summaryrefslogtreecommitdiff
path: root/Build/source/utils/asymptote/base/syzygy.asy
blob: dce87805c9d8c5ad64b23f94e43a00ed6fc821d3 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
/***** syzygy.asy {{{1
 * Andy Hammerlindl  2006/12/02
 *
 * Automates the drawing of braids, relations, and syzygies, along with the
 * corresponding equations.
 *
 * See
 *   http://katlas.math.toronto.edu/drorbn/index.php?title=06-1350/Syzygies_in_Asymptote
 * For more information.
 *****/
struct Component { // {{{1
  // The number of strings coming in or out of the component.
  int in;
  int out;

  // Which 'out' string each 'in' string is connected to.  For deriving
  // equations.
  int[] connections;

  string symbol;    // For pullback notation.
  string lsym;      // For linear equations.
  string codename;  // For Mathematica code.

  guide[] draw(picture pic, guide[] ins);
}

// Utility functions {{{1
pair[] endpoints(guide[] a) {
  pair[] z;
  for (int i=0; i<a.length; ++i)
    z.push(endpoint(a[i]));
  return z;
}

pair min(pair[] z) {
  pair m=(infinity, infinity);
  for (int i=0; i<z.length; ++i) {
    if (z[i].x < m.x)
      m=(z[i].x,m.y);
    if (z[i].y < m.y)
      m=(m.x,z[i].y);
  }
  return m;
}

pair max(pair[] z) {
  pair M=(-infinity, -infinity);
  for (int i=0; i<z.length; ++i) {
    if (z[i].x > M.x)
      M=(z[i].x,M.y);
    if (z[i].y > M.y)
      M=(M.x,z[i].y);
  }
  return M;
}

// Component Definitions {{{1
real hwratio=1.4;
real gapfactor=6;

Component bp=new Component;
bp.in=2; bp.out=2;
bp.connections=new int[] {1,0};
bp.symbol="B^+"; bp.lsym="b^+"; bp.codename="bp";
bp.draw=new guide[] (picture pic, guide[] ins) {
  pair[] z=endpoints(ins);
  pair m=min(z), M=max(z);
  real w=M.x-m.x, h=hwratio*w;
  pair centre=(0.5(m.x+M.x),M.y+h/2);

  /*
    return new guide[] {ins[1]..centre{NW}..z[0]+h*N,
    ins[0]..centre{NE}..z[1]+h*N};
  */

  real offset=gapfactor*linewidth(currentpen);
  draw(pic, ins[1]..(centre-offset*NW){NW});
  return new guide[] {(centre+offset*NW){NW}..z[0]+h*N,
                                                ins[0]..centre{NE}..z[1]+h*N};
};
    
Component bm=new Component;
bm.in=2; bm.out=2;
bm.connections=new int[] {1,0};
bm.symbol="B^-"; bm.lsym="b^-"; bm.codename="bm";
bm.draw=new guide[] (picture pic, guide[] ins) {
  pair[] z=endpoints(ins);
  pair m=min(z), M=max(z);
  real w=M.x-m.x, h=hwratio*w;
  pair centre=(0.5(m.x+M.x),M.y+h/2);

  /*
    return new guide[] {ins[1]..centre{NW}..z[0]+h*N,
    ins[0]..centre{NE}..z[1]+h*N};
  */

  real offset=gapfactor*linewidth(currentpen);
  draw(pic, ins[0]..(centre-offset*NE){NE});
  return new guide[] {ins[1]..centre{NW}..z[0]+h*N,
                                            (centre+offset*NE){NE}..z[1]+h*N};
};
    
Component phi=new Component;
phi.in=2; phi.out=1;
phi.connections=new int[] {0,0};
phi.symbol="\Phi"; phi.lsym="\phi"; phi.codename="phi";
phi.draw=new guide[] (picture pic, guide[] ins) {
  pair[] z=endpoints(ins);
  pair m=min(z), M=max(z);
  real w=M.x-m.x, h=hwratio*w;
  pair centre=(0.5(m.x+M.x),M.y+h/2);


  //real offset=4*linewidth(currentpen);
  draw(pic, ins[0]..centre{NE});
  draw(pic, ins[1]..centre{NW});
  draw(pic, centre,linewidth(5*linewidth(currentpen)));
  dot(pic, centre);
  return new guide[] {centre..centre+0.5h*N};
};

Component wye=new Component;
wye.in=1; wye.out=2;
wye.connections=null; // TODO: Fix this!
wye.symbol="Y"; wye.lsym="y"; wye.codename="wye";
wye.draw=new guide[] (picture pic, guide[] ins) {
  pair z=endpoint(ins[0]);
  real w=10, h=hwratio*w; // The 10 is a guess here, and may produce badness.
  pair centre=(z.x,z.y+h/2);


  draw(pic, ins[0]..centre);
  draw(pic, centre,linewidth(5*linewidth(currentpen)));
  return new guide[] {centre{NW}..centre+(-0.5w,0.5h),
                                    centre{NE}..centre+(0.5w,0.5h)};
};


struct Braid { // {{{1
  // Members {{{2
  // Number of lines initially.
  int n;

  struct Placement {
    Component c;
    int place;

    Placement copy() {
      Placement p=new Placement;
      p.c=this.c; p.place=this.place;
      return p;
    }
  }
  Placement[] places;

  void add(Component c, int place) {
    Placement p=new Placement;
    p.c=c; p.place=place;
    places.push(p);
  }

  void add(Braid sub, int place) {
    for (int i=0; i<sub.places.length; ++i)
      add(sub.places[i].c,sub.places[i].place+place);
  }

  // Drawing {{{2
  guide[] drawStep(picture pic, Placement p, guide[] ins) {
    int i=0,j=0;

    // Draw the component.
    Component c=p.c;
    //write("drawing "+c.symbol+" at place "+(string)p.place);
    guide[] couts=c.draw(pic, ins[sequence(c.in)+p.place]);

    pair M=max(endpoints(couts));

    // Extend lines not in the component.
    guide[] outs;
    pair[] z=endpoints(ins);
    while (i<p.place) {
      outs.push(ins[i]..(z[i].x,M.y));
      ++i;
    }

    outs.append(couts);
    i+=c.in;

    while (i<ins.length) {
      outs.push(ins[i]..(z[i].x,M.y));
      ++i;
    }

    return outs;
  }

  void drawEnd(picture pic, guide[] ins, real minheight=0) {
    pair[] z=endpoints(ins);
    for (int i=0; i<ins.length; ++i) {
      draw(pic, z[i].y >= minheight ? ins[i] : ins[i]..(z[i].x,minheight));
    }
  }

  void draw(picture pic, guide[] ins, real minheight=0) {
    int steps=places.length;

    guide[] nodes=ins;
    for (int i=0; i<steps; ++i) {
      Placement p=places[i];
      nodes=drawStep(pic, places[i], nodes);
    }

    drawEnd(pic, nodes, minheight);
  }

  void draw(picture pic=currentpicture, real spacing=15,
            real minheight=2hwratio*spacing) {
    pair[] ins;
    for (int i=0; i<n; ++i)
      ins.push((spacing*i,0));

    draw(pic, ins, minheight);
  }

  // Utilities {{{2
  int in() {
    return n;
  }
  int out() {
    int steps=places.length;
    int num=n; // The number of nodes at this step.

    for (int i=0; i<steps; ++i) {
      Placement p=places[i];
      int nextNum=num-p.c.in+p.c.out;
      num=nextNum;
    }
    return num;
  }

  // Deep copy of a braid.
  Braid copy() {
    Braid b=new Braid;
    b.n=this.n;
    for (int i=0; i<this.places.length; ++i)
      b.add(this.places[i].c,this.places[i].place);
    return b;
  }

  // Matching {{{2
  // Tests if a component p can be swapped with a component q which is assumed
  // to be directly above it.
  static bool swapable(Placement p, Placement q) {
    return  p.place + p.c.out <= q.place || // p is left of q or
      q.place + q.c.in <= p.place;    // q is left of p
  }

  // Creates a new braid with a transposition of two components.
  Braid swap(int i, int j) {
    if (i>j)
      return swap(j,i);
    else {
      assert(j==i+1); assert(swapable(places[i],places[j]));

      Placement p=places[i].copy();
      Placement q=places[j].copy();
      /*write("swap:");
        write("p originally at " + (string)p.place);
        write("q originally at " + (string)q.place);
        write("p.c.in: " + (string)p.c.in + " p.c.out: " + (string)p.c.out);
        write("q.c.in: " + (string)q.c.in + " q.c.out: " + (string)q.c.out);*/
      if (q.place + q.c.in <= p.place)
        // q is left of p - adjust for q renumbering strings.
        p.place+=q.c.out-q.c.in;
      else if (p.place + p.c.out <= q.place)
        // q is right of p - adjust for p renumbering strings.
        q.place+=p.c.in-p.c.out;
      else
        // q is directly on top of p
        assert(false, "swapable");

      /*write("q now at " + (string)q.place);
        write("p now at " + (string)p.place);*/

      Braid b=this.copy();
      b.places[i]=q;
      b.places[j]=p;
      return b;
    }
  }

  // Tests if the component at index 'start' can be moved to index 'end'
  // without interfering with other components.
  bool moveable(int start, int end) {
    assert(start<places.length); assert(end<places.length);
    if (start==end)
      return true;
    else if (end<start)
      return moveable(end,start);
    else {
      assert(start<end);
      Placement p=places[start].copy();
      for (int step=start; step<end; ++step) {
        Placement q=places[step+1];
        if (q.place + q.c.in <= p.place)
          // q is left of p - adjust for q renumbering strings.
          p.place+=q.c.out-q.c.in;
        else if (p.place + p.c.out <= q.place)
          // q is right of p - nothing to do.
          continue;
        else
          // q is directly on top of p
          return false;
      }
      return true;
    }
  }

  bool matchComponent(Braid sub, int subindex, int place, int step) {
    int i=subindex;
    return sub.places[i].c == this.places[step].c &&
      sub.places[i].place + place == this.places[step].place;
  }

  // Returns true if a sub-braid occurs within the one at the specified
  // coordinates with no component occuring anywhere inbetween.
  bool exactMatch(Braid sub, int place, int step) {
    for (int i=0; i<sub.places.length; ++i) {
      if (!matchComponent(sub, i, place, i+step)) {
        write("match failed at iteration: ", i);
        return false;
      }
    }
    return true;
  }

  /*
    bool findSubsequence(Braid sub, int place, int size, int[] acc) {
    // If we've matched all the components, we've won.
    if (acc.length >= sub.places.length)
    return true;

    // The next component to match.
    Placement p=sub.places[acc.length];

    // Start looking immediately after the last match.
    for (int step=acc[acc.length-1]+1; step<this.places.length; ++step) {
    Placement q=this.places[step];
  */

  bool tryMatch(Braid sub, int place, int size, int[] acc) {
    // If we've matched all the components, we've won.
    if (acc.length >= sub.places.length)
      return true;

    // The next component to match.
    Placement p=sub.places[acc.length];

    // Start looking immediately after the last match.
    for (int step=acc[acc.length-1]+1; step<this.places.length; ++step) {
      Placement q=this.places[step];
      // Check if the next component is in the set of strings used by the
      // subbraid.
      if (q.place + q.c.in > place && q.place < place + size) {
        // It's in the window, so it must match the next component in the
        // subbraid.
        if (p.c==q.c && p.place+place==q.place) {
          // A match - go on to the next component.
          acc.push(step);
          return tryMatch(sub, place, size, acc); // TODO: Adjust place/size.
        }
        else
          return false;
      }

      // TODO: Adjust place and size.
    }

    // We've run out of components to match.
    return false;
  }


  // This attempts to find a subbraid within the braid.  It allows other
  // components to be interspersed with the components of the subbraid so long
  // as they don't occur on the same string as the ones the subbraid lies on.
  // Returns null on failure.
  int[] match(Braid sub, int place) {
    for (int i=0; i<=this.places.length-sub.places.length; ++i) {
      // Find where the first component of the subbraid matches and try to
      // match the rest of the braid starting from there.
      if (matchComponent(sub, 0, place, i)) {
        int[] result;
        result.push(i);
        if (tryMatch(sub,place,sub.n,result))
          return result;
      }
    }
    return null;
  }

  // Equations {{{2
  // Returns the string that 'place' moves to when going through the section
  // with Placement p.
  static int advancePast(Placement p, int place) {
    // If it's to the left of the component, it is unaffected.
    return place<p.place ? place :
      // If it's to the right of the component, adjust the numbering due
      // to the change of the number of strings in the component.
      p.place+p.c.in <= place ? place - p.c.in + p.c.out :
      // If it's in the component, ask the component to do the work.
      p.place + p.c.connections[place-p.place];
  }

  // Adjust the place (at step 0) to the step given, to find which string it is
  // on in that part of the diagram.
  int advanceToStep(int step, int place) {
    assert(place>=0 && place<n);
    assert(step>=0 && step<places.length);

    for (int i=0; i<step; ++i)
      place=advancePast(places[i], place);

    return place;
  }

  int pullbackWindowPlace(int step, int place,
                          int w_place, int w_size) {
    place=advanceToStep(step,place);
    return place < w_place           ? 1 : // The shielding.
      w_place + w_size <= place ? 0 : // The string doesn't touch it.
      place-w_place+2;
  }

  int pullbackPlace(int step, int place) {
    // Move to the right step.
    //write("advance: ", step, place, advanceToStep(step,place));
    //place=advanceToStep(step,place);
    Placement p=places[step];
    return pullbackWindowPlace(step,place, p.place, p.c.in);
    /*return place < p.place           ? 1 : // The shielding.
      p.place + p.c.in <= place ? 0 : // The string doesn't touch it.
      place-p.place+2;*/
  }
                
  int[] pullbackWindow(int step, int w_place, int w_size) {
    int[] a={1};
    for (int place=0; place<n; ++place)
      a.push(pullbackWindowPlace(step, place, w_place, w_size));
    return a;
  }

  int[] pullback(int step) {
    Placement p=places[step];
    return pullbackWindow(step, p.place, p.c.in);
    /*int[] a={1};
      for (int place=0; place<n; ++place)
      a.push(pullbackPlace(step, place));
      return a;*/
  }

  string stepToFormula(int step) {
    // Determine the pullbacks.
    string s="(1";
    for (int place=0; place<n; ++place)
      //write("pullback: ", step, place, pullbackString(step,place));
      s+=(string)pullbackPlace(step, place);
    s+=")^\star "+places[step].c.symbol;
    return s;
  }

  // Write it as a formula with pullback notation.
  string toFormula() {
    if (places.length==0)
      return "1";
    else {
      string s;
      for (int step=0; step<places.length; ++step) {
        if (step>0)
          s+=" ";
        s+=stepToFormula(step);
      }
      return s;
    }
  }

  string windowToLinear(int step, int w_place, int w_size) {
    int[] a=pullbackWindow(step, w_place, w_size);
    string s="(";
    for (int arg=1; arg<=w_size+1; ++arg) {
      if (arg>1)
        s+=",";
      bool first=true;
      for (int var=0; var<a.length; ++var) {
        if (a[var]==arg) {
          if (first)
            first=false;
          else
            s+="+";
          s+="x_"+(string)(var+1);
        }
      }
    }
    return s+")";
  }

  string windowToCode(int step, int w_place, int w_size) {
    int[] a=pullbackWindow(step, w_place, w_size);
    string s="[";
    for (int arg=1; arg<=w_size+1; ++arg) {
      if (arg>1)
        s+=", ";
      bool first=true;
      for (int var=0; var<a.length; ++var) {
        if (a[var]==arg) {
          if (first)
            first=false;
          else
            s+=" + ";
          s+="x"+(string)(var+1);
        }
      }
    }
    return s+"]";
  }

  string stepToLinear(int step) {
    //int[] a=pullback(step);
    Placement p=places[step];
    return p.c.lsym+windowToLinear(step, p.place, p.c.in);

    /*string s=p.c.lsym+"(";
      for (int arg=1; arg<=p.c.in+1; ++arg) {
      if (arg>1)
      s+=",";
      bool first=true;
      for (int var=0; var<a.length; ++var) {
      if (a[var]==arg) {
      if (first)
      first=false;
      else
      s+="+";
      s+="x_"+(string)(var+1);
      }
      }
      }
      return s+")";*/
  }

  string stepToCode(int step) {
    Placement p=places[step];
    return p.c.codename+windowToCode(step, p.place, p.c.in);
  }

  string toLinear(bool subtract=false) {
    if (places.length==0)
      return subtract ? "0" : "";  // or "1" ?
    else {
      string s = subtract ? " - " : "";
      for (int step=0; step<places.length; ++step) {
        if (step>0)
          s+= subtract ? " - " : " + ";
        s+=stepToLinear(step);
      }
      return s;
    }
  }

  string toCode(bool subtract=false) {
    if (places.length==0)
      return subtract ? "0" : "";  // or "1" ?
    else {
      string s = subtract ? " - " : "";
      for (int step=0; step<places.length; ++step) {
        if (step>0)
          s+= subtract ? " - " : " + ";
        s+=stepToCode(step);
      }
      return s;
    }
  }
}

struct Relation { // {{{1
  Braid lhs, rhs;

  string lsym, codename;
  bool inverted=false;

  string toFormula() {
    return lhs.toFormula() + " = " + rhs.toFormula();
  }

  string linearName() {
    assert(lhs.n==rhs.n);
    assert(lsym!="");

    string s=(inverted ? "-" : "") + lsym+"(";
    for (int i=1; i<=lhs.n+1; ++i) {
      if (i>1)
        s+=",";
      s+="x_"+(string)i;
    }
    return s+")";
  }

  string fullCodeName() {
    assert(lhs.n==rhs.n);
    assert(codename!="");

    string s=(inverted ? "minus" : "") + codename+"[";
    for (int i=1; i<=lhs.n+1; ++i) {
      if (i>1)
        s+=", ";
      s+="x"+(string)i+"_";
    }
    return s+"]";
  }

  string toLinear() {
    return linearName() + " = " + lhs.toLinear() + rhs.toLinear(true);
  }

  string toCode() {
    return fullCodeName() + " :> " + lhs.toCode() + rhs.toCode(true);
  }

  void draw(picture pic=currentpicture) {
    picture left; lhs.draw(left);
    frame l=left.fit();
    picture right; rhs.draw(right);
    frame r=right.fit();

    real xpad=30;

    add(pic, l);
    label(pic, "=", (max(l).x + 0.5xpad, 0.25(max(l).y+max(r).y)));
    add(pic, r, (max(l).x+xpad,0));
  }
}

Relation operator- (Relation r) {
  Relation opposite;
  opposite.lhs=r.rhs;
  opposite.rhs=r.lhs;
  opposite.lsym=r.lsym;
  opposite.codename=r.codename;
  opposite.inverted=!r.inverted;
  return opposite;
}


Braid apply(Relation r, Braid b, int step, int place) {
  bool valid=b.exactMatch(r.lhs,place,step);
  if (valid) {
    Braid result=new Braid;
    result.n=b.n;
    for (int i=0; i<step; ++i)
      result.places.push(b.places[i]);
    result.add(r.rhs,place);
    for (int i=step+r.lhs.places.length; i<b.places.length; ++i)
      result.places.push(b.places[i]);
    return result;
  }
  else {
    write("Invalid match!");
    return null;
  }
}

// Tableau {{{1
frame[] fit(picture[] pics) {
  frame[] f;
  for (int i=0; i<pics.length; ++i) {
    frame ff=pics[i].fit();
    //label(ff, (string)i, (10,10));
    //f.push(pics[i].fit());
    f.push(ff);
  }
  return f;
}

// Draw a number of frames in a nice circular arrangement.
picture tableau(frame[] cards, bool number=false) {
  int n=cards.length;

  // Calculate the max height and width of the frames (assuming min(f)=(0,0)).
  pair M=(0,0);
  for (int i=0; i<n; ++i) {
    pair z=max(cards[i]);
    if (z.x > M.x)
      M=(z.x,M.y);
    if (z.y > M.y)
      M=(M.x,z.y);
  }

  picture pic;
  real xpad=2.0, ypad=1.3;
  void place(int index, real row, real column) {
    pair z=((M.x*xpad)*column,(M.y*ypad)*row);
    add(pic, cards[index], z);
    if (number) {
      label(pic,(string)index, z+(0.5M.x,0), S);
    }
  }

  // Handle small collections.
  if (n<=4) {
    for (int i=0; i<n; ++i)
      place(i,0,i);
  }
  else {
    int rows=quotient(n-1,2), columns=3;

    // Add the top middle card.
    place(0,rows-1,1);

    // place cards down the right side.
    for (int i=1; i<rows; ++i)
      place(i, rows-i,2);

    // place cards at the bottom.
    if (n%2==0) {
      place(rows,0,2);
      place(rows+1,0,1);
      place(rows+2,0,0);
    }
    else {
      place(rows,0,1.5);
      place(rows+1,0,0.5);
    }

    // place cards up the left side.
    for (int i=1; i<rows; ++i)
      place(i+n-rows,i,0);
  }

  return pic;
}

struct Syzygy { // {{{1
  // Setup {{{2
  Braid initial=null;
  bool cyclic=true;
  bool showall=false;
  bool number=false;  // Number the diagrams when drawn.

  string lsym, codename; 

  bool watched=false;
  bool uptodate=true;

  struct Move {
    Braid action(Braid);
    Relation rel;
    int place, step;
  }

  Move[] moves;

  void apply(Relation r, int step, int place) {
    Move m=new Move;
    m.rel=r;
    m.place=place; m.step=step;
    m.action=new Braid (Braid b) {
      return apply(r, b, step, place);
    };
    moves.push(m);

    uptodate = false;
  }

  void swap(int i, int j) {
    Move m=new Move;
    m.rel=null;
    m.action=new Braid (Braid b) {
      return b.swap(i, j);
    };
    moves.push(m);

    uptodate = false;
  }

  // Drawing {{{2
  picture[] drawMoves() {
    picture[] pics;

    assert(initial!=null, "must set initial braid");
    Braid b=initial;

    picture pic;
    b.draw(pic);
    pics.push(pic);

    for (int i=0; i<moves.length; ++i) {
      b=moves[i].action(b);
      if (showall || moves[i].rel != null) {
        picture pic;
        b.draw(pic);
        pics.push(pic);
      }
    }

    // Remove the last picture.
    if (this.cyclic)
      pics.pop();

    return pics;
  }

  void draw(picture pic=currentpicture) {
    pic.add(tableau(fit(drawMoves()), this.number));
  }

  void updatefunction() {
    if (!uptodate) {
      picture pic; this.draw(pic);
      shipout(pic);
      uptodate = true;
    }
  }

  void oldupdatefunction() = null;

  void watch() {
    if (!watched) {
      watched = true;
      oldupdatefunction = atupdate();
      atupdate(this.updatefunction);
      uptodate = false;
    }
  }

  void unwatch() {
    assert(watched == true);
    atupdate(oldupdatefunction);
    uptodate = false;
  }

  // Writing {{{2
  string linearName() {
    assert(lsym!="");

    string s=lsym+"(";
    for (int i=1; i<=initial.n+1; ++i) {
      if (i>1)
        s+=",";
      s+="x_"+(string)i;
    }
    return s+")";
  }

  string fullCodeName() {
    assert(codename!="");

    string s=codename+"[";
    for (int i=1; i<=initial.n+1; ++i) {
      if (i>1)
        s+=", ";
      s+="x"+(string)i+"_";
    }
    return s+"]";
  }

  string toLinear() {
    string s=linearName()+" = ";

    Braid b=initial;
    bool first=true;
    for (int i=0; i<moves.length; ++i) {
      Move m=moves[i];
      if (m.rel != null) {
        if (first) {
          first=false;
          if (m.rel.inverted)
            s+=" - ";
        }
        else
          s+=m.rel.inverted ? " - " : " + ";
        s+=m.rel.lsym+b.windowToLinear(m.step, m.place, m.rel.lhs.n);
      }
      b=m.action(b);
    }

    return s;
  }

  string toCode() {
    string s=fullCodeName()+" :> ";

    Braid b=initial;
    bool first=true;
    for (int i=0; i<moves.length; ++i) {
      Move m=moves[i];
      if (m.rel != null) {
        if (first) {
          first=false;
          if (m.rel.inverted)
            s+=" - ";
        }
        else
          s+=m.rel.inverted ? " - " : " + ";
        s+=m.rel.codename+b.windowToCode(m.step, m.place, m.rel.lhs.n);
      }
      b=m.action(b);
    }

    return s;
  }

}

// Relation definitions {{{1
// If you define more relations that you think would be useful, please email
// them to me, and I'll add them to the script.  --Andy.
Relation r3;
r3.lhs.n=3;
r3.lsym="\rho_3"; r3.codename="rho3";
r3.lhs.add(bp,0); r3.lhs.add(bp,1); r3.lhs.add(bp,0);
r3.rhs.n=3;
r3.rhs.add(bp,1); r3.rhs.add(bp,0); r3.rhs.add(bp,1);

Relation r4a;
r4a.lhs.n=3;
r4a.lsym="\rho_{4a}"; r4a.codename="rho4a";
r4a.lhs.add(bp,0); r4a.lhs.add(bp,1); r4a.lhs.add(phi,0);
r4a.rhs.n=3;
r4a.rhs.add(phi,1); r4a.rhs.add(bp,0);

Relation r4b;
r4b.lhs.n=3;
r4b.lsym="\rho_{4b}"; r4b.codename="rho4b";
r4b.lhs.add(bp,1); r4b.lhs.add(bp,0); r4b.lhs.add(phi,1);
r4b.rhs.n=3;
r4b.rhs.add(phi,0); r4b.rhs.add(bp,0);