ways todo as illustrated in Figure 5. (cid:3) or Figure 5.
Twofold ruleFinally we get D , = 4 , ,
954 which is twice of the total number ofelements of Q (3 , by Twofold rule.4. Partition matrices A partition matrix P ( p,q ) for the set Q ( p,q ) of all ( p, q )-quasimosaics is a2 q × p matrix ( N ij ) where every row (or column) is related to the pres-ence of connection points on the q bottom (or p rightmost, respectively)edges. Roughly speaking, each N ij is the number of all ( p, q )-quasimosaicswhose bottom edges and rightmost edges have specific presences of connec-tion points associated to the i -th and the j -th in some order, respectively.In this section, we introduce two partition matrices P (1 , and P (2 , whichwould play an important role in finding the precise values of D m,n for m, n =4 , , D , . In Section 5. we build (2 , , , , , , , , D , , D , , D , , D , and D , (but not D , ).First we establish a partition matrix P (1 , for Q (1 , . For an (1 , b , b and r as the left figure in Figure 6. A partition matrix P (1 , is a 4 × N ij ) where every row is related to b b and every columnis related to r as follows; N ij is the number of all (1 , i -th b b in the order of xx, xo, ox and oo, and the j -th r in theorder of x and o. For an example, the family of four (1 , N where b b = xx and r = o are illustrated on the right in Figure 6. Notethat the sum of all entries of P (1 , is the number of elements of Q (1 , . Lemma 6. P (1 , = MALL KNOT MOSAICS AND PARTITION MATRICES 9 b b r xxxx xx xx Figure 6. (1 , N where b b = xx and r = o. Proof.
The proof follows from Lemma 4 directly considering types P , P , P and P . (cid:3) Next we establish another partition matrix P (2 , for Q (2 , . For an (2 , b , b , r and r , and two interior edges by c and c as in Figure 7. Apartition matrix P (2 , is a 4 × N ′ ij ) where every row is related to b b and every column is related to r r as follows; N ′ ij is the number of all(2 , i -th b b , and the j -th r r in the sameorder as previous. b b r r c c M * c M * Figure 7. (2 , Lemma 7. P (2 , =
22 22 43 4322 55 43 13943 43 109 6443 139 64 403
Proof.
Let M ∗ denote the bottom and the right side mosaic tile, and M c ∗ therest quasimosaic consisting of three mosaic tiles. First, we consider the case b r = xx for N ′ , N ′ , N ′ and N ′ . In this case, M c ∗ always has type P .For every four choices of b r , the related c c has two choices. For example,when b r = ox, c c must be either ox or xo. Furthermore for each choiceof c c , M ∗ is uniquely determined by Choice rule, except when b r = ooand c c = oo. M ∗ in the exceptional case has four choices of mosaic tiles.Thus we get N ′ , N ′ , N ′ = 11 × N ′ = 11 + 11 × b r = xo for N ′ , N ′ , N ′ and N ′ . Similarlyfor every four choices of b r , the related c c has two choices. In this case, M c ∗ has type P if c = x, and type P if c = o. For each choiceof c c , M ∗ is uniquely determined, except when b r = oo and c c = oo,implying that M ∗ has four choices. Thus we get N ′ , N ′ , N ′ = 11 + 32and N ′ = 11 + 32 ×
4. The case b r = ox for N ′ , N ′ , N ′ and N ′ willbe handled in the same manner.Finally, consider the case b r = oo for the rest four entries of P (2 , . Inthis case, M c ∗ possibly has three types P , P P according to c c . And M ∗ is uniquely determined, except when b r c c = oooo. Thus we get N ′ = 11 + 98, N ′ = 32 × N ′ = 32 × N ′ = 11 + 98 × (cid:3) Proof of Theorem 3
In this section, we apply partition matrices P (1 , and P (2 , to find theprecise values of D m,n for m, n = 4 , ,
6, except D , . For a matrix P = ( N ij ), k P k denote the sum of all entries of P , and [ P ] = ( N ij ).Note that k P ( p,q ) k is the total number of elements of Q ( p,q ) for ( p, q ) =(1 ,
2) or (2 , Q ( p,q ) can be extended to exactlytwo knot ( p + 2 , q + 2)-mosaics by Twofold rule. Conversely, every knot ( p +2 , q + 2)-mosaics can be obtained by extending a proper ( p, q )-quasimosaic in Q ( p,q ) . Thus we can conclude D m,n = 2 k P ( m − ,n − k . So D , = 2 k P (2 , k =2594.5.1. Partition matrix multiplying argument.Let P (1 , = ( N ij ) and P (2 , = ( N ′ ij ). Consider a (2 , Q in Q (2 , . We name three boundary edges on the bottom by b , b and r ′ asupper figures in Figure 8. Let Q l and Q r be the (2 , Q and the (2 , Q l on the right by r and r , and other two boundary edges of Q r on the leftby b ′ and b ′ . Then r r of Q l must be the same as b ′ b ′ of Q r . Remark that Q l is an element of Q (2 , and Q r , after rotating 90 ◦ counter-clockwise, isan element of Q (1 , . N ik is the number of elements of Q (2 , which have the i -th b b and the k -th r r in the order of xx, xo, ox and oo, and N ′ kj is thenumber of elements of Q (1 , which have the k -th b ′ b ′ in the order of xx,xo, ox and oo, and the j -th r ′ in the order of x and o. Thus P k =1 N ik N ′ kj is the number of elements of Q (2 , which have the i -th b b and the j -th r ′ .Indeed it is the i -th row and the j -th column entry of P (2 , · P (1 , . Thisimplies that k P (2 , · P (1 , k is the total number of Q (2 , . Now we concludethat D , = 2 k P (2 , · P (1 , k = 54 , , Q ′ in Q (2 , . We name four boundaryedges on the bottom by b , b , r ′ and r ′ as lower figures in Figure 8. Let Q ′ l and Q ′ r be the (2 , Q ′ and from the right two columns, respectively. We name again fourboundary edges of Q ′ l and Q ′ r as previous. Again r r of Q ′ l must be thesame as b ′ b ′ of Q ′ r . Remark that Q ′ l and Q ′ r is elements of Q (2 , . Similarlywe rotate Q ′ r ◦ counter-clockwise. Thus P k =1 N ik N kj is the number ofelements of Q (2 , which have the i -th b b and the j -th r ′ r ′ . This impliesthat D , = 2 k P (2 , · P (2 , k = 1 , , MALL KNOT MOSAICS AND PARTITION MATRICES 11 b b r Q r b b r r r r b b Q l Q r b b Q b b r r b b Q l Q r r r Figure 8.
Dividing a (2 , , , , P (2 , · P (1 , = ( N ij ). Consider a (4 , Q in Q (4 , . Wename three interior edges on the middle by b , b and r as upper figures inFigure 9. Let Q u and Q l be the (2 , Q and from the lower two rows, respectively. We name againthree boundary edges of Q u on the bottom by b ′ , b ′ and r ′ , and other threeboundary edges of Q l on the top by b ′′ , b ′′ and r ′′ , so that b ′ b ′ r ′ = b ′′ b ′′ r ′′ .We reflect Q l through a horizontal line. Remark that Q u and Q l is elementsof Q (2 , . N ij is the number of elements of Q (2 , which have the i -th b ′ b ′ andthe j -th r ′ . Thus N ij is the number of elements of Q (4 , which have the i -th b b and the j -th r . This implies that D , = D , = 2 k [ P (2 , · P (1 , ] k =331 , , P (2 , · P (2 , = ( N ij ). Consider a (4 , Q in Q (4 , .We name four interior edges on the middle by b , b , r and r as lowerfigures in Figure 9. The similar argument as previous guarantees that N ij isthe number of elements of Q (4 , which have the i -th b b and the j -th r r .This implies that D , = 2 k [ P (2 , · P (2 , ] k = 101 , , , Conclusion
In this paper, we found the cardinality of knot ( m, n )-mosaics D m,n for m, n = 4 , ,
6. Mainly we build sets of quasimosaics of nine types to calcu-late two partition matrices related to (1 , , D m,n increases rapidly so that D , is larger than10 . In Section 5. we introduce partition matrix multiplying argument andsquaring argument to find D m,n for only m, n = 4 , , D m,n for larger m, n by applyingthe arguments here. For example, partition matrices for (1 , , Q b rb Q u Q l b rb b b rQ b r r b Q u Q l b r r b b b r r Figure 9.
Dividing a (4 , , , , , D m,n for m, n = 4 , , , ,
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Estimating Jones polynomials is a complete problemfor one clean qubit , Quantum Inform. Comput. (2008) 681–714. Department of Mathematics, Korea University, Anam-dong, Sungbuk-ku,Seoul 136-701, Korea
E-mail address : [email protected] Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu,Daejeon 305-701, Korea
E-mail address : [email protected] Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu,Daejeon 305-701, Korea
E-mail address : [email protected] Department of Mathematics, Korea University, Anam-dong, Sungbuk-ku,Seoul 136-701, Korea
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