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RIBBON TILE INVARIANTS from SIGNED AREA Cristopher Moore

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RIBBON TILE INVARIANTS from SIGNED AREA Cristopher Moore RIBBON TILE INVARIANTS FROM SIGNED AREA Cristopher Moore Computer Science Department and Department of Physics and Astronomy University of New Mexico Albuquerque NM mo orecsunmedu Igor Pak Department of Mathematics MIT Cambridge MA pakmathmitedu May Abstract Ribb on tiles are p olyomino es consisting of n squares laid out in a path each step of which go es north or east Tile invariants were rst intro duced in P where a full basis of invariants of ribb on tiles was conjectured Here we present a complete pro of of the conjecture which works by asso ciating ribb on tiles with certain p olygons in the complex plane and deriving invariants from the signed area of these p olygons Introduction Polyomino tilings have b een an ob ject of attention of serious mathematicians as well as amateurs for many decades G Recently however the interest in tiling problems has grown as some imp ortant ideas and techniques have b een intro duced In P the second author intro duced a tile counting group which app ears to enco de a large amount of information concerning the combinatorics of tilings He made a conjecture on the group structure and obtained several partial results A sp ecial case of the conjecture was later resolved in MP In this pap er we continue this study and complete the pro of of the conjecture Consider the set of ribbon tiles T dened as connected nsquare tiles with no n two squares in the same diagonal x y c as in the gures b elow It is easy to n1 see that jT j as each tile can b e asso ciated with a path of length n in n the square lattice each step of which go es east or north Recording these moves by n1 and resp ectively we obtain a sequence f g which 1 n1 uniquely enco des a ribb on tile We will refer to this tile as Key words and phrases Polyomino tilings tile invariants Conway group height representation Typ eset by A ST X M E CRISTOPHER MOORE IGOR PAK 0 1 Figure Two domino es 00 10 01 11 Figure Four ribb on tromino es 000 010 001 011 100 001 101 111 Figure Eight ribb on tetromino es Now let b e a nite simply connected region and let b e a tiling of by ribb on tiles in T n We denote by a the numb er of times the ribb on tile n is used in Conjecture P Let and be as above Then for every i i n we have X X a a c i : =0 =1 : =1 =0 i ni i ni where the c depend only on and are independent of the tiling of Fur i thermore when n is even we have X a c mo d : =1 n2 RIBBON TILE INVARIANTS FROM SIGNED AREA where c is also independent of The main result of the pap er is a pro of of this conjecture for all n Theorem Conjecture holds for tilings by ribbon tiles T for al l n n and for al l simply connected regions A few words ab out the history of this conjecture For n it implies that for every domino tiling of the parity of the numb er of vertical domino es is always the same This in fact holds for every region not just the simply connected ones and follows from a folklore coloring argument see GP for details For n the conjecture gives only one relation a a c 01 10 1 This is the celebrated ConwayLagarias relation for tromino es CL Recently the conjecture was established for n MP using a combinatorial technique similar to CL In this notation it was shown in MP that a a a a c 001 011 101 111 1 a a a a c mo d 010 011 110 111 It was shown in CL in a certain rigorous sense that even for n the conjecture cant b e proved by means of coloring arguments This was extended by the second author to all n P It was observed in P that for n there exists a nonsimply connected region for which the relations in the conjecture do not hold Thus there is little hop e of generalizing the conjecture to all regions The conjecture originated in P where the author considered only row or column convex regions and proved the linear relations in Conjecture for all such P Theorem The technique used a connection with combinatorics of Young tableaux which could not b e extended to all simply connected regions see P for details The author in P also showed that the linear relations in the conjecture are the only relations which can o ccur b etween the a even for this smaller set of regions see section b elow Ab out the pro of technique We use notion of tile invariants intro duced in P but here we dene new realvalued invariants which we call adele invariants As it turns out these invariants imply all the integervalued invariants that we need to establish We then show the validity of the adele invariants by presenting them as a signed area of a certain p olygon corresp onding to each tile These two results together imply Theorem The rest of the pap er is structured as follows In section we intro duce tile invariants and compute the tile counting group based on Theorem Much of the material follows P so we present only sketches of the pro ofs for completeness In section we dene and study the adele invariants Small examples are computed in section We exhibit the relationship b etween the adele invariants and integer invariants in section This completes the pro of of Theorem We conclude with nal remarks in section CRISTOPHER MOORE IGOR PAK Tile invariants Let us start by dening tilings and tile invariants Let b e a set of closed 2 squares of a square grid Z on a plane A region is a nite subset Region is called simply connected if its b oundary is connected We say that two regions and are equivalent denoted if is a parallel translation of e rotations and reections are not allowed Let f g b e the set of regions equivalent to Let T f g b e a nite set of simply connected regions which we call 1 r e tiles By e we denote the set of their parallel translations and let T e A i i i e tiling of denoted is a set of tiles T such that their disjoint union is G Here we ignore the intersection of the b oundaries Let G b e an ab elian group and let T G b e any map We extend the e denition of to all T by setting for all We say that the i i map is a tile invariant of T if for every simply connected region and every tiling by the set of tiles T we have X c where the constant on the rhs dep ends only on the region and is indep endent of In this pap er G is either Z or Z ZnZ or R with addition as the group n op eration Tile invariants are directly related to numerical relations b etween the resp ective numb ers of times dierentlyshap ed tiles o ccur in a tiling Indeed let a j e j i i b e the numb er of tiles in the tiling We immediately have i r X X a c i i i=1 In P we intro duced a tile counting group G T which is dened as a quotient r G T Z a a a a 1 1 r r where are tilings of the same simply connected region by the set of tiles T Computing the tile counting group G T is a dicult task even in simple cases The main result of this pap er is a computation of G T for the case of ribb on n tiles m+1 Theorem If n m then G T Z If n m then G T n n m Z Z 2 Theorem was stated as a conjecture in P It was shown in P that it follows from Theorem For completeness we sketch the pro of b elow RIBBON TILE INVARIANTS FROM SIGNED AREA m+1 Sketch of proof Indeed in P Theorem it was shown that G T Z m for n m and G T Z Z for n m Observe that one can view n 2 the relations in Conjecture as elements of G T Recall that these relations n together with the trivial area invariant f dened by f for all T 0 0 n n are indep endent in Z see the pro of of Theorem in P x Now Theorem implies the result Before we conclude this section let us make a nal observation on the relations in Conjecture implied by previous work Following P x dene the shade invariant as follows n1 X f k mo d n H k k =1 where The fact that it is an invariant follows easily from an 1 n1 extended coloring argument P x Namely consider a coloring of the squares 2 Z Z dened by x y y mo d n Note that the sum of the colors in each n ribb on tile is equal to f C where C C n Z is a constant which H n 1 dep ends only on n We omit the easy details Prop osition When n is even the relations in the rst part of Conjecture imply that in the second part Proof We will show that the mo d relation follows from the m n relations in the rst part and the shade invariant In the language of invariants consider the k convexity invariants f intro duced in P k f where k k nk 1 n1 We need to show that the shade invariant and the k convexity invariants generate the parity invariant f f mo d where n m m But this is immediate since f mo d f f m f f mo d H 1 2 m1 cf P x This completes the pro of New ribbon tile invariants and the signed area Let T b e the set of ribb on tiles dened as ab ove From now on we will also use n n1 a dierent enco ding of T by sequences fg T f g n 1 n n where if for all i n ie and i i 1 In contrast with other ribb on tile invariants we intro duce the shade invariant can b e extended to al l regions not just the simply connected ones P Theorem CRISTOPHER MOORE IGOR PAK For every n we dene a function T R as follows n n1 X k sin k n k =1 where fg as ab ove The main result of this section is 1 n1 k the following key observation Theorem The function T R is a tile invariant for the set T of n n ribbon tiles for al l n We will call the th adele invariant Note that when n m we have for
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