9 21

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9 20.gif

9_20

9 22.gif

9_22

9 21.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

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[edit 9 21 Quick Notes]
[edit 9 21 Further Notes and Views]


Knot presentations

Planar diagram presentation X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X13,1,14,18 X5,15,6,14 X17,7,18,6 X7,17,8,16 X15,9,16,8
Gauss code -1, 4, -3, 1, -6, 7, -8, 9, -2, 3, -4, 2, -5, 6, -9, 8, -7, 5
Dowker-Thistlethwaite code 4 10 14 16 12 2 18 8 6
Conway Notation [31122]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif

Length is 10, width is 5,

Braid index is 5

9 21 ML.gif 9 21 AP.gif
[{11, 6}, {7, 5}, {6, 10}, {1, 7}, {8, 11}, {10, 4}, {5, 2}, {3, 1}, {4, 9}, {2, 8}, {9, 3}]

[edit Notes on presentations of 9 21]


Computer Talk
The above data is available with the Mathematica package 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.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["9 21"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X13,1,14,18 X5,15,6,14 X17,7,18,6 X7,17,8,16 X15,9,16,8
In[5]:= GaussCode[K]
Out[5]= -1, 4, -3, 1, -6, 7, -8, 9, -2, 3, -4, 2, -5, 6, -9, 8, -7, 5
In[6]:= DTCode[K]
Out[6]= 4 10 14 16 12 2 18 8 6

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [31122]
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]= [math]\displaystyle{ \textrm{BR}(5,\{1,1,2,-1,2,-3,2,4,-3,4\}) }[/math]
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]= { 5, 10, 5 }
In[11]:= Show[BraidPlot[br]]
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
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."
9 21 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{11, 6}, {7, 5}, {6, 10}, {1, 7}, {8, 11}, {10, 4}, {5, 2}, {3, 1}, {4, 9}, {2, 8}, {9, 3}]
In[14]:= Draw[ap]
9 21 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 2
Super bridge index [math]\displaystyle{ \{4,7\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [-1][-10]
Hyperbolic Volume 10.1833
A-Polynomial See Data:9 21/A-polynomial

[edit Notes for 9 21's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math]
Topological 4 genus [math]\displaystyle{ 1 }[/math]
Concordance genus [math]\displaystyle{ 2 }[/math]
Rasmussen s-Invariant 2

[edit Notes for 9 21's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -2 t^2+11 t-17+11 t^{-1} -2 t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ -2 z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 43, 2 }
Jones polynomial [math]\displaystyle{ -q^8+2 q^7-4 q^6+6 q^5-7 q^4+8 q^3-6 q^2+5 q-3+ q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^4 a^{-2} -z^4 a^{-4} +2 z^2 a^{-6} +z^2+ a^{-2} + a^{-6} - a^{-8} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^8 a^{-4} +z^8 a^{-6} +3 z^7 a^{-3} +5 z^7 a^{-5} +2 z^7 a^{-7} +4 z^6 a^{-2} +4 z^6 a^{-4} +2 z^6 a^{-6} +2 z^6 a^{-8} +3 z^5 a^{-1} -3 z^5 a^{-3} -10 z^5 a^{-5} -3 z^5 a^{-7} +z^5 a^{-9} -6 z^4 a^{-2} -9 z^4 a^{-4} -7 z^4 a^{-6} -5 z^4 a^{-8} +z^4-4 z^3 a^{-1} +2 z^3 a^{-3} +9 z^3 a^{-5} -3 z^3 a^{-9} +3 z^2 a^{-2} +6 z^2 a^{-4} +5 z^2 a^{-6} +3 z^2 a^{-8} -z^2-z a^{-3} -3 z a^{-5} +2 z a^{-9} - a^{-2} - a^{-6} - a^{-8} }[/math]
The A2 invariant [math]\displaystyle{ q^4-q^2-1+2 q^{-2} - q^{-4} +2 q^{-6} + q^{-8} + q^{-12} - q^{-14} +2 q^{-16} - q^{-20} + q^{-22} - q^{-24} - q^{-26} }[/math]
The G2 invariant [math]\displaystyle{ q^{18}-2 q^{16}+4 q^{14}-6 q^{12}+4 q^{10}-q^8-4 q^6+13 q^4-17 q^2+22-19 q^{-2} +5 q^{-4} +10 q^{-6} -27 q^{-8} +38 q^{-10} -39 q^{-12} +28 q^{-14} -7 q^{-16} -17 q^{-18} +36 q^{-20} -40 q^{-22} +30 q^{-24} -9 q^{-26} -13 q^{-28} +24 q^{-30} -21 q^{-32} +8 q^{-34} +18 q^{-36} -33 q^{-38} +40 q^{-40} -24 q^{-42} -4 q^{-44} +36 q^{-46} -58 q^{-48} +63 q^{-50} -46 q^{-52} +17 q^{-54} +18 q^{-56} -45 q^{-58} +57 q^{-60} -50 q^{-62} +27 q^{-64} -25 q^{-68} +32 q^{-70} -23 q^{-72} +7 q^{-74} +16 q^{-76} -28 q^{-78} +26 q^{-80} -10 q^{-82} -14 q^{-84} +36 q^{-86} -44 q^{-88} +36 q^{-90} -17 q^{-92} -8 q^{-94} +27 q^{-96} -37 q^{-98} +35 q^{-100} -23 q^{-102} +6 q^{-104} +6 q^{-106} -16 q^{-108} +16 q^{-110} -14 q^{-112} +9 q^{-114} -3 q^{-116} -2 q^{-118} +3 q^{-120} -4 q^{-122} +3 q^{-124} - q^{-126} + q^{-128} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_21.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^3-2 q+2 q^{-1} - q^{-3} +2 q^{-5} + q^{-7} - q^{-9} +2 q^{-11} -2 q^{-13} + q^{-15} - q^{-17} }[/math]
2 [math]\displaystyle{ q^{10}-2 q^8-q^6+6 q^4-4 q^2-5+11 q^{-2} -3 q^{-4} -9 q^{-6} +11 q^{-8} + q^{-10} -8 q^{-12} +5 q^{-14} +4 q^{-16} -2 q^{-18} -5 q^{-20} +6 q^{-22} +5 q^{-24} -11 q^{-26} +3 q^{-28} +8 q^{-30} -11 q^{-32} - q^{-34} +8 q^{-36} -5 q^{-38} -2 q^{-40} +4 q^{-42} - q^{-44} - q^{-46} + q^{-48} }[/math]
3 [math]\displaystyle{ q^{21}-2 q^{19}-q^{17}+3 q^{15}+3 q^{13}-4 q^{11}-8 q^9+8 q^7+13 q^5-9 q^3-21 q+7 q^{-1} +32 q^{-3} -3 q^{-5} -40 q^{-7} -5 q^{-9} +46 q^{-11} +14 q^{-13} -41 q^{-15} -22 q^{-17} +38 q^{-19} +26 q^{-21} -25 q^{-23} -28 q^{-25} +13 q^{-27} +25 q^{-29} +2 q^{-31} -21 q^{-33} -16 q^{-35} +20 q^{-37} +26 q^{-39} -12 q^{-41} -38 q^{-43} +6 q^{-45} +42 q^{-47} + q^{-49} -46 q^{-51} -10 q^{-53} +41 q^{-55} +19 q^{-57} -36 q^{-59} -24 q^{-61} +24 q^{-63} +27 q^{-65} -13 q^{-67} -23 q^{-69} +5 q^{-71} +17 q^{-73} + q^{-75} -11 q^{-77} -2 q^{-79} +6 q^{-81} +2 q^{-83} -3 q^{-85} - q^{-87} + q^{-89} + q^{-91} - q^{-93} }[/math]
4 [math]\displaystyle{ q^{36}-2 q^{34}-q^{32}+3 q^{30}+3 q^{26}-7 q^{24}-3 q^{22}+9 q^{20}+q^{18}+9 q^{16}-22 q^{14}-16 q^{12}+22 q^{10}+21 q^8+29 q^6-50 q^4-59 q^2+16+62 q^{-2} +97 q^{-4} -54 q^{-6} -135 q^{-8} -51 q^{-10} +79 q^{-12} +191 q^{-14} +6 q^{-16} -168 q^{-18} -144 q^{-20} +28 q^{-22} +231 q^{-24} +92 q^{-26} -118 q^{-28} -179 q^{-30} -51 q^{-32} +173 q^{-34} +128 q^{-36} -27 q^{-38} -136 q^{-40} -97 q^{-42} +68 q^{-44} +109 q^{-46} +52 q^{-48} -59 q^{-50} -104 q^{-52} -38 q^{-54} +77 q^{-56} +114 q^{-58} +5 q^{-60} -107 q^{-62} -126 q^{-64} +47 q^{-66} +162 q^{-68} +69 q^{-70} -94 q^{-72} -198 q^{-74} -2 q^{-76} +178 q^{-78} +135 q^{-80} -42 q^{-82} -224 q^{-84} -75 q^{-86} +123 q^{-88} +168 q^{-90} +50 q^{-92} -170 q^{-94} -123 q^{-96} +20 q^{-98} +125 q^{-100} +110 q^{-102} -65 q^{-104} -96 q^{-106} -50 q^{-108} +42 q^{-110} +91 q^{-112} +4 q^{-114} -33 q^{-116} -47 q^{-118} -8 q^{-120} +39 q^{-122} +13 q^{-124} + q^{-126} -19 q^{-128} -11 q^{-130} +11 q^{-132} +3 q^{-134} +4 q^{-136} -4 q^{-138} -4 q^{-140} +3 q^{-142} + q^{-146} - q^{-148} - q^{-150} + q^{-152} }[/math]
5 [math]\displaystyle{ q^{55}-2 q^{53}-q^{51}+3 q^{49}-2 q^{41}-2 q^{39}+5 q^{37}+4 q^{35}-6 q^{33}-8 q^{31}-5 q^{29}+6 q^{27}+18 q^{25}+24 q^{23}-3 q^{21}-46 q^{19}-51 q^{17}-10 q^{15}+62 q^{13}+109 q^{11}+65 q^9-75 q^7-195 q^5-154 q^3+44 q+271 q^{-1} +307 q^{-3} +54 q^{-5} -326 q^{-7} -489 q^{-9} -214 q^{-11} +313 q^{-13} +652 q^{-15} +448 q^{-17} -214 q^{-19} -771 q^{-21} -680 q^{-23} +38 q^{-25} +782 q^{-27} +884 q^{-29} +188 q^{-31} -703 q^{-33} -984 q^{-35} -407 q^{-37} +529 q^{-39} +991 q^{-41} +568 q^{-43} -320 q^{-45} -875 q^{-47} -655 q^{-49} +101 q^{-51} +710 q^{-53} +657 q^{-55} +70 q^{-57} -492 q^{-59} -598 q^{-61} -214 q^{-63} +299 q^{-65} +509 q^{-67} +303 q^{-69} -122 q^{-71} -417 q^{-73} -373 q^{-75} -36 q^{-77} +353 q^{-79} +446 q^{-81} +147 q^{-83} -302 q^{-85} -520 q^{-87} -281 q^{-89} +266 q^{-91} +622 q^{-93} +398 q^{-95} -228 q^{-97} -701 q^{-99} -552 q^{-101} +156 q^{-103} +774 q^{-105} +703 q^{-107} -39 q^{-109} -782 q^{-111} -844 q^{-113} -126 q^{-115} +722 q^{-117} +932 q^{-119} +326 q^{-121} -568 q^{-123} -957 q^{-125} -506 q^{-127} +348 q^{-129} +865 q^{-131} +645 q^{-133} -88 q^{-135} -699 q^{-137} -689 q^{-139} -133 q^{-141} +458 q^{-143} +632 q^{-145} +303 q^{-147} -218 q^{-149} -501 q^{-151} -364 q^{-153} +17 q^{-155} +324 q^{-157} +340 q^{-159} +111 q^{-161} -160 q^{-163} -263 q^{-165} -154 q^{-167} +40 q^{-169} +161 q^{-171} +142 q^{-173} +28 q^{-175} -80 q^{-177} -102 q^{-179} -46 q^{-181} +26 q^{-183} +60 q^{-185} +38 q^{-187} - q^{-189} -27 q^{-191} -27 q^{-193} -5 q^{-195} +13 q^{-197} +12 q^{-199} +3 q^{-201} - q^{-203} -6 q^{-205} -4 q^{-207} +3 q^{-209} +2 q^{-211} - q^{-213} - q^{-219} + q^{-221} + q^{-223} - q^{-225} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^4-q^2-1+2 q^{-2} - q^{-4} +2 q^{-6} + q^{-8} + q^{-12} - q^{-14} +2 q^{-16} - q^{-20} + q^{-22} - q^{-24} - q^{-26} }[/math]
1,1 [math]\displaystyle{ q^{12}-4 q^{10}+10 q^8-20 q^6+34 q^4-52 q^2+78-104 q^{-2} +124 q^{-4} -142 q^{-6} +152 q^{-8} -140 q^{-10} +113 q^{-12} -60 q^{-14} +2 q^{-16} +74 q^{-18} -150 q^{-20} +220 q^{-22} -268 q^{-24} +298 q^{-26} -297 q^{-28} +272 q^{-30} -228 q^{-32} +162 q^{-34} -89 q^{-36} +14 q^{-38} +50 q^{-40} -100 q^{-42} +136 q^{-44} -152 q^{-46} +144 q^{-48} -130 q^{-50} +106 q^{-52} -82 q^{-54} +56 q^{-56} -36 q^{-58} +23 q^{-60} -12 q^{-62} +6 q^{-64} -2 q^{-66} + q^{-68} }[/math]
2,0 [math]\displaystyle{ q^{12}-q^{10}-2 q^8+q^6+4 q^4+q^2-7- q^{-2} +8 q^{-4} -7 q^{-8} +2 q^{-10} +8 q^{-12} -4 q^{-16} +2 q^{-18} +4 q^{-20} -3 q^{-22} + q^{-24} +2 q^{-26} -2 q^{-28} +2 q^{-30} +7 q^{-32} - q^{-34} -6 q^{-36} + q^{-38} +5 q^{-40} -4 q^{-42} -9 q^{-44} + q^{-46} +5 q^{-48} - q^{-50} -4 q^{-52} +3 q^{-56} -2 q^{-60} + q^{-64} + q^{-66} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^8-2 q^6+4 q^2-5+ q^{-2} +8 q^{-4} -8 q^{-6} - q^{-8} +9 q^{-10} -6 q^{-12} -2 q^{-14} +8 q^{-16} +2 q^{-22} +5 q^{-24} - q^{-26} -6 q^{-28} +6 q^{-30} + q^{-32} -10 q^{-34} +6 q^{-36} +2 q^{-38} -9 q^{-40} +3 q^{-42} + q^{-44} -4 q^{-46} +2 q^{-48} + q^{-50} - q^{-52} + q^{-54} }[/math]
1,0,0 [math]\displaystyle{ q^5-q^3- q^{-1} +2 q^{-3} - q^{-5} +2 q^{-7} + q^{-9} + q^{-11} + q^{-17} - q^{-19} +2 q^{-21} + q^{-25} - q^{-27} + q^{-29} - q^{-31} - q^{-33} - q^{-35} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^8-2 q^6+4 q^4-6 q^2+7-9 q^{-2} +10 q^{-4} -8 q^{-6} +7 q^{-8} -3 q^{-10} +6 q^{-14} -10 q^{-16} +14 q^{-18} -16 q^{-20} +18 q^{-22} -17 q^{-24} +15 q^{-26} -10 q^{-28} +6 q^{-30} - q^{-32} -2 q^{-34} +6 q^{-36} -8 q^{-38} +9 q^{-40} -9 q^{-42} +7 q^{-44} -6 q^{-46} +4 q^{-48} -3 q^{-50} + q^{-52} - q^{-54} }[/math]
1,0 [math]\displaystyle{ q^{14}-2 q^{10}-2 q^8+2 q^6+5 q^4-6-4 q^{-2} +6 q^{-4} +9 q^{-6} -2 q^{-8} -10 q^{-10} -4 q^{-12} +8 q^{-14} +7 q^{-16} -4 q^{-18} -8 q^{-20} +2 q^{-22} +8 q^{-24} +2 q^{-26} -6 q^{-28} - q^{-30} +7 q^{-32} +5 q^{-34} -4 q^{-36} -4 q^{-38} +4 q^{-40} +6 q^{-42} -3 q^{-44} -7 q^{-46} + q^{-48} +8 q^{-50} + q^{-52} -9 q^{-54} -7 q^{-56} +6 q^{-58} +9 q^{-60} -2 q^{-62} -10 q^{-64} -4 q^{-66} +6 q^{-68} +5 q^{-70} -2 q^{-72} -5 q^{-74} - q^{-76} +3 q^{-78} +2 q^{-80} - q^{-82} - q^{-84} + q^{-88} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{18}-2 q^{16}+4 q^{14}-6 q^{12}+4 q^{10}-q^8-4 q^6+13 q^4-17 q^2+22-19 q^{-2} +5 q^{-4} +10 q^{-6} -27 q^{-8} +38 q^{-10} -39 q^{-12} +28 q^{-14} -7 q^{-16} -17 q^{-18} +36 q^{-20} -40 q^{-22} +30 q^{-24} -9 q^{-26} -13 q^{-28} +24 q^{-30} -21 q^{-32} +8 q^{-34} +18 q^{-36} -33 q^{-38} +40 q^{-40} -24 q^{-42} -4 q^{-44} +36 q^{-46} -58 q^{-48} +63 q^{-50} -46 q^{-52} +17 q^{-54} +18 q^{-56} -45 q^{-58} +57 q^{-60} -50 q^{-62} +27 q^{-64} -25 q^{-68} +32 q^{-70} -23 q^{-72} +7 q^{-74} +16 q^{-76} -28 q^{-78} +26 q^{-80} -10 q^{-82} -14 q^{-84} +36 q^{-86} -44 q^{-88} +36 q^{-90} -17 q^{-92} -8 q^{-94} +27 q^{-96} -37 q^{-98} +35 q^{-100} -23 q^{-102} +6 q^{-104} +6 q^{-106} -16 q^{-108} +16 q^{-110} -14 q^{-112} +9 q^{-114} -3 q^{-116} -2 q^{-118} +3 q^{-120} -4 q^{-122} +3 q^{-124} - q^{-126} + q^{-128} }[/math]

.

Computer Talk
The above data is available with the Mathematica package 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.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["9 21"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -2 t^2+11 t-17+11 t^{-1} -2 t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -2 z^4+3 z^2+1 }[/math]
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]= [math]\displaystyle{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 43, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^8+2 q^7-4 q^6+6 q^5-7 q^4+8 q^3-6 q^2+5 q-3+ q^{-1} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -z^4 a^{-2} -z^4 a^{-4} +2 z^2 a^{-6} +z^2+ a^{-2} + a^{-6} - a^{-8} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^8 a^{-4} +z^8 a^{-6} +3 z^7 a^{-3} +5 z^7 a^{-5} +2 z^7 a^{-7} +4 z^6 a^{-2} +4 z^6 a^{-4} +2 z^6 a^{-6} +2 z^6 a^{-8} +3 z^5 a^{-1} -3 z^5 a^{-3} -10 z^5 a^{-5} -3 z^5 a^{-7} +z^5 a^{-9} -6 z^4 a^{-2} -9 z^4 a^{-4} -7 z^4 a^{-6} -5 z^4 a^{-8} +z^4-4 z^3 a^{-1} +2 z^3 a^{-3} +9 z^3 a^{-5} -3 z^3 a^{-9} +3 z^2 a^{-2} +6 z^2 a^{-4} +5 z^2 a^{-6} +3 z^2 a^{-8} -z^2-z a^{-3} -3 z a^{-5} +2 z a^{-9} - a^{-2} - a^{-6} - a^{-8} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n129,}

Computer Talk
The above data is available with the Mathematica package 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.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["9 21"];
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]= { [math]\displaystyle{ -2 t^2+11 t-17+11 t^{-1} -2 t^{-2} }[/math], [math]\displaystyle{ -q^8+2 q^7-4 q^6+6 q^5-7 q^4+8 q^3-6 q^2+5 q-3+ q^{-1} }[/math] }
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]= {}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11n129,}

Vassiliev invariants

V2 and V3: (3, 6)
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
[math]\displaystyle{ 12 }[/math] [math]\displaystyle{ 48 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 238 }[/math] [math]\displaystyle{ 50 }[/math] [math]\displaystyle{ 576 }[/math] [math]\displaystyle{ 1248 }[/math] [math]\displaystyle{ 192 }[/math] [math]\displaystyle{ 272 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 1152 }[/math] [math]\displaystyle{ 2856 }[/math] [math]\displaystyle{ 600 }[/math] [math]\displaystyle{ \frac{65311}{10} }[/math] [math]\displaystyle{ -\frac{3182}{5} }[/math] [math]\displaystyle{ \frac{18154}{5} }[/math] [math]\displaystyle{ \frac{235}{2} }[/math] [math]\displaystyle{ \frac{5631}{10} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]2 is the signature of 9 21. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -2-101234567χ
17         1-1
15        1 1
13       31 -2
11      31  2
9     43   -1
7    43    1
5   24     2
3  34      -1
1 13       2
-1 2        -2
-31         1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

 [math]\displaystyle{ n }[/math]  [math]\displaystyle{ J_n }[/math]
2 [math]\displaystyle{ q^{23}-2 q^{22}+6 q^{20}-8 q^{19}-3 q^{18}+19 q^{17}-17 q^{16}-13 q^{15}+38 q^{14}-22 q^{13}-27 q^{12}+54 q^{11}-21 q^{10}-38 q^9+57 q^8-15 q^7-37 q^6+44 q^5-6 q^4-27 q^3+24 q^2-13+8 q^{-1} + q^{-2} -3 q^{-3} + q^{-4} }[/math]
3 [math]\displaystyle{ -q^{45}+2 q^{44}-2 q^{42}-3 q^{41}+7 q^{40}+4 q^{39}-10 q^{38}-12 q^{37}+19 q^{36}+20 q^{35}-22 q^{34}-40 q^{33}+29 q^{32}+60 q^{31}-25 q^{30}-88 q^{29}+17 q^{28}+115 q^{27}-3 q^{26}-139 q^{25}-19 q^{24}+162 q^{23}+38 q^{22}-175 q^{21}-63 q^{20}+188 q^{19}+76 q^{18}-181 q^{17}-99 q^{16}+183 q^{15}+99 q^{14}-158 q^{13}-111 q^{12}+142 q^{11}+102 q^{10}-107 q^9-99 q^8+82 q^7+83 q^6-52 q^5-67 q^4+31 q^3+48 q^2-15 q-32+6 q^{-1} +20 q^{-2} -3 q^{-3} -10 q^{-4} + q^{-5} +4 q^{-6} + q^{-7} -3 q^{-8} + q^{-9} }[/math]
4 [math]\displaystyle{ q^{74}-2 q^{73}+2 q^{71}-q^{70}+4 q^{69}-9 q^{68}+10 q^{66}-2 q^{65}+12 q^{64}-31 q^{63}-8 q^{62}+30 q^{61}+10 q^{60}+38 q^{59}-78 q^{58}-47 q^{57}+44 q^{56}+47 q^{55}+125 q^{54}-127 q^{53}-139 q^{52}-2 q^{51}+78 q^{50}+300 q^{49}-112 q^{48}-244 q^{47}-145 q^{46}+31 q^{45}+520 q^{44}+6 q^{43}-289 q^{42}-343 q^{41}-118 q^{40}+702 q^{39}+183 q^{38}-246 q^{37}-523 q^{36}-314 q^{35}+806 q^{34}+346 q^{33}-153 q^{32}-638 q^{31}-487 q^{30}+825 q^{29}+458 q^{28}-44 q^{27}-675 q^{26}-602 q^{25}+759 q^{24}+503 q^{23}+67 q^{22}-618 q^{21}-643 q^{20}+594 q^{19}+464 q^{18}+176 q^{17}-463 q^{16}-598 q^{15}+370 q^{14}+336 q^{13}+237 q^{12}-253 q^{11}-459 q^{10}+167 q^9+164 q^8+213 q^7-79 q^6-274 q^5+55 q^4+34 q^3+129 q^2+2 q-123+20 q^{-1} -12 q^{-2} +54 q^{-3} +11 q^{-4} -44 q^{-5} +12 q^{-6} -11 q^{-7} +16 q^{-8} +5 q^{-9} -13 q^{-10} +4 q^{-11} -3 q^{-12} +4 q^{-13} + q^{-14} -3 q^{-15} + q^{-16} }[/math]
5 [math]\displaystyle{ -q^{110}+2 q^{109}-2 q^{107}+q^{106}-2 q^{104}+5 q^{103}+q^{102}-9 q^{101}-q^{100}+5 q^{99}+2 q^{98}+14 q^{97}+2 q^{96}-27 q^{95}-23 q^{94}+5 q^{93}+28 q^{92}+53 q^{91}+24 q^{90}-61 q^{89}-95 q^{88}-51 q^{87}+50 q^{86}+161 q^{85}+138 q^{84}-42 q^{83}-216 q^{82}-245 q^{81}-59 q^{80}+264 q^{79}+409 q^{78}+187 q^{77}-232 q^{76}-552 q^{75}-440 q^{74}+127 q^{73}+692 q^{72}+708 q^{71}+97 q^{70}-726 q^{69}-1031 q^{68}-429 q^{67}+682 q^{66}+1319 q^{65}+830 q^{64}-506 q^{63}-1548 q^{62}-1283 q^{61}+231 q^{60}+1708 q^{59}+1724 q^{58}+100 q^{57}-1758 q^{56}-2131 q^{55}-487 q^{54}+1770 q^{53}+2467 q^{52}+842 q^{51}-1687 q^{50}-2749 q^{49}-1195 q^{48}+1621 q^{47}+2940 q^{46}+1468 q^{45}-1463 q^{44}-3105 q^{43}-1742 q^{42}+1382 q^{41}+3158 q^{40}+1917 q^{39}-1164 q^{38}-3198 q^{37}-2131 q^{36}+1045 q^{35}+3114 q^{34}+2212 q^{33}-739 q^{32}-2992 q^{31}-2341 q^{30}+532 q^{29}+2730 q^{28}+2318 q^{27}-177 q^{26}-2405 q^{25}-2288 q^{24}-77 q^{23}+1974 q^{22}+2098 q^{21}+378 q^{20}-1517 q^{19}-1865 q^{18}-539 q^{17}+1038 q^{16}+1521 q^{15}+659 q^{14}-626 q^{13}-1169 q^{12}-641 q^{11}+294 q^{10}+803 q^9+568 q^8-69 q^7-507 q^6-437 q^5-45 q^4+276 q^3+293 q^2+94 q-127-184 q^{-1} -81 q^{-2} +49 q^{-3} +95 q^{-4} +53 q^{-5} -7 q^{-6} -44 q^{-7} -37 q^{-8} +2 q^{-9} +23 q^{-10} +12 q^{-11} -2 q^{-12} - q^{-13} -10 q^{-14} -4 q^{-15} +11 q^{-16} + q^{-17} -5 q^{-18} + q^{-19} -3 q^{-21} +4 q^{-22} + q^{-23} -3 q^{-24} + q^{-25} }[/math]
6 [math]\displaystyle{ q^{153}-2 q^{152}+2 q^{150}-q^{149}-2 q^{147}+6 q^{146}-6 q^{145}-2 q^{144}+11 q^{143}-3 q^{142}-4 q^{141}-13 q^{140}+15 q^{139}-12 q^{138}-2 q^{137}+40 q^{136}+5 q^{135}-13 q^{134}-55 q^{133}+11 q^{132}-40 q^{131}+123 q^{129}+70 q^{128}+10 q^{127}-140 q^{126}-54 q^{125}-175 q^{124}-67 q^{123}+257 q^{122}+283 q^{121}+212 q^{120}-147 q^{119}-148 q^{118}-548 q^{117}-432 q^{116}+213 q^{115}+571 q^{114}+754 q^{113}+268 q^{112}+108 q^{111}-1003 q^{110}-1279 q^{109}-499 q^{108}+387 q^{107}+1340 q^{106}+1291 q^{105}+1343 q^{104}-779 q^{103}-2139 q^{102}-2047 q^{101}-1012 q^{100}+990 q^{99}+2286 q^{98}+3647 q^{97}+947 q^{96}-1858 q^{95}-3616 q^{94}-3609 q^{93}-1140 q^{92}+2000 q^{91}+6025 q^{90}+4038 q^{89}+347 q^{88}-3910 q^{87}-6293 q^{86}-4749 q^{85}-209 q^{84}+7174 q^{83}+7295 q^{82}+4009 q^{81}-2419 q^{80}-7843 q^{79}-8568 q^{78}-3712 q^{77}+6702 q^{76}+9565 q^{75}+7832 q^{74}+172 q^{73}-7986 q^{72}-11495 q^{71}-7246 q^{70}+5260 q^{69}+10601 q^{68}+10808 q^{67}+2762 q^{66}-7292 q^{65}-13269 q^{64}-9958 q^{63}+3692 q^{62}+10823 q^{61}+12730 q^{60}+4755 q^{59}-6367 q^{58}-14165 q^{57}-11759 q^{56}+2300 q^{55}+10589 q^{54}+13842 q^{53}+6259 q^{52}-5291 q^{51}-14366 q^{50}-12922 q^{49}+804 q^{48}+9790 q^{47}+14251 q^{46}+7601 q^{45}-3686 q^{44}-13634 q^{43}-13488 q^{42}-1121 q^{41}+7974 q^{40}+13616 q^{39}+8730 q^{38}-1279 q^{37}-11490 q^{36}-13007 q^{35}-3237 q^{34}+4967 q^{33}+11417 q^{32}+9006 q^{31}+1493 q^{30}-7898 q^{29}-10923 q^{28}-4636 q^{27}+1467 q^{26}+7737 q^{25}+7748 q^{24}+3485 q^{23}-3817 q^{22}-7419 q^{21}-4482 q^{20}-1153 q^{19}+3722 q^{18}+5140 q^{17}+3793 q^{16}-754 q^{15}-3722 q^{14}-2957 q^{13}-2008 q^{12}+860 q^{11}+2376 q^{10}+2690 q^9+525 q^8-1187 q^7-1203 q^6-1489 q^5-297 q^4+600 q^3+1306 q^2+522 q-136-164 q^{-1} -670 q^{-2} -356 q^{-3} -44 q^{-4} +448 q^{-5} +191 q^{-6} +44 q^{-7} +128 q^{-8} -191 q^{-9} -149 q^{-10} -105 q^{-11} +126 q^{-12} +19 q^{-13} +3 q^{-14} +97 q^{-15} -35 q^{-16} -35 q^{-17} -46 q^{-18} +42 q^{-19} -11 q^{-20} -13 q^{-21} +34 q^{-22} -7 q^{-23} -4 q^{-24} -14 q^{-25} +18 q^{-26} -4 q^{-27} -9 q^{-28} +9 q^{-29} -3 q^{-30} -3 q^{-32} +4 q^{-33} + q^{-34} -3 q^{-35} + q^{-36} }[/math]

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.

9 20.gif

9_20

9 22.gif

9_22

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