The file below shows the image in the Temperley-Lieb algebra
of the following 5-string 95-crossing braid:
dCCbAbaBddddabCBABAcDDBDcdCbaddabCCCdcBDBAABaaaBccDbaabccbAAbaabAbdddCDccDDBADcABaCabdddCCdcDDD
This braid is algebraically right 1-split which means that the image in
TL_{5,1} looks like the last string of the braid is split.
More precisely: this means that the coefficients of (seven) Kauffman diagrams
with the last (rightmost) string n o t going straight down are non-zero only
for those with at least two cups (and for the unit). In fact, the braid
is not right split which is easily seen from the fact that the image has
many non-zero coefficients for non-split generators with more than 1 cup.
The following braid is obtained from the one above in a manner desribed in
Piwocki, Traczyk "Representations of small braid groups in Temperley-Lieb algebra".
The image of the braid below in TL_{9,2} is trivial.
dCCbAbaBddddabCBABAcDDBDcdCbaddabCCCdcBDBAABaaaBccDbaabccbAAbaabAbdddCDccDDBADcABaCabdddCCdcDDD
eFFgHghGeeeehgFGHGHfEEGEfeFgheehgFFFefGEGHHGhhhGffEghhgffgHHghhgHgeeeFEffEEGHEfHGhFhgeeeFFefEEE
dddCDccDDDBAcAbaCdabddCCdcDDDBaBAABaaBCCBAABdCCbAAAbaabdbCDcccBADDABcDCdbddCababcBADDDDbABaBccD
eeeFEffEEEGHfHghFehgeeFFefEEEGhGHHGhhGFFGHHGeFFgHHHghhgegFEfffGHEEHGfEFegeeFhghgfGHEEEEgHGhGffE
The image in TL_{5} is given in the following manner:
generator number/number of cups/first term degree/coefficients with step 4.
The generators of TL_{5}, the Kauffman diagrams, are listed as
described in Piwocki, Traczyk
"Representations of small braid groups in Temperley-Lieb algebra"
Lists of generators are given as TLgensxx.txt
1 0 9 1
2 1 3
3 1 1
4 1 1
5 1 3 -1 1
6 1 3
7 1 -15 1 -3 4 -3 1 -1 1
8 2 -135 -1 11 -65 274 -921 2611 -6441 14080 -27554 48464 -76411 106653 -127497 118527 -51227 -106492 381221 -781502 1284964 -1829979 2316910 -2622164 2624428 -2237736 1442180 -301936 -1037200 2372786 -3488043 4200336 -4402202 4083402 -3328416 2291394 -1157072 99399 751185 -1324948 1613184 -1655181 1518168 -1276868 997588 -729153 500510 -323205 196366 -112104 59975 -29939 13857 -5893 2273 -780 231 -56 10 -1
9 2 -133 -1 12 -77 351 -1272 3882 -10311 24316 -51542 98876 -172029 270550 -380127 463407 -452490 248060 269294 -1211794 2642154 -4524116 6680803 -8779788 10361455 -10917258 10008493 -7398983 3163303 2268164 -8152478 13566126 -17597961 19553594 -19114908 16409757 -11973129 6613136 -1222117 -3413925 6758760 -8592231 8994800 -8268696 6827662 -5088993 3393180 -1962860 899372 -206031 -176170 333078 -350207 296840 -220386 147845 -90841 51401 -26799 12833 -5605 2207 -769 230 -56 10 -1
10 1 3
11 1 -7 1 -1 -1 1
12 1 -25 -1 2 -1 -2 4 -2 0 -1 2
13 2 -141 1 -11 66 -284 976 -2831 7153 -16042 32286 -58615 95953 -140513 180096 -190730 135568 31680 -357069 870543 -1566642 2387447 -3215137 3880818 -4192529 3979157 -3140316 1687692 236749 -2375065 4400156 -5984632 6873516 -6940558 6212617 -4856336 3132686 -1333893 -279079 1522678 -2311614 2653964 -2627160 2345143 -1926799 1472756 -1053403 707220 -446198 264489 -147073 76502 -37063 16620 -6838 2549 -845 242 -57 10 -1
14 2 -139 1 -12 78 -362 1338 -4168 11309 -27275 59222 -116643 209073 -340577 500299 -649799 710293 -555669 19054 1079869 -2883969 5429670 -8575754 11954714 -14973960 16886718 -16934417 14537634 -9489258 2091068 6819219 -15977729 23929096 -29333678 31267884 -29435099 24228161 -16627141 7963934 378228 -7248461 11925336 -14189440 14272544 -12717590 10201247 -7370049 4725944 -2575271 1034957 -76790 -413160 579138 -555696 446314 -317907 205600 -122003 66693 -33581 15520 -6538 2482 -834 241 -57 10 -1
15 1 1
16 1 -9 -1 3 -4 3 -2 1
17 2 -131 -1 13 -90 442 -1725 5674 -16283 41675 -96545 204518 -398969 720044 -1205294 1872137 -2692824 3566687 -4299225 4601055 -4119417 2508791 464900 -4807436 10201170 -15971023 21142655 -24601406 25329043 -22664688 16518297 -7468564 -3296018 14190852 -23555483 29999037 -32681017 31456891 -26858156 19923893 -11939604 4157778 2430451 -7227332 10051879 -11072423 10685070 -9379328 7624409 -5795101 4140755 -2789445 1773757 -1064517 602178 -320276 159545 -74033 31758 -12462 4407 -1374 365 -78 12 -1
18 1 -19 1 -2 1 2 -3 2 -1 1
19 2 -137 1 -13 91 -454 1803 -6039 17656 -46064 108856 -235441 469484 -867412 1489384 -2379404 3533542 -4859937 6140777 -7012766 6985498 -5513290 2123474 3415825 -10927524 19711986 -28543715 35814387 -39819742 39140278 -33024810 21668190 -6290202 -11028963 27733357 -41306998 49790382 -52165416 48514570 -39925509 28183217 -15344938 3313896 6487176 -13283044 16945175 -17858928 16719754 -14322344 11391640 -8480562 5935596 -3914695 2434800 -1427513 787771 -408129 197746 -89126 37093 -14111 4837 -1463 378 -79 12 -1
20 2 -115 -1 11 -64 263 -856 2338 -5532 11543 -21430 35447 -51822 65308 -66098 39986 29350 -155557 342910 -579531 833268 -1053011 1177012 -1147469 927962 -518393 -38110 659911 -1243178 1685748 -1911660 1888990 -1635765 1212687 -705635 203851 219614 -520223 685065 -727870 679146 -575255 449496 -326786 222031 -141250 84129 -46825 24265 -11639 5123 -2043 724 -221 55 -10 1
21 2 -133 -1 12 -77 352 -1283 3949 -10610 25404 -54947 108325 -195733 325013 -495796 691983 -874767 979785 -922463 614357 10015 -961104 2166833 -3461605 4601571 -5308990 5338484 -4548344 2954867 -749937 -1728060 4065337 -5867109 6848368 -6895000 6078964 -4626127 2849781 -1071787 -445869 1542869 -2170063 2371369 -2249440 1928078 -1521643 1116696 -766125 492611 -297015 167736 -88481 43400 -19664 8153 -3052 1011 -287 66 -11 1
22 2 -121 1 -11 65 -273 910 -2548 6189 -13286 25461 -43699 66887 -89837 101372 -83525 13136 133162 -370574 695967 -1079089 1459282 -1751217 1860857 -1709249 1258097 -528944 -391571 1364279 -2226151 2827736 -3068625 2920442 -2430783 1707682 -890259 114986 512686 -934523 1141414 -1163436 1053410 -869838 663704 -471239 312449 -193707 112237 -60653 30455 -14127 6003 -2308 788 -232 56 -10 1
23 2 -139 1 -12 78 -363 1349 -4236 11618 -28420 62870 -126948 235391 -402153 633525 -918166 1216087 -1450729 1509756 -1259858 577831 605487 -2258777 4217148 -6179234 7745733 -8498796 8106791 -6426453 3569869 89612 -3981524 7459509 -9947486 11069553 -10727094 9103423 -6599089 3721411 -962950 -1297506 2849419 -3654986 3813364 -3503407 2924935 -2253471 1615280 -1081899 678436 -398377 218764 -112034 53273 -23371 9374 -3393 1087 -299 67 -11 1
24 2 -119 -1 10 -53 200 -603 1536 -3402 6646 -11504 17551 -23110 24739 -17156 -6005 50385 -118225 205552 -300383 383144 -429991 418721 -335802 182240 24014 -250489 457223 -606490 672256 -646409 539915 -378799 196545 -25508 -110074 197770 -236966 236000 -207900 166240 -122172 83027 -52304 30539 -16488 8194 -3721 1527 -557 176 -46 9 -1
25 2 -117 -1 11 -64 264 -867 2402 -5793 12377 -23638 40439 -61610 82013 -90648 69852 3086 -150845 387510 -708171 1080057 -1439541 1698734 -1762852 1555619 -1046030 267572 678161 -1637366 2437039 -2926497 3015219 -2695394 2042304 -1192851 309458 459286 -1009857 1299280 -1343032 1199074 -945128 656795 -392095 184738 -45294 -32572 64158 -66755 55074 -39273 24969 -14330 7449 -3498 1472 -547 175 -46 9 -1
26 2 -115 -1 12 -76 341 -1218 3675 -9690 22799 -48515 94186 -167699 274490 -412753 567614 -706182 776346 -712299 448476 59190 -809409 1736878 -2707546 3534087 -4012147 3970491 -3321717 2097562 -455689 -1347172 3010286 -4257371 4899024 -4871594 4242687 -3184087 1922480 -683756 -353557 1085215 -1485468 1592461 -1483047 1246088 -961268 686892 -456638 282863 -163191 87490 -43411 19812 -8242 3085 -1019 288 -66 11 -1
27 2 -99 -1 10 -52 190 -550 1337 -2810 5171 -8335 11600 -13329 10856 -858 -19667 51923 -93772 138808 -176734 195377 -184052 137322 -57815 -43124 147513 -235123 288849 -299279 266942 -201587 118890 -35861 -33461 80764 -104126 106839 -95190 76122 -55474 37106 -22841 12932 -6712 3173 -1352 511 -167 45 -9 1
28 2 -117 -1 11 -64 265 -877 2457 -6016 13120 -25780 45936 -74389 109238 -144227 167790 -163788 114710 -7330 -160379 373418 -597854 785000 -881974 846393 -660793 341625 61072 -474230 819812 -1034822 1086907 -980743 753729 -463204 169956 76563 -247277 335310 -351689 317684 -256870 189195 -128019 79875 -45979 24371 -11842 5236 -2083 733 -222 55 -10 1
29 1 1
30 1 -5 1 -2 1
31 1 -23 -1 3 -4 3 0 -1 0 0 1
32 2 -135 -1 11 -66 284 -977 2840 -7197 16198 -32734 59711 -98293 144920 -187423 201362 -148570 -19658 352778 -884424 1611746 -2476534 3356082 -4071279 4415615 -4202175 3318234 -1771669 -287233 2579650 -4748628 6435010 -7358745 7382621 -6540326 5021687 -3121031 1165191 557785 -1850725 2629758 -2916691 2809411 -2442741 1951847 -1446445 998802 -643921 387600 -217505 113440 -54730 24261 -9788 3546 -1131 308 -68 11 -1
33 2 -133 -1 12 -78 362 -1339 4178 -11363 27485 -59880 118396 -213156 349015 -515873 675445 -747553 601962 -63979 -1058254 2921119 -5572483 8874457 -12448155 15670304 -17744530 17851324 -15353844 10010327 -2128656 -7396749 17198625 -25690954 31407512 -33336933 31160896 -25326929 16935180 -7473245 -1523934 8795072 -13573591 15664536 -15379012 13366847 -10408878 7229032 -4367737 2131633 -610183 -265346 648770 -714440 613743 -454933 301499 -181350 99584 -49951 22808 -9411 3467 -1119 307 -68 11 -1
34 1 -17 1 -3 4 -3 1 0 0 1
35 2 -131 -1 13 -91 454 -1804 6050 -17721 46340 -109799 238183 -476485 883408 -1522483 2441888 -3641569 5031069 -6388345 7336788 -7361304 5880947 -2382874 -3398741 11299434 -20595785 29992950 -37770367 42086635 -41392214 34852008 -22657650 6123974 12483917 -30371131 44784202 -53587519 55694654 -51245893 41497372 -28468748 14458627 -1561629 -8696130 15534915 -18904914 19319732 -17613310 14695985 -11366105 8203835 -5545155 3514664 -2088513 1161765 -603292 291257 -129979 53201 -19759 6560 -1905 468 -92 13 -1
36 2 -115 -1 11 -65 273 -911 2557 -6232 13433 -25865 44640 -68789 93199 -106541 90264 -20029 -129282 374844 -714972 1119333 -1524726 1840443 -1964855 1810984 -1334703 556601 431077 -1477222 2402981 -3043354 3287260 -3103986 2548153 -1742327 843523 -4464 -658977 1086040 -1272488 1258115 -1106823 886422 -653544 446054 -282449 165914 -90223 45237 -20782 8668 -3239 1063 -297 67 -11 1
37 2 -133 -1 12 -78 363 -1350 4246 -11672 28631 -63537 128747 -239651 411158 -650682 947762 -1262296 1515624 -1590409 1345169 -646359 -585025 2322885 -4398397 6493930 -8181723 9008017 -8606940 6814020 -3743106 -203223 4401472 -8140368 10784154 -11919513 11443871 -9571461 6759689 -3582418 590375 1801919 -3376678 4114501 -4149076 3697791 -2992474 2228009 -1536395 984303 -586289 324322 -166158 78482 -33948 13320 -4676 1439 -376 79 -12 1
38 2 -129 1 -11 65 -275 932 -2673 6686 -14849 29589 -53145 85921 -123832 155040 -157193 96975 65990 -369518 833958 -1445048 2140151 -2804707 3284457 -3414861 3063032 -2171525 790026 917638 -2697372 4256176 -5328136 5736493 -5433729 4508320 -3157318 1634375 -189485 -982461 1771967 -2159750 2199816 -1989917 1639965 -1246875 880189 -578583 354376 -202005 106864 -52225 23420 -9546 3489 -1121 307 -68 11 -1
39 2 -127 1 -12 77 -352 1284 -3956 10630 -25404 54664 -106667 189258 -304662 440830 -560217 589100 -413006 -114704 1145052 -2785021 5035151 -7729394 10500740 -12796685 13959411 -13366897 10607803 -5643152 -1099423 8741987 -16097450 21911284 -25144395 25219400 -22159116 16576600 -9520535 2222688 4179205 -8867851 11468606 -12050022 11022092 -8976940 6524188 -4163608 2217454 -822843 -31858 449778 -568744 518814 -399098 271709 -166946 93308 -47513 21978 -9170 3410 -1109 306 -68 11 -1
40 2 -125 1 -13 90 -443 1738 -5761 16690 -43181 101237 -217270 429868 -787743 1340662 -2120606 3112736 -4220269 5233341 -5817328 5537851 -3934766 644482 4449177 -11074991 18496578 -25544459 30782490 -32794221 30530979 -23630658 12608909 1150279 -15624351 28556763 -37935499 42419136 -41598052 36027471 -27036281 16378013 -5827988 -3166537 9705461 -13493964 14757747 -14067360 12135478 -9642026 7119950 -4909269 3167245 -1912373 1079330 -567939 277504 -125194 51742 -19381 6481 -1893 467 -92 13 -1
41 2 -109 1 -11 64 -264 867 -2399 5766 -12250 23221 -39371 59368 -78091 85019 -63769 -6045 143816 -360027 647148 -972541 1277189 -1483175 1510298 -1298665 830868 -145980 -660939 1455936 -2096346 2465650 -2502433 2214479 -1673971 996000 -307842 -281208 700964 -927944 979854 -901298 746728 -565972 395449 -255571 152863 -84478 42973 -19998 8436 -3183 1053 -296 67 -11 1
42 2 -127 1 -12 77 -353 1295 -4023 10930 -26503 58133 -116386 213892 -361887 563785 -806182 1049385 -1221455 1220921 -932670 258282 843535 -2317145 3986002 -5559310 6677401 -6991693 6261160 -4437099 1706653 1524819 -4714077 7303688 -8854146 9145556 -8219531 6351518 -3966020 1523799 584300 -2108759 2964716 -3210554 2997577 -2511379 1922617 -1357199 887268 -537975 302333 -157090 75137 -32865 13021 -4609 1428 -375 79 -12 1
1526