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{{Template:Basic Knot Invariants|name=9_25}}
<!-- This page was generated from the splice base [[Rolfsen_Splice_Base]]. Please do not edit!
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{{Rolfsen Knot Page|
n = 9 |
k = 25 |
KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,9,-2,1,-3,8,-9,2,-4,7,-8,3,-5,6,-7,4,-6,5/goTop.html |
braid_table = <table cellspacing=0 cellpadding=0 border=0>
<tr><td>[[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr>
<tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]]</td></tr>
</table> |
braid_crossings = 10 |
braid_width = 5 |
braid_index = 5 |
same_alexander = [[K11n134]], |
same_jones = [[K11n25]], |
khovanov_table = <table border=1>
<tr align=center>
<td width=14.2857%><table cellpadding=0 cellspacing=0>
<tr><td>\</td><td>&nbsp;</td><td>r</td></tr>
<tr><td>&nbsp;</td><td>&nbsp;\&nbsp;</td><td>&nbsp;</td></tr>
<tr><td>j</td><td>&nbsp;</td><td>\</td></tr>
</table></td>
<td width=7.14286%>-7</td ><td width=7.14286%>-6</td ><td width=7.14286%>-5</td ><td width=7.14286%>-4</td ><td width=7.14286%>-3</td ><td width=7.14286%>-2</td ><td width=7.14286%>-1</td ><td width=7.14286%>0</td ><td width=7.14286%>1</td ><td width=7.14286%>2</td ><td width=14.2857%>&chi;</td></tr>
<tr align=center><td>3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td>1</td></tr>
<tr align=center><td>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>&nbsp;</td><td>-1</td></tr>
<tr align=center><td>-1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>4</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>3</td></tr>
<tr align=center><td>-3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>4</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>&nbsp;</td><td>-2</td></tr>
<tr align=center><td>-5</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>4</td><td bgcolor=yellow>3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
<tr align=center><td>-7</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>4</td><td bgcolor=yellow>4</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
<tr align=center><td>-9</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>3</td><td bgcolor=yellow>4</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
<tr align=center><td>-11</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>2</td><td bgcolor=yellow>4</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>2</td></tr>
<tr align=center><td>-13</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-2</td></tr>
<tr align=center><td>-15</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>2</td></tr>
<tr align=center><td>-17</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
</table> |
coloured_jones_2 = <math>q^4-2 q^3+q^2+5 q-11+4 q^{-1} +19 q^{-2} -31 q^{-3} +4 q^{-4} +43 q^{-5} -50 q^{-6} -3 q^{-7} +62 q^{-8} -56 q^{-9} -12 q^{-10} +65 q^{-11} -45 q^{-12} -19 q^{-13} +52 q^{-14} -25 q^{-15} -20 q^{-16} +30 q^{-17} -7 q^{-18} -12 q^{-19} +10 q^{-20} -3 q^{-22} + q^{-23} </math> |
coloured_jones_3 = <math>q^9-2 q^8+q^7+q^6+q^5-7 q^4+4 q^3+11 q^2-5 q-28+14 q^{-1} +46 q^{-2} -8 q^{-3} -87 q^{-4} +10 q^{-5} +121 q^{-6} +11 q^{-7} -167 q^{-8} -34 q^{-9} +204 q^{-10} +67 q^{-11} -234 q^{-12} -98 q^{-13} +248 q^{-14} +130 q^{-15} -251 q^{-16} -155 q^{-17} +240 q^{-18} +173 q^{-19} -218 q^{-20} -184 q^{-21} +185 q^{-22} +188 q^{-23} -145 q^{-24} -184 q^{-25} +101 q^{-26} +173 q^{-27} -60 q^{-28} -148 q^{-29} +20 q^{-30} +120 q^{-31} +6 q^{-32} -87 q^{-33} -20 q^{-34} +55 q^{-35} +23 q^{-36} -29 q^{-37} -20 q^{-38} +14 q^{-39} +12 q^{-40} -5 q^{-41} -5 q^{-42} +3 q^{-44} - q^{-45} </math> |
coloured_jones_4 = <math>q^{16}-2 q^{15}+q^{14}+q^{13}-3 q^{12}+5 q^{11}-7 q^{10}+6 q^9+6 q^8-17 q^7+8 q^6-17 q^5+33 q^4+33 q^3-59 q^2-23 q-57+113 q^{-1} +145 q^{-2} -104 q^{-3} -133 q^{-4} -223 q^{-5} +215 q^{-6} +416 q^{-7} -43 q^{-8} -287 q^{-9} -588 q^{-10} +216 q^{-11} +784 q^{-12} +204 q^{-13} -347 q^{-14} -1067 q^{-15} +38 q^{-16} +1077 q^{-17} +553 q^{-18} -244 q^{-19} -1456 q^{-20} -235 q^{-21} +1174 q^{-22} +837 q^{-23} -32 q^{-24} -1631 q^{-25} -477 q^{-26} +1088 q^{-27} +974 q^{-28} +198 q^{-29} -1587 q^{-30} -637 q^{-31} +861 q^{-32} +974 q^{-33} +421 q^{-34} -1361 q^{-35} -725 q^{-36} +526 q^{-37} +850 q^{-38} +617 q^{-39} -977 q^{-40} -715 q^{-41} +140 q^{-42} +597 q^{-43} +712 q^{-44} -516 q^{-45} -559 q^{-46} -155 q^{-47} +266 q^{-48} +615 q^{-49} -134 q^{-50} -293 q^{-51} -243 q^{-52} +2 q^{-53} +373 q^{-54} +40 q^{-55} -67 q^{-56} -158 q^{-57} -90 q^{-58} +146 q^{-59} +46 q^{-60} +23 q^{-61} -53 q^{-62} -61 q^{-63} +35 q^{-64} +12 q^{-65} +20 q^{-66} -7 q^{-67} -19 q^{-68} +5 q^{-69} +5 q^{-71} -3 q^{-73} + q^{-74} </math> |
coloured_jones_5 = <math>q^{25}-2 q^{24}+q^{23}+q^{22}-3 q^{21}+q^{20}+5 q^{19}-5 q^{18}+q^{17}+4 q^{16}-12 q^{15}-3 q^{14}+18 q^{13}+4 q^{12}+9 q^{11}-3 q^{10}-48 q^9-39 q^8+34 q^7+81 q^6+91 q^5+3 q^4-182 q^3-226 q^2-33 q+261+448 q^{-1} +221 q^{-2} -379 q^{-3} -782 q^{-4} -504 q^{-5} +347 q^{-6} +1199 q^{-7} +1083 q^{-8} -245 q^{-9} -1645 q^{-10} -1763 q^{-11} -188 q^{-12} +2009 q^{-13} +2710 q^{-14} +801 q^{-15} -2245 q^{-16} -3608 q^{-17} -1711 q^{-18} +2213 q^{-19} +4540 q^{-20} +2724 q^{-21} -1967 q^{-22} -5262 q^{-23} -3790 q^{-24} +1500 q^{-25} +5784 q^{-26} +4773 q^{-27} -920 q^{-28} -6046 q^{-29} -5602 q^{-30} +289 q^{-31} +6106 q^{-32} +6206 q^{-33} +325 q^{-34} -5975 q^{-35} -6615 q^{-36} -884 q^{-37} +5728 q^{-38} +6824 q^{-39} +1369 q^{-40} -5355 q^{-41} -6884 q^{-42} -1814 q^{-43} +4896 q^{-44} +6809 q^{-45} +2224 q^{-46} -4328 q^{-47} -6602 q^{-48} -2625 q^{-49} +3640 q^{-50} +6267 q^{-51} +3003 q^{-52} -2837 q^{-53} -5780 q^{-54} -3320 q^{-55} +1944 q^{-56} +5103 q^{-57} +3543 q^{-58} -1002 q^{-59} -4285 q^{-60} -3580 q^{-61} +116 q^{-62} +3314 q^{-63} +3398 q^{-64} +662 q^{-65} -2310 q^{-66} -3008 q^{-67} -1176 q^{-68} +1331 q^{-69} +2422 q^{-70} +1442 q^{-71} -512 q^{-72} -1754 q^{-73} -1427 q^{-74} -79 q^{-75} +1094 q^{-76} +1215 q^{-77} +402 q^{-78} -532 q^{-79} -895 q^{-80} -515 q^{-81} +157 q^{-82} +564 q^{-83} +457 q^{-84} +53 q^{-85} -283 q^{-86} -340 q^{-87} -134 q^{-88} +115 q^{-89} +209 q^{-90} +119 q^{-91} -19 q^{-92} -98 q^{-93} -94 q^{-94} -16 q^{-95} +49 q^{-96} +50 q^{-97} +12 q^{-98} -9 q^{-99} -21 q^{-100} -20 q^{-101} +7 q^{-102} +12 q^{-103} +2 q^{-104} -5 q^{-107} +3 q^{-109} - q^{-110} </math> |
coloured_jones_6 = <math>q^{36}-2 q^{35}+q^{34}+q^{33}-3 q^{32}+q^{31}+q^{30}+7 q^{29}-10 q^{28}-q^{27}+9 q^{26}-13 q^{25}+q^{24}+9 q^{23}+28 q^{22}-24 q^{21}-19 q^{20}+13 q^{19}-46 q^{18}-4 q^{17}+51 q^{16}+119 q^{15}-15 q^{14}-69 q^{13}-52 q^{12}-220 q^{11}-82 q^{10}+166 q^9+468 q^8+248 q^7-36 q^6-281 q^5-904 q^4-671 q^3+129 q^2+1296 q+1406+843 q^{-1} -286 q^{-2} -2463 q^{-3} -2814 q^{-4} -1280 q^{-5} +2022 q^{-6} +3948 q^{-7} +4080 q^{-8} +1692 q^{-9} -3997 q^{-10} -7111 q^{-11} -6048 q^{-12} +341 q^{-13} +6591 q^{-14} +10268 q^{-15} +7942 q^{-16} -2671 q^{-17} -11812 q^{-18} -14534 q^{-19} -6166 q^{-20} +6068 q^{-21} +17110 q^{-22} +18345 q^{-23} +3858 q^{-24} -13313 q^{-25} -23788 q^{-26} -16818 q^{-27} +302 q^{-28} +20740 q^{-29} +29289 q^{-30} +14325 q^{-31} -9763 q^{-32} -29726 q^{-33} -27651 q^{-34} -8996 q^{-35} +19526 q^{-36} +36660 q^{-37} +24647 q^{-38} -2982 q^{-39} -30850 q^{-40} -34893 q^{-41} -17980 q^{-42} +15180 q^{-43} +39209 q^{-44} +31596 q^{-45} +3769 q^{-46} -28659 q^{-47} -37739 q^{-48} -24145 q^{-49} +10297 q^{-50} +38309 q^{-51} +34815 q^{-52} +8783 q^{-53} -25138 q^{-54} -37536 q^{-55} -27577 q^{-56} +5901 q^{-57} +35503 q^{-58} +35603 q^{-59} +12527 q^{-60} -20802 q^{-61} -35488 q^{-62} -29521 q^{-63} +1281 q^{-64} +30969 q^{-65} +34848 q^{-66} +16129 q^{-67} -14860 q^{-68} -31473 q^{-69} -30505 q^{-70} -4448 q^{-71} +23891 q^{-72} +32060 q^{-73} +19627 q^{-74} -6779 q^{-75} -24549 q^{-76} -29514 q^{-77} -10684 q^{-78} +14058 q^{-79} +25993 q^{-80} +21279 q^{-81} +2141 q^{-82} -14620 q^{-83} -24835 q^{-84} -14902 q^{-85} +3352 q^{-86} +16539 q^{-87} +18845 q^{-88} +8669 q^{-89} -3934 q^{-90} -16341 q^{-91} -14471 q^{-92} -4504 q^{-93} +6245 q^{-94} +12251 q^{-95} +10001 q^{-96} +3585 q^{-97} -6897 q^{-98} -9567 q^{-99} -6822 q^{-100} -880 q^{-101} +4656 q^{-102} +6712 q^{-103} +5644 q^{-104} -498 q^{-105} -3620 q^{-106} -4667 q^{-107} -2986 q^{-108} -165 q^{-109} +2389 q^{-110} +3804 q^{-111} +1480 q^{-112} -58 q^{-113} -1619 q^{-114} -1905 q^{-115} -1366 q^{-116} -17 q^{-117} +1418 q^{-118} +950 q^{-119} +732 q^{-120} -54 q^{-121} -510 q^{-122} -820 q^{-123} -452 q^{-124} +252 q^{-125} +207 q^{-126} +391 q^{-127} +194 q^{-128} +39 q^{-129} -248 q^{-130} -225 q^{-131} +3 q^{-132} -31 q^{-133} +92 q^{-134} +80 q^{-135} +76 q^{-136} -45 q^{-137} -57 q^{-138} - q^{-139} -29 q^{-140} +9 q^{-141} +12 q^{-142} +29 q^{-143} -7 q^{-144} -12 q^{-145} +5 q^{-146} -7 q^{-147} +5 q^{-150} -3 q^{-152} + q^{-153} </math> |
coloured_jones_7 = <math>q^{49}-2 q^{48}+q^{47}+q^{46}-3 q^{45}+q^{44}+q^{43}+3 q^{42}+2 q^{41}-12 q^{40}+4 q^{39}+8 q^{38}-9 q^{37}+2 q^{36}+2 q^{35}+15 q^{34}+8 q^{33}-47 q^{32}-4 q^{31}+20 q^{30}-8 q^{29}+23 q^{28}+17 q^{27}+58 q^{26}+23 q^{25}-143 q^{24}-96 q^{23}-36 q^{22}-9 q^{21}+152 q^{20}+191 q^{19}+277 q^{18}+144 q^{17}-377 q^{16}-540 q^{15}-567 q^{14}-327 q^{13}+414 q^{12}+954 q^{11}+1430 q^{10}+1109 q^9-402 q^8-1739 q^7-2768 q^6-2573 q^5-380 q^4+2316 q^3+5012 q^2+5548 q+2388-2370 q^{-1} -7721 q^{-2} -10214 q^{-3} -6877 q^{-4} +623 q^{-5} +10484 q^{-6} +16861 q^{-7} +14375 q^{-8} +4244 q^{-9} -11540 q^{-10} -24697 q^{-11} -25884 q^{-12} -13868 q^{-13} +9592 q^{-14} +32412 q^{-15} +40381 q^{-16} +28983 q^{-17} -1848 q^{-18} -37526 q^{-19} -57242 q^{-20} -49960 q^{-21} -12323 q^{-22} +38102 q^{-23} +73166 q^{-24} +74995 q^{-25} +34149 q^{-26} -31469 q^{-27} -86279 q^{-28} -102321 q^{-29} -61840 q^{-30} +17521 q^{-31} +93437 q^{-32} +128109 q^{-33} +93677 q^{-34} +3851 q^{-35} -93581 q^{-36} -150173 q^{-37} -126202 q^{-38} -30131 q^{-39} +86429 q^{-40} +165930 q^{-41} +156317 q^{-42} +58993 q^{-43} -73228 q^{-44} -174769 q^{-45} -181702 q^{-46} -87249 q^{-47} +56185 q^{-48} +176989 q^{-49} +200831 q^{-50} +112483 q^{-51} -37527 q^{-52} -173938 q^{-53} -213557 q^{-54} -133252 q^{-55} +19431 q^{-56} +167410 q^{-57} +220610 q^{-58} +149010 q^{-59} -3390 q^{-60} -159021 q^{-61} -223169 q^{-62} -160099 q^{-63} -10179 q^{-64} +150009 q^{-65} +222740 q^{-66} +167515 q^{-67} +21258 q^{-68} -141101 q^{-69} -220251 q^{-70} -172291 q^{-71} -30665 q^{-72} +132140 q^{-73} +216507 q^{-74} +175603 q^{-75} +39315 q^{-76} -122795 q^{-77} -211642 q^{-78} -178042 q^{-79} -48156 q^{-80} +112108 q^{-81} +205276 q^{-82} +180019 q^{-83} +58062 q^{-84} -99274 q^{-85} -196848 q^{-86} -181260 q^{-87} -69252 q^{-88} +83491 q^{-89} +185321 q^{-90} +181081 q^{-91} +81626 q^{-92} -64344 q^{-93} -169860 q^{-94} -178470 q^{-95} -94293 q^{-96} +42096 q^{-97} +149911 q^{-98} +172010 q^{-99} +105726 q^{-100} -17597 q^{-101} -125246 q^{-102} -160638 q^{-103} -114329 q^{-104} -7281 q^{-105} +96865 q^{-106} +143639 q^{-107} +117833 q^{-108} +30299 q^{-109} -66111 q^{-110} -121194 q^{-111} -115123 q^{-112} -49018 q^{-113} +35809 q^{-114} +94658 q^{-115} +105401 q^{-116} +61050 q^{-117} -8534 q^{-118} -66080 q^{-119} -89559 q^{-120} -65447 q^{-121} -12985 q^{-122} +38731 q^{-123} +69413 q^{-124} +62059 q^{-125} +26964 q^{-126} -15169 q^{-127} -47756 q^{-128} -52635 q^{-129} -33026 q^{-130} -2300 q^{-131} +27660 q^{-132} +39557 q^{-133} +32200 q^{-134} +12759 q^{-135} -11470 q^{-136} -25676 q^{-137} -26572 q^{-138} -16839 q^{-139} +373 q^{-140} +13644 q^{-141} +18824 q^{-142} +16002 q^{-143} +5406 q^{-144} -4743 q^{-145} -11038 q^{-146} -12491 q^{-147} -7209 q^{-148} -564 q^{-149} +5046 q^{-150} +8191 q^{-151} +6327 q^{-152} +2770 q^{-153} -1113 q^{-154} -4371 q^{-155} -4446 q^{-156} -3090 q^{-157} -820 q^{-158} +1866 q^{-159} +2551 q^{-160} +2323 q^{-161} +1311 q^{-162} -374 q^{-163} -1079 q^{-164} -1465 q^{-165} -1226 q^{-166} -177 q^{-167} +368 q^{-168} +717 q^{-169} +757 q^{-170} +279 q^{-171} +71 q^{-172} -258 q^{-173} -484 q^{-174} -237 q^{-175} -99 q^{-176} +92 q^{-177} +198 q^{-178} +94 q^{-179} +115 q^{-180} +37 q^{-181} -100 q^{-182} -75 q^{-183} -62 q^{-184} -4 q^{-185} +42 q^{-186} -3 q^{-187} +29 q^{-188} +29 q^{-189} -9 q^{-190} -12 q^{-191} -20 q^{-192} -2 q^{-193} +12 q^{-194} -5 q^{-195} +7 q^{-197} -5 q^{-200} +3 q^{-202} - q^{-203} </math> |
computer_talk =
<table>
<tr valign=top>
<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:=&nbsp;&nbsp;&nbsp;&nbsp;</pre></td>
<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
</tr>
<tr valign=top><td colspan=2><nowiki>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</nowiki></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[2]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[Knot[9, 25]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[2]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 12, 6, 13], X[9, 17, 10, 16],
X[13, 18, 14, 1], X[17, 14, 18, 15], X[15, 11, 16, 10],
X[11, 6, 12, 7], X[7, 2, 8, 3]]</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[3]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[Knot[9, 25]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[3]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[-1, 9, -2, 1, -3, 8, -9, 2, -4, 7, -8, 3, -5, 6, -7, 4, -6, 5]</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[4]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[Knot[9, 25]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[4]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[4, 8, 12, 2, 16, 6, 18, 10, 14]</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[5]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>br = BR[Knot[9, 25]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[5]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BR[5, {-1, -1, 2, -1, -3, -2, -2, 4, -3, 4}]</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[6]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{First[br], Crossings[br]}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[6]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{5, 10}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[7]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BraidIndex[Knot[9, 25]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[7]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>5</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[8]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Show[DrawMorseLink[Knot[9, 25]]]</nowiki></code></td></tr>
<tr align=left><td></td><td>[[Image:9_25_ML.gif]]</td></tr><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[8]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>-Graphics-</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[9]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> (#[Knot[9, 25]]&) /@ {
SymmetryType, UnknottingNumber, ThreeGenus,
BridgeIndex, SuperBridgeIndex, NakanishiIndex
}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[9]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Reversible, 2, 2, 3, {4, 7}, 1}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[10]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>alex = Alexander[Knot[9, 25]][t]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[10]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 3 12 2
-17 - -- + -- + 12 t - 3 t
2 t
t</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[11]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Conway[Knot[9, 25]][z]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[11]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 4
1 - 3 z</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[12]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[12]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[9, 25], Knot[11, NonAlternating, 134]}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[13]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{KnotDet[Knot[9, 25]], KnotSignature[Knot[9, 25]]}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[13]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{47, -2}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[14]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Jones[Knot[9, 25]][q]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[14]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -8 3 5 7 8 8 7 5
-2 - q + -- - -- + -- - -- + -- - -- + - + q
7 6 5 4 3 2 q
q q q q q q</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[15]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[15]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[9, 25], Knot[11, NonAlternating, 25]}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[16]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>A2Invariant[Knot[9, 25]][q]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[16]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -26 -24 2 -18 2 2 2 -6 -4 3 4
-q - q + --- + q + --- - --- - --- + q - q + -- + q
22 16 14 10 2
q q q q q</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[17]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>HOMFLYPT[Knot[9, 25]][a, z]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[17]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 4 6 8 2 4 2 6 2 2 4 4 4
1 + a - 3 a + 3 a - a + z - 4 a z + 3 a z - a z - 2 a z</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[18]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kauffman[Knot[9, 25]][a, z]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[18]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 4 6 8 3 5 7 9 2
1 - a - 3 a - 3 a - a - a z - a z + a z + a z - 2 z +
2 2 4 2 6 2 8 2 3 3 3
2 a z + 13 a z + 13 a z + 4 a z - 2 a z + 3 a z +
5 3 7 3 9 3 4 2 4 4 4 6 4
5 a z - 2 a z - 2 a z + z - 3 a z - 15 a z - 18 a z -
8 4 5 3 5 5 5 7 5 9 5 2 6
7 a z + 2 a z - 3 a z - 10 a z - 4 a z + a z + 3 a z +
4 6 6 6 8 6 3 7 5 7 7 7 4 8
6 a z + 6 a z + 3 a z + 3 a z + 6 a z + 3 a z + a z +
6 8
a z</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[19]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Vassiliev[2][Knot[9, 25]], Vassiliev[3][Knot[9, 25]]}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[19]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{0, -1}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[20]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kh[Knot[9, 25]][q, t]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[20]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>2 4 1 2 1 3 2 4 3
-- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- +
3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4
q q t q t q t q t q t q t q t
4 4 4 4 3 4 t 3 2
----- + ----- + ----- + ----- + ---- + ---- + - + q t + q t
9 3 7 3 7 2 5 2 5 3 q
q t q t q t q t q t q t</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[21]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>ColouredJones[Knot[9, 25], 2][q]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[21]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -23 3 10 12 7 30 20 25 52 19
-11 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- -
22 20 19 18 17 16 15 14 13
q q q q q q q q q
45 65 12 56 62 3 50 43 4 31 19 4
--- + --- - --- - -- + -- - -- - -- + -- + -- - -- + -- + - + 5 q +
12 11 10 9 8 7 6 5 4 3 2 q
q q q q q q q q q q q
2 3 4
q - 2 q + q</nowiki></code></td></tr>
</table> }}

Latest revision as of 16:58, 1 September 2005

9 24.gif

9_24

9 26.gif

9_26

9 25.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 9 25's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 25 at Knotilus!


Knot presentations

Planar diagram presentation X1425 X3849 X5,12,6,13 X9,17,10,16 X13,18,14,1 X17,14,18,15 X15,11,16,10 X11,6,12,7 X7283
Gauss code -1, 9, -2, 1, -3, 8, -9, 2, -4, 7, -8, 3, -5, 6, -7, 4, -6, 5
Dowker-Thistlethwaite code 4 8 12 2 16 6 18 10 14
Conway Notation [22,21,2]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif

Length is 10, width is 5,

Braid index is 5

9 25 ML.gif 9 25 AP.gif
[{12, 4}, {3, 10}, {8, 11}, {10, 12}, {9, 5}, {4, 8}, {5, 2}, {1, 3}, {6, 9}, {2, 7}, {11, 6}, {7, 1}]

[edit Notes on presentations of 9 25]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 3
Super bridge index
Nakanishi index 1
Maximal Thurston-Bennequin number [-10][-1]
Hyperbolic Volume 11.3903
A-Polynomial See Data:9 25/A-polynomial

[edit Notes for 9 25's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus
Topological 4 genus
Concordance genus
Rasmussen s-Invariant -2

[edit Notes for 9 25's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 47, -2 }
Jones polynomial
HOMFLY-PT polynomial (db, data sources)
Kauffman polynomial (db, data sources)
The A2 invariant
The G2 invariant

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n134,}

Same Jones Polynomial (up to mirroring, ): {K11n25,}

Vassiliev invariants

V2 and V3: (0, -1)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where -2 is the signature of 9 25. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-7-6-5-4-3-2-1012χ
3         11
1        1 -1
-1       41 3
-3      42  -2
-5     43   1
-7    44    0
-9   34     -1
-11  24      2
-13 13       -2
-15 2        2
-171         -1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials