8 2: Difference between revisions
DrorsRobot (talk | contribs) No edit summary |
No edit summary |
||
Line 1: | Line 1: | ||
<!-- This page was generated from the splice template "Rolfsen_Splice_Template". Please do not edit! --> |
<!-- This page was generated from the splice template "Rolfsen_Splice_Template". Please do not edit! --> |
||
<!-- --> |
<!-- --> <!-- |
||
--> |
|||
{{Rolfsen Knot Page| |
|||
<!-- --> |
|||
n = 8 | |
|||
<!-- --> |
|||
k = 2 | |
|||
<!-- provide an anchor so we can return to the top of the page --> |
|||
KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,4,-3,1,-2,7,-5,8,-6,3,-4,2,-7,5,-8,6/goTop.html | |
|||
<span id="top"></span> |
|||
braid_table = <table cellspacing=0 cellpadding=0 border=0> |
|||
<!-- --> |
|||
<!-- this relies on transclusion for next and previous links --> |
|||
{{Knot Navigation Links|ext=gif}} |
|||
{{Rolfsen Knot Page Header|n=8|k=2|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,4,-3,1,-2,7,-5,8,-6,3,-4,2,-7,5,-8,6/goTop.html}} |
|||
<br style="clear:both" /> |
|||
{{:{{PAGENAME}} Further Notes and Views}} |
|||
{{Knot Presentations}} |
|||
<center><table border=1 cellpadding=10><tr align=center valign=top> |
|||
<td> |
|||
[[Braid Representatives|Minimum Braid Representative]]: |
|||
<table cellspacing=0 cellpadding=0 border=0> |
|||
<tr><td>[[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]]</td></tr> |
<tr><td>[[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]]</td></tr> |
||
<tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]]</td></tr> |
<tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]]</td></tr> |
||
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]]</td></tr> |
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]]</td></tr> |
||
</table> |
</table> | |
||
braid_crossings = 8 | |
|||
braid_width = 3 | |
|||
[[Invariants from Braid Theory|Length]] is 8, width is 3. |
|||
braid_index = 3 | |
|||
same_alexander = [[K11n6]], | |
|||
[[Invariants from Braid Theory|Braid index]] is 3. |
|||
same_jones = | |
|||
</td> |
|||
khovanov_table = <table border=1> |
|||
<td> |
|||
[[Lightly Documented Features|A Morse Link Presentation]]: |
|||
[[Image:{{PAGENAME}}_ML.gif]] |
|||
</td> |
|||
</tr></table></center> |
|||
{{3D Invariants}} |
|||
{{4D Invariants}} |
|||
{{Polynomial Invariants}} |
|||
=== "Similar" Knots (within the Atlas) === |
|||
Same [[The Alexander-Conway Polynomial|Alexander/Conway Polynomial]]: |
|||
{[[K11n6]], ...} |
|||
Same [[The Jones Polynomial|Jones Polynomial]] (up to mirroring, <math>q\leftrightarrow q^{-1}</math>): |
|||
{...} |
|||
{{Vassiliev Invariants}} |
|||
{{Khovanov Homology|table=<table border=1> |
|||
<tr align=center> |
<tr align=center> |
||
<td width=15.3846%><table cellpadding=0 cellspacing=0> |
<td width=15.3846%><table cellpadding=0 cellspacing=0> |
||
<tr><td>\</td><td> </td><td>r</td></tr> |
|||
<tr><td> </td><td> \ </td><td> </td></tr> |
<tr><td> </td><td> \ </td><td> </td></tr> |
||
<tr><td>j</td><td> </td><td>\</td></tr> |
<tr><td>j</td><td> </td><td>\</td></tr> |
||
</table></td> |
</table></td> |
||
<td width=7.69231%>-6</td ><td width=7.69231%>-5</td ><td width=7.69231%>-4</td ><td width=7.69231%>-3</td ><td width=7.69231%>-2</td ><td width=7.69231%>-1</td ><td width=7.69231%>0</td ><td width=7.69231%>1</td ><td width=7.69231%>2</td ><td width=15.3846%>χ</td></tr> |
|||
<tr align=center><td>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td>1</td></tr> |
<tr align=center><td>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td>1</td></tr> |
||
<tr align=center><td>-1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td>0</td></tr> |
<tr align=center><td>-1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td>0</td></tr> |
||
Line 70: | Line 34: | ||
<tr align=center><td>-15</td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> |
<tr align=center><td>-15</td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> |
||
<tr align=center><td>-17</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> |
<tr align=center><td>-17</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> |
||
</table> |
</table> | |
||
coloured_jones_2 = <math>q^2-q-1+3 q^{-1} - q^{-2} -3 q^{-3} +5 q^{-4} -5 q^{-6} +5 q^{-7} + q^{-8} -7 q^{-9} +5 q^{-10} +2 q^{-11} -7 q^{-12} +4 q^{-13} +3 q^{-14} -6 q^{-15} +3 q^{-16} +2 q^{-17} -4 q^{-18} +3 q^{-19} -2 q^{-21} + q^{-22} </math> | |
|||
coloured_jones_3 = <math>q^6-q^5-q^4+3 q^2-3-2 q^{-1} +5 q^{-2} +2 q^{-3} -3 q^{-4} -5 q^{-5} +5 q^{-6} +4 q^{-7} -2 q^{-8} -6 q^{-9} +3 q^{-10} +4 q^{-11} -6 q^{-13} +2 q^{-14} +3 q^{-15} -4 q^{-17} + q^{-18} + q^{-19} + q^{-20} - q^{-21} - q^{-22} + q^{-24} +2 q^{-25} -2 q^{-26} - q^{-27} +2 q^{-29} - q^{-30} + q^{-31} -2 q^{-32} + q^{-34} +2 q^{-35} -2 q^{-36} -2 q^{-37} +2 q^{-38} + q^{-39} -2 q^{-41} + q^{-42} </math> | |
|||
{{Display Coloured Jones|J2=<math>q^2-q-1+3 q^{-1} - q^{-2} -3 q^{-3} +5 q^{-4} -5 q^{-6} +5 q^{-7} + q^{-8} -7 q^{-9} +5 q^{-10} +2 q^{-11} -7 q^{-12} +4 q^{-13} +3 q^{-14} -6 q^{-15} +3 q^{-16} +2 q^{-17} -4 q^{-18} +3 q^{-19} -2 q^{-21} + q^{-22} </math>|J3=<math>q^6-q^5-q^4+3 q^2-3-2 q^{-1} +5 q^{-2} +2 q^{-3} -3 q^{-4} -5 q^{-5} +5 q^{-6} +4 q^{-7} -2 q^{-8} -6 q^{-9} +3 q^{-10} +4 q^{-11} -6 q^{-13} +2 q^{-14} +3 q^{-15} -4 q^{-17} + q^{-18} + q^{-19} + q^{-20} - q^{-21} - q^{-22} + q^{-24} +2 q^{-25} -2 q^{-26} - q^{-27} +2 q^{-29} - q^{-30} + q^{-31} -2 q^{-32} + q^{-34} +2 q^{-35} -2 q^{-36} -2 q^{-37} +2 q^{-38} + q^{-39} -2 q^{-41} + q^{-42} </math>|J4=<math>q^{12}-q^{11}-q^{10}+4 q^7-q^6-2 q^5-2 q^4-3 q^3+8 q^2+q-1-3 q^{-1} -8 q^{-2} +9 q^{-3} +2 q^{-4} +2 q^{-5} - q^{-6} -12 q^{-7} +9 q^{-8} +3 q^{-10} +2 q^{-11} -12 q^{-12} +10 q^{-13} -3 q^{-14} + q^{-15} +3 q^{-16} -12 q^{-17} +14 q^{-18} -4 q^{-19} -3 q^{-20} + q^{-21} -11 q^{-22} +20 q^{-23} -3 q^{-24} -7 q^{-25} -2 q^{-26} -11 q^{-27} +25 q^{-28} -2 q^{-29} -11 q^{-30} -4 q^{-31} -10 q^{-32} +29 q^{-33} - q^{-34} -15 q^{-35} -6 q^{-36} -8 q^{-37} +32 q^{-38} + q^{-39} -18 q^{-40} -9 q^{-41} -7 q^{-42} +31 q^{-43} +3 q^{-44} -14 q^{-45} -10 q^{-46} -10 q^{-47} +25 q^{-48} +5 q^{-49} -8 q^{-50} -7 q^{-51} -11 q^{-52} +17 q^{-53} +3 q^{-54} -3 q^{-55} -2 q^{-56} -10 q^{-57} +10 q^{-58} + q^{-59} -6 q^{-62} +4 q^{-63} + q^{-65} -2 q^{-67} + q^{-68} </math>|J5=<math>q^{20}-q^{19}-q^{18}+q^{15}+3 q^{14}-3 q^{12}-2 q^{11}-2 q^{10}+6 q^8+4 q^7-q^6-4 q^5-5 q^4-4 q^3+5 q^2+7 q+3- q^{-1} -6 q^{-2} -7 q^{-3} +2 q^{-4} +5 q^{-5} +4 q^{-6} +2 q^{-7} -3 q^{-8} -7 q^{-9} +3 q^{-10} +3 q^{-11} + q^{-12} -3 q^{-14} -5 q^{-15} +6 q^{-16} +7 q^{-17} -5 q^{-19} -8 q^{-20} -8 q^{-21} +11 q^{-22} +14 q^{-23} +5 q^{-24} -7 q^{-25} -18 q^{-26} -14 q^{-27} +12 q^{-28} +22 q^{-29} +12 q^{-30} -6 q^{-31} -25 q^{-32} -22 q^{-33} +9 q^{-34} +29 q^{-35} +20 q^{-36} -3 q^{-37} -29 q^{-38} -29 q^{-39} +4 q^{-40} +33 q^{-41} +27 q^{-42} -32 q^{-44} -33 q^{-45} +35 q^{-47} +34 q^{-48} + q^{-49} -36 q^{-50} -37 q^{-51} -2 q^{-52} +39 q^{-53} +41 q^{-54} + q^{-55} -40 q^{-56} -42 q^{-57} -4 q^{-58} +39 q^{-59} +46 q^{-60} +5 q^{-61} -39 q^{-62} -43 q^{-63} -10 q^{-64} +33 q^{-65} +45 q^{-66} +9 q^{-67} -29 q^{-68} -36 q^{-69} -13 q^{-70} +23 q^{-71} +36 q^{-72} +7 q^{-73} -19 q^{-74} -24 q^{-75} -9 q^{-76} +14 q^{-77} +22 q^{-78} +5 q^{-79} -13 q^{-80} -14 q^{-81} -3 q^{-82} +8 q^{-83} +10 q^{-84} +3 q^{-85} -6 q^{-86} -8 q^{-87} - q^{-88} +5 q^{-89} +2 q^{-90} +3 q^{-91} -2 q^{-92} -4 q^{-93} +2 q^{-95} + q^{-97} -2 q^{-99} + q^{-100} </math>|J6=<math>q^{30}-q^{29}-q^{28}+q^{25}+4 q^{23}-q^{22}-3 q^{21}-2 q^{20}-2 q^{19}-q^{17}+10 q^{16}+2 q^{15}-q^{14}-3 q^{13}-5 q^{12}-4 q^{11}-8 q^{10}+13 q^9+5 q^8+4 q^7+q^6-3 q^5-6 q^4-16 q^3+11 q^2+2 q+6+3 q^{-1} +3 q^{-2} -2 q^{-3} -19 q^{-4} +12 q^{-5} -2 q^{-6} +4 q^{-7} - q^{-8} +3 q^{-9} -19 q^{-11} +18 q^{-12} + q^{-13} +8 q^{-14} -5 q^{-15} -2 q^{-16} -6 q^{-17} -26 q^{-18} +20 q^{-19} +8 q^{-20} +21 q^{-21} -3 q^{-23} -14 q^{-24} -42 q^{-25} +11 q^{-26} +10 q^{-27} +35 q^{-28} +13 q^{-29} +6 q^{-30} -16 q^{-31} -57 q^{-32} -5 q^{-33} +3 q^{-34} +42 q^{-35} +25 q^{-36} +21 q^{-37} -8 q^{-38} -65 q^{-39} -20 q^{-40} -10 q^{-41} +40 q^{-42} +30 q^{-43} +35 q^{-44} +6 q^{-45} -65 q^{-46} -30 q^{-47} -23 q^{-48} +33 q^{-49} +29 q^{-50} +45 q^{-51} +22 q^{-52} -60 q^{-53} -35 q^{-54} -34 q^{-55} +23 q^{-56} +25 q^{-57} +51 q^{-58} +38 q^{-59} -54 q^{-60} -40 q^{-61} -42 q^{-62} +15 q^{-63} +23 q^{-64} +55 q^{-65} +48 q^{-66} -49 q^{-67} -47 q^{-68} -49 q^{-69} +11 q^{-70} +25 q^{-71} +62 q^{-72} +56 q^{-73} -49 q^{-74} -56 q^{-75} -57 q^{-76} +7 q^{-77} +29 q^{-78} +70 q^{-79} +64 q^{-80} -45 q^{-81} -60 q^{-82} -64 q^{-83} - q^{-84} +25 q^{-85} +70 q^{-86} +68 q^{-87} -33 q^{-88} -50 q^{-89} -61 q^{-90} -8 q^{-91} +13 q^{-92} +56 q^{-93} +59 q^{-94} -23 q^{-95} -32 q^{-96} -45 q^{-97} -4 q^{-98} +3 q^{-99} +37 q^{-100} +40 q^{-101} -23 q^{-102} -18 q^{-103} -27 q^{-104} +7 q^{-105} +3 q^{-106} +22 q^{-107} +22 q^{-108} -26 q^{-109} -9 q^{-110} -13 q^{-111} +11 q^{-112} +4 q^{-113} +13 q^{-114} +10 q^{-115} -22 q^{-116} -3 q^{-117} -7 q^{-118} +9 q^{-119} +2 q^{-120} +8 q^{-121} +4 q^{-122} -14 q^{-123} + q^{-124} -4 q^{-125} +5 q^{-126} +4 q^{-128} + q^{-129} -6 q^{-130} +2 q^{-131} -2 q^{-132} +2 q^{-133} + q^{-135} -2 q^{-137} + q^{-138} </math>|J7=<math>q^{42}-q^{41}-q^{40}+q^{37}+q^{35}+3 q^{34}-q^{33}-3 q^{32}-2 q^{31}-3 q^{30}+q^{29}+q^{27}+9 q^{26}+3 q^{25}-q^{24}-3 q^{23}-8 q^{22}-3 q^{21}-4 q^{20}-4 q^{19}+11 q^{18}+8 q^{17}+6 q^{16}+5 q^{15}-9 q^{14}-4 q^{13}-8 q^{12}-13 q^{11}+6 q^{10}+5 q^9+8 q^8+13 q^7-4 q^6-4 q^4-16 q^3+4 q^2-2 q+2+14 q^{-1} -5 q^{-2} + q^{-3} - q^{-4} -14 q^{-5} +10 q^{-6} +2 q^{-7} + q^{-8} +16 q^{-9} -9 q^{-10} -6 q^{-11} -8 q^{-12} -23 q^{-13} +12 q^{-14} +8 q^{-15} +12 q^{-16} +30 q^{-17} - q^{-18} -7 q^{-19} -17 q^{-20} -43 q^{-21} -7 q^{-22} +2 q^{-23} +17 q^{-24} +50 q^{-25} +20 q^{-26} +10 q^{-27} -11 q^{-28} -58 q^{-29} -32 q^{-30} -22 q^{-31} +3 q^{-32} +55 q^{-33} +41 q^{-34} +36 q^{-35} +11 q^{-36} -52 q^{-37} -44 q^{-38} -46 q^{-39} -25 q^{-40} +40 q^{-41} +44 q^{-42} +54 q^{-43} +36 q^{-44} -30 q^{-45} -35 q^{-46} -54 q^{-47} -49 q^{-48} +14 q^{-49} +28 q^{-50} +54 q^{-51} +52 q^{-52} -5 q^{-53} -12 q^{-54} -44 q^{-55} -57 q^{-56} -9 q^{-57} + q^{-58} +37 q^{-59} +55 q^{-60} +16 q^{-61} +12 q^{-62} -22 q^{-63} -52 q^{-64} -25 q^{-65} -27 q^{-66} +13 q^{-67} +49 q^{-68} +30 q^{-69} +36 q^{-70} + q^{-71} -42 q^{-72} -36 q^{-73} -49 q^{-74} -11 q^{-75} +38 q^{-76} +41 q^{-77} +57 q^{-78} +21 q^{-79} -33 q^{-80} -43 q^{-81} -67 q^{-82} -31 q^{-83} +30 q^{-84} +50 q^{-85} +73 q^{-86} +33 q^{-87} -28 q^{-88} -52 q^{-89} -80 q^{-90} -40 q^{-91} +30 q^{-92} +62 q^{-93} +85 q^{-94} +38 q^{-95} -32 q^{-96} -68 q^{-97} -90 q^{-98} -41 q^{-99} +36 q^{-100} +76 q^{-101} +97 q^{-102} +45 q^{-103} -37 q^{-104} -84 q^{-105} -103 q^{-106} -49 q^{-107} +33 q^{-108} +84 q^{-109} +108 q^{-110} +59 q^{-111} -25 q^{-112} -80 q^{-113} -109 q^{-114} -66 q^{-115} +13 q^{-116} +67 q^{-117} +103 q^{-118} +70 q^{-119} +2 q^{-120} -50 q^{-121} -92 q^{-122} -70 q^{-123} -12 q^{-124} +35 q^{-125} +73 q^{-126} +59 q^{-127} +17 q^{-128} -14 q^{-129} -54 q^{-130} -49 q^{-131} -16 q^{-132} +4 q^{-133} +36 q^{-134} +30 q^{-135} +12 q^{-136} +6 q^{-137} -19 q^{-138} -16 q^{-139} -7 q^{-140} -11 q^{-141} +9 q^{-142} +6 q^{-143} -2 q^{-144} +11 q^{-145} +5 q^{-147} +4 q^{-148} -15 q^{-149} - q^{-150} -7 q^{-151} -7 q^{-152} +11 q^{-153} +3 q^{-154} +10 q^{-155} +8 q^{-156} -10 q^{-157} -4 q^{-158} -8 q^{-159} -6 q^{-160} +8 q^{-161} +2 q^{-162} +5 q^{-163} +6 q^{-164} -6 q^{-165} - q^{-166} -4 q^{-167} -4 q^{-168} +5 q^{-169} +2 q^{-171} +2 q^{-172} -3 q^{-173} -2 q^{-176} +2 q^{-177} + q^{-179} -2 q^{-181} + q^{-182} </math>}} |
|||
coloured_jones_4 = <math>q^{12}-q^{11}-q^{10}+4 q^7-q^6-2 q^5-2 q^4-3 q^3+8 q^2+q-1-3 q^{-1} -8 q^{-2} +9 q^{-3} +2 q^{-4} +2 q^{-5} - q^{-6} -12 q^{-7} +9 q^{-8} +3 q^{-10} +2 q^{-11} -12 q^{-12} +10 q^{-13} -3 q^{-14} + q^{-15} +3 q^{-16} -12 q^{-17} +14 q^{-18} -4 q^{-19} -3 q^{-20} + q^{-21} -11 q^{-22} +20 q^{-23} -3 q^{-24} -7 q^{-25} -2 q^{-26} -11 q^{-27} +25 q^{-28} -2 q^{-29} -11 q^{-30} -4 q^{-31} -10 q^{-32} +29 q^{-33} - q^{-34} -15 q^{-35} -6 q^{-36} -8 q^{-37} +32 q^{-38} + q^{-39} -18 q^{-40} -9 q^{-41} -7 q^{-42} +31 q^{-43} +3 q^{-44} -14 q^{-45} -10 q^{-46} -10 q^{-47} +25 q^{-48} +5 q^{-49} -8 q^{-50} -7 q^{-51} -11 q^{-52} +17 q^{-53} +3 q^{-54} -3 q^{-55} -2 q^{-56} -10 q^{-57} +10 q^{-58} + q^{-59} -6 q^{-62} +4 q^{-63} + q^{-65} -2 q^{-67} + q^{-68} </math> | |
|||
coloured_jones_5 = <math>q^{20}-q^{19}-q^{18}+q^{15}+3 q^{14}-3 q^{12}-2 q^{11}-2 q^{10}+6 q^8+4 q^7-q^6-4 q^5-5 q^4-4 q^3+5 q^2+7 q+3- q^{-1} -6 q^{-2} -7 q^{-3} +2 q^{-4} +5 q^{-5} +4 q^{-6} +2 q^{-7} -3 q^{-8} -7 q^{-9} +3 q^{-10} +3 q^{-11} + q^{-12} -3 q^{-14} -5 q^{-15} +6 q^{-16} +7 q^{-17} -5 q^{-19} -8 q^{-20} -8 q^{-21} +11 q^{-22} +14 q^{-23} +5 q^{-24} -7 q^{-25} -18 q^{-26} -14 q^{-27} +12 q^{-28} +22 q^{-29} +12 q^{-30} -6 q^{-31} -25 q^{-32} -22 q^{-33} +9 q^{-34} +29 q^{-35} +20 q^{-36} -3 q^{-37} -29 q^{-38} -29 q^{-39} +4 q^{-40} +33 q^{-41} +27 q^{-42} -32 q^{-44} -33 q^{-45} +35 q^{-47} +34 q^{-48} + q^{-49} -36 q^{-50} -37 q^{-51} -2 q^{-52} +39 q^{-53} +41 q^{-54} + q^{-55} -40 q^{-56} -42 q^{-57} -4 q^{-58} +39 q^{-59} +46 q^{-60} +5 q^{-61} -39 q^{-62} -43 q^{-63} -10 q^{-64} +33 q^{-65} +45 q^{-66} +9 q^{-67} -29 q^{-68} -36 q^{-69} -13 q^{-70} +23 q^{-71} +36 q^{-72} +7 q^{-73} -19 q^{-74} -24 q^{-75} -9 q^{-76} +14 q^{-77} +22 q^{-78} +5 q^{-79} -13 q^{-80} -14 q^{-81} -3 q^{-82} +8 q^{-83} +10 q^{-84} +3 q^{-85} -6 q^{-86} -8 q^{-87} - q^{-88} +5 q^{-89} +2 q^{-90} +3 q^{-91} -2 q^{-92} -4 q^{-93} +2 q^{-95} + q^{-97} -2 q^{-99} + q^{-100} </math> | |
|||
{{Computer Talk Header}} |
|||
coloured_jones_6 = <math>q^{30}-q^{29}-q^{28}+q^{25}+4 q^{23}-q^{22}-3 q^{21}-2 q^{20}-2 q^{19}-q^{17}+10 q^{16}+2 q^{15}-q^{14}-3 q^{13}-5 q^{12}-4 q^{11}-8 q^{10}+13 q^9+5 q^8+4 q^7+q^6-3 q^5-6 q^4-16 q^3+11 q^2+2 q+6+3 q^{-1} +3 q^{-2} -2 q^{-3} -19 q^{-4} +12 q^{-5} -2 q^{-6} +4 q^{-7} - q^{-8} +3 q^{-9} -19 q^{-11} +18 q^{-12} + q^{-13} +8 q^{-14} -5 q^{-15} -2 q^{-16} -6 q^{-17} -26 q^{-18} +20 q^{-19} +8 q^{-20} +21 q^{-21} -3 q^{-23} -14 q^{-24} -42 q^{-25} +11 q^{-26} +10 q^{-27} +35 q^{-28} +13 q^{-29} +6 q^{-30} -16 q^{-31} -57 q^{-32} -5 q^{-33} +3 q^{-34} +42 q^{-35} +25 q^{-36} +21 q^{-37} -8 q^{-38} -65 q^{-39} -20 q^{-40} -10 q^{-41} +40 q^{-42} +30 q^{-43} +35 q^{-44} +6 q^{-45} -65 q^{-46} -30 q^{-47} -23 q^{-48} +33 q^{-49} +29 q^{-50} +45 q^{-51} +22 q^{-52} -60 q^{-53} -35 q^{-54} -34 q^{-55} +23 q^{-56} +25 q^{-57} +51 q^{-58} +38 q^{-59} -54 q^{-60} -40 q^{-61} -42 q^{-62} +15 q^{-63} +23 q^{-64} +55 q^{-65} +48 q^{-66} -49 q^{-67} -47 q^{-68} -49 q^{-69} +11 q^{-70} +25 q^{-71} +62 q^{-72} +56 q^{-73} -49 q^{-74} -56 q^{-75} -57 q^{-76} +7 q^{-77} +29 q^{-78} +70 q^{-79} +64 q^{-80} -45 q^{-81} -60 q^{-82} -64 q^{-83} - q^{-84} +25 q^{-85} +70 q^{-86} +68 q^{-87} -33 q^{-88} -50 q^{-89} -61 q^{-90} -8 q^{-91} +13 q^{-92} +56 q^{-93} +59 q^{-94} -23 q^{-95} -32 q^{-96} -45 q^{-97} -4 q^{-98} +3 q^{-99} +37 q^{-100} +40 q^{-101} -23 q^{-102} -18 q^{-103} -27 q^{-104} +7 q^{-105} +3 q^{-106} +22 q^{-107} +22 q^{-108} -26 q^{-109} -9 q^{-110} -13 q^{-111} +11 q^{-112} +4 q^{-113} +13 q^{-114} +10 q^{-115} -22 q^{-116} -3 q^{-117} -7 q^{-118} +9 q^{-119} +2 q^{-120} +8 q^{-121} +4 q^{-122} -14 q^{-123} + q^{-124} -4 q^{-125} +5 q^{-126} +4 q^{-128} + q^{-129} -6 q^{-130} +2 q^{-131} -2 q^{-132} +2 q^{-133} + q^{-135} -2 q^{-137} + q^{-138} </math> | |
|||
coloured_jones_7 = <math>q^{42}-q^{41}-q^{40}+q^{37}+q^{35}+3 q^{34}-q^{33}-3 q^{32}-2 q^{31}-3 q^{30}+q^{29}+q^{27}+9 q^{26}+3 q^{25}-q^{24}-3 q^{23}-8 q^{22}-3 q^{21}-4 q^{20}-4 q^{19}+11 q^{18}+8 q^{17}+6 q^{16}+5 q^{15}-9 q^{14}-4 q^{13}-8 q^{12}-13 q^{11}+6 q^{10}+5 q^9+8 q^8+13 q^7-4 q^6-4 q^4-16 q^3+4 q^2-2 q+2+14 q^{-1} -5 q^{-2} + q^{-3} - q^{-4} -14 q^{-5} +10 q^{-6} +2 q^{-7} + q^{-8} +16 q^{-9} -9 q^{-10} -6 q^{-11} -8 q^{-12} -23 q^{-13} +12 q^{-14} +8 q^{-15} +12 q^{-16} +30 q^{-17} - q^{-18} -7 q^{-19} -17 q^{-20} -43 q^{-21} -7 q^{-22} +2 q^{-23} +17 q^{-24} +50 q^{-25} +20 q^{-26} +10 q^{-27} -11 q^{-28} -58 q^{-29} -32 q^{-30} -22 q^{-31} +3 q^{-32} +55 q^{-33} +41 q^{-34} +36 q^{-35} +11 q^{-36} -52 q^{-37} -44 q^{-38} -46 q^{-39} -25 q^{-40} +40 q^{-41} +44 q^{-42} +54 q^{-43} +36 q^{-44} -30 q^{-45} -35 q^{-46} -54 q^{-47} -49 q^{-48} +14 q^{-49} +28 q^{-50} +54 q^{-51} +52 q^{-52} -5 q^{-53} -12 q^{-54} -44 q^{-55} -57 q^{-56} -9 q^{-57} + q^{-58} +37 q^{-59} +55 q^{-60} +16 q^{-61} +12 q^{-62} -22 q^{-63} -52 q^{-64} -25 q^{-65} -27 q^{-66} +13 q^{-67} +49 q^{-68} +30 q^{-69} +36 q^{-70} + q^{-71} -42 q^{-72} -36 q^{-73} -49 q^{-74} -11 q^{-75} +38 q^{-76} +41 q^{-77} +57 q^{-78} +21 q^{-79} -33 q^{-80} -43 q^{-81} -67 q^{-82} -31 q^{-83} +30 q^{-84} +50 q^{-85} +73 q^{-86} +33 q^{-87} -28 q^{-88} -52 q^{-89} -80 q^{-90} -40 q^{-91} +30 q^{-92} +62 q^{-93} +85 q^{-94} +38 q^{-95} -32 q^{-96} -68 q^{-97} -90 q^{-98} -41 q^{-99} +36 q^{-100} +76 q^{-101} +97 q^{-102} +45 q^{-103} -37 q^{-104} -84 q^{-105} -103 q^{-106} -49 q^{-107} +33 q^{-108} +84 q^{-109} +108 q^{-110} +59 q^{-111} -25 q^{-112} -80 q^{-113} -109 q^{-114} -66 q^{-115} +13 q^{-116} +67 q^{-117} +103 q^{-118} +70 q^{-119} +2 q^{-120} -50 q^{-121} -92 q^{-122} -70 q^{-123} -12 q^{-124} +35 q^{-125} +73 q^{-126} +59 q^{-127} +17 q^{-128} -14 q^{-129} -54 q^{-130} -49 q^{-131} -16 q^{-132} +4 q^{-133} +36 q^{-134} +30 q^{-135} +12 q^{-136} +6 q^{-137} -19 q^{-138} -16 q^{-139} -7 q^{-140} -11 q^{-141} +9 q^{-142} +6 q^{-143} -2 q^{-144} +11 q^{-145} +5 q^{-147} +4 q^{-148} -15 q^{-149} - q^{-150} -7 q^{-151} -7 q^{-152} +11 q^{-153} +3 q^{-154} +10 q^{-155} +8 q^{-156} -10 q^{-157} -4 q^{-158} -8 q^{-159} -6 q^{-160} +8 q^{-161} +2 q^{-162} +5 q^{-163} +6 q^{-164} -6 q^{-165} - q^{-166} -4 q^{-167} -4 q^{-168} +5 q^{-169} +2 q^{-171} +2 q^{-172} -3 q^{-173} -2 q^{-176} +2 q^{-177} + q^{-179} -2 q^{-181} + q^{-182} </math> | |
|||
<table> |
|||
computer_talk = |
|||
<tr valign=top> |
|||
<table> |
|||
<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:= </pre></td> |
|||
<tr valign=top> |
|||
<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td> |
|||
<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:= </pre></td> |
|||
</tr> |
|||
<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td> |
|||
</tr> |
|||
<tr valign=top><td colspan=2>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</td></tr> |
|||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[Knot[8, 2]]</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[Knot[8, 2]]</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[1, 4, 2, 5], X[5, 12, 6, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], |
|||
X[7, 14, 8, 15], X[9, 16, 10, 1], X[13, 6, 14, 7], X[15, 8, 16, 9]]</nowiki></pre></td></tr> |
X[7, 14, 8, 15], X[9, 16, 10, 1], X[13, 6, 14, 7], X[15, 8, 16, 9]]</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[3]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>GaussCode[Knot[8, 2]]</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[-1, 4, -3, 1, -2, 7, -5, 8, -6, 3, -4, 2, -7, 5, -8, 6]</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[4]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>DTCode[Knot[8, 2]]</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>DTCode[4, 10, 12, 14, 16, 2, 6, 8]</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[ |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[5]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>br = BR[Knot[8, 2]]</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[ |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[3, {-1, -1, -1, -1, -1, 2, -1, 2}]</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[6]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{First[br], Crossings[br]}</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{3, 8}</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[7]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>BraidIndex[Knot[8, 2]]</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>3</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Show[DrawMorseLink[Knot[8, 2]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:8_2_ML.gif]]</td></tr><tr valign=top><td><tt><font color=blue>Out[8]=</font></tt><td><tt><font color=black>-Graphics-</font></tt></td></tr> |
||
<tr valign=top><td><pre |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>(#[Knot[8, 2]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Reversible, 2, 3, 2, {4, 5}, 1}</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[ |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[10]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>alex = Alexander[Knot[8, 2]][t]</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[ |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -3 3 3 2 3 |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Show[DrawMorseLink[Knot[8, 2]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:8_2_ML.gif]]</td></tr><tr valign=top><td><tt><font color=blue>Out[8]=</font></tt><td><tt><font color=black>-Graphics-</font></tt></td></tr> |
|||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>(#[Knot[8, 2]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Reversible, 2, 3, 2, {4, 5}, 1}</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[10]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>alex = Alexander[Knot[8, 2]][t]</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -3 3 3 2 3 |
|||
3 - t + -- - - - 3 t + 3 t - t |
3 - t + -- - - - 3 t + 3 t - t |
||
2 t |
2 t |
||
t</nowiki></pre></td></tr> |
t</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[11]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Conway[Knot[8, 2]][z]</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 4 6 |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 4 6 |
|||
1 - 3 z - z</nowiki></pre></td></tr> |
1 - 3 z - z</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[12]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[8, 2], Knot[11, NonAlternating, 6]}</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[13]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{KnotDet[Knot[8, 2]], KnotSignature[Knot[8, 2]]}</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{17, -4}</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[ |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[14]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Jones[Knot[8, 2]][q]</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[ |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -8 2 2 3 3 2 2 1 |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[14]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Jones[Knot[8, 2]][q]</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -8 2 2 3 3 2 2 1 |
|||
1 + q - -- + -- - -- + -- - -- + -- - - |
1 + q - -- + -- - -- + -- - -- + -- - - |
||
7 6 5 4 3 2 q |
7 6 5 4 3 2 q |
||
q q q q q q</nowiki></pre></td></tr> |
q q q q q q</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[15]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[8, 2]}</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[16]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Knot[8, 2]][q]</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[16]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -24 -18 -16 -12 -10 -6 -4 -2 |
|||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[16]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Knot[8, 2]][q]</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[16]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -24 -18 -16 -12 -10 -6 -4 -2 |
|||
1 + q - q - q - q + q + q + q + q</nowiki></pre></td></tr> |
1 + q - q - q - q + q + q + q + q</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[17]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>HOMFLYPT[Knot[8, 2]][a, z]</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[17]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 4 6 2 2 4 2 6 2 2 4 4 4 |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[17]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 4 6 2 2 4 2 6 2 2 4 4 4 |
|||
3 a - 3 a + a + 4 a z - 7 a z + 3 a z + a z - 5 a z + |
3 a - 3 a + a + 4 a z - 7 a z + 3 a z + a z - 5 a z + |
||
6 4 4 6 |
6 4 4 6 |
||
a z - a z</nowiki></pre></td></tr> |
a z - a z</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[18]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Knot[8, 2]][a, z]</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[18]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 4 6 3 5 7 9 2 2 4 2 |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[18]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 4 6 3 5 7 9 2 2 4 2 |
|||
-3 a - 3 a - a + a z + a z - a z - a z + 7 a z + 12 a z + |
-3 a - 3 a - a + a z + a z - a z - a z + 7 a z + 12 a z + |
||
Line 156: | Line 105: | ||
7 5 2 6 4 6 6 6 3 7 5 7 |
7 5 2 6 4 6 6 6 3 7 5 7 |
||
2 a z + a z + 3 a z + 2 a z + a z + a z</nowiki></pre></td></tr> |
2 a z + a z + 3 a z + 2 a z + a z + a z</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[19]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{Vassiliev[2][Knot[8, 2]], Vassiliev[3][Knot[8, 2]]}</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[19]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{0, 1}</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[20]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Knot[8, 2]][q, t]</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[20]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -5 2 1 1 1 1 1 2 |
|||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[20]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Knot[8, 2]][q, t]</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[20]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -5 2 1 1 1 1 1 2 |
|||
q + -- + ------ + ------ + ------ + ------ + ------ + ------ + |
q + -- + ------ + ------ + ------ + ------ + ------ + ------ + |
||
3 17 6 15 5 13 5 13 4 11 4 11 3 |
3 17 6 15 5 13 5 13 4 11 4 11 3 |
||
Line 170: | Line 117: | ||
9 3 9 2 7 2 7 5 3 |
9 3 9 2 7 2 7 5 3 |
||
q t q t q t q t q t q</nowiki></pre></td></tr> |
q t q t q t q t q t q</nowiki></pre></td></tr> |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[21]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>ColouredJones[Knot[8, 2], 2][q]</nowiki></pre></td></tr> |
|||
<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[21]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -22 2 3 4 2 3 6 3 4 7 2 |
||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[21]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -22 2 3 4 2 3 6 3 4 7 2 |
|||
-1 + q - --- + --- - --- + --- + --- - --- + --- + --- - --- + --- + |
-1 + q - --- + --- - --- + --- + --- - --- + --- + --- - --- + --- + |
||
21 19 18 17 16 15 14 13 12 11 |
21 19 18 17 16 15 14 13 12 11 |
||
Line 181: | Line 127: | ||
10 9 7 6 4 3 q |
10 9 7 6 4 3 q |
||
q q q q q q</nowiki></pre></td></tr> |
q q q q q q</nowiki></pre></td></tr> |
||
</table> }} |
|||
</table> |
|||
{| width=100% |
|||
|align=left|See/edit the [[Rolfsen_Splice_Template]]. |
|||
Back to the [[#top|top]]. |
|||
|align=right|{{Knot Navigation Links|ext=gif}} |
|||
|} |
|||
[[Category:Knot Page]] |
Revision as of 09:35, 30 August 2005
|
|
(KnotPlot image) |
See the full Rolfsen Knot Table. Visit 8 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
Planar diagram presentation | X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X7,14,8,15 X9,16,10,1 X13,6,14,7 X15,8,16,9 |
Gauss code | -1, 4, -3, 1, -2, 7, -5, 8, -6, 3, -4, 2, -7, 5, -8, 6 |
Dowker-Thistlethwaite code | 4 10 12 14 16 2 6 8 |
Conway Notation | [512] |
Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||
Length is 8, width is 3, Braid index is 3 |
[{10, 2}, {1, 8}, {9, 3}, {2, 4}, {8, 10}, {3, 5}, {4, 6}, {5, 7}, {6, 9}, {7, 1}] |
[edit Notes on presentations of 8 2]
KnotTheory`
. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
|
K = Knot["8 2"];
|
In[4]:=
|
PD[K]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X7,14,8,15 X9,16,10,1 X13,6,14,7 X15,8,16,9 |
In[5]:=
|
GaussCode[K]
|
Out[5]=
|
-1, 4, -3, 1, -2, 7, -5, 8, -6, 3, -4, 2, -7, 5, -8, 6 |
In[6]:=
|
DTCode[K]
|
Out[6]=
|
4 10 12 14 16 2 6 8 |
(The path below may be different on your system)
In[7]:=
|
AppendTo[$Path, "C:/bin/LinKnot/"];
|
In[8]:=
|
ConwayNotation[K]
|
Out[8]=
|
[512] |
In[9]:=
|
br = BR[K]
|
KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
|
Out[9]=
|
In[10]:=
|
{First[br], Crossings[br], BraidIndex[K]}
|
KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
|
KnotTheory::loading: Loading precomputed data in IndianaData`.
|
Out[10]=
|
{ 3, 8, 3 } |
In[11]:=
|
Show[BraidPlot[br]]
|
Out[11]=
|
-Graphics- |
In[12]:=
|
Show[DrawMorseLink[K]]
|
KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
|
KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
|
Out[12]=
|
-Graphics- |
In[13]:=
|
ap = ArcPresentation[K]
|
Out[13]=
|
ArcPresentation[{10, 2}, {1, 8}, {9, 3}, {2, 4}, {8, 10}, {3, 5}, {4, 6}, {5, 7}, {6, 9}, {7, 1}] |
In[14]:=
|
Draw[ap]
|
Out[14]=
|
-Graphics- |
Three dimensional invariants
|
Four dimensional invariants
|
Polynomial invariants
A1 Invariants.
Weight | Invariant |
---|---|
1 | |
2 | |
3 | |
4 | |
5 | |
6 |
A2 Invariants.
Weight | Invariant |
---|---|
1,0 | |
1,1 | |
2,0 | |
3,0 |
A3 Invariants.
Weight | Invariant |
---|---|
0,1,0 | |
1,0,0 | |
1,0,1 |
A4 Invariants.
Weight | Invariant |
---|---|
0,1,0,0 | |
1,0,0,0 |
B2 Invariants.
Weight | Invariant |
---|---|
0,1 | |
1,0 |
D4 Invariants.
Weight | Invariant |
---|---|
1,0,0,0 |
G2 Invariants.
Weight | Invariant |
---|---|
1,0 |
.
KnotTheory`
, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
|
K = Knot["8 2"];
|
In[4]:=
|
Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
In[5]:=
|
Conway[K][z]
|
Out[5]=
|
In[6]:=
|
Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
|
In[7]:=
|
{KnotDet[K], KnotSignature[K]}
|
Out[7]=
|
{ 17, -4 } |
In[8]:=
|
Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
|
In[9]:=
|
HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
|
In[10]:=
|
Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
|
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11n6,}
Same Jones Polynomial (up to mirroring, ): {}
KnotTheory`
. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
|
K = Knot["8 2"];
|
In[4]:=
|
{A = Alexander[K][t], J = Jones[K][q]}
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[4]=
|
{ , } |
In[5]:=
|
DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
|
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
|
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
|
Out[5]=
|
{K11n6,} |
In[6]:=
|
DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
Out[6]=
|
{} |
Vassiliev invariants
V2 and V3: | (0, 1) |
V2,1 through 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 -4 is the signature of 8 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
|
Integral Khovanov Homology
(db, data source) |
|
The Coloured Jones Polynomials
2 | |
3 | |
4 | |
5 | |
6 | |
7 |
Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`
. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.
Modifying This Page
Read me first: Modifying Knot Pages
See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top. |
|