8 9

From Knot Atlas
Jump to navigationJump to search

8 8.gif

8_8

8 10.gif

8_10

8 9.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 8 9's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 8 9 at Knotilus!

[edit 8 9 Quick Notes]
[edit 8 9 Further Notes and Views]


Knot presentations

Planar diagram presentation X6271 X14,8,15,7 X10,3,11,4 X2,13,3,14 X12,5,13,6 X4,11,5,12 X16,10,1,9 X8,16,9,15
Gauss code 1, -4, 3, -6, 5, -1, 2, -8, 7, -3, 6, -5, 4, -2, 8, -7
Dowker-Thistlethwaite code 6 10 12 14 16 4 2 8
Conway Notation [3113]


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

Length is 8, width is 3,

Braid index is 3

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

[edit Notes on presentations of 8 9]

Knot 8_9.
A graph, knot 8_9.
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["8 9"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X6271 X14,8,15,7 X10,3,11,4 X2,13,3,14 X12,5,13,6 X4,11,5,12 X16,10,1,9 X8,16,9,15
In[5]:= GaussCode[K]
Out[5]= 1, -4, 3, -6, 5, -1, 2, -8, 7, -3, 6, -5, 4, -2, 8, -7
In[6]:= DTCode[K]
Out[6]= 6 10 12 14 16 4 2 8

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [3113]
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}(3,\{-1,-1,-1,2,-1,2,2,2\}) }[/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]= { 3, 8, 3 }
In[11]:= Show[BraidPlot[br]]
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.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."
8 9 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{2, 10}, {1, 5}, {9, 4}, {10, 6}, {5, 3}, {4, 2}, {3, 7}, {6, 8}, {7, 9}, {8, 1}]
In[14]:= Draw[ap]
8 9 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Fully amphicheiral
Unknotting number 1
3-genus 3
Bridge index 2
Super bridge index [math]\displaystyle{ \{3,6\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [-5][-5]
Hyperbolic Volume 7.58818
A-Polynomial See Data:8 9/A-polynomial

[edit Notes for 8 9's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 0 }[/math]
Topological 4 genus [math]\displaystyle{ 0 }[/math]
Concordance genus [math]\displaystyle{ 0 }[/math]
Rasmussen s-Invariant 0

[edit Notes for 8 9's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^3+3 t^2-5 t+7-5 t^{-1} +3 t^{-2} - t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ -z^6-3 z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 25, 0 }
Jones polynomial [math]\displaystyle{ q^4-2 q^3+3 q^2-4 q+5-4 q^{-1} +3 q^{-2} -2 q^{-3} + q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^6+a^2 z^4+z^4 a^{-2} -5 z^4+3 a^2 z^2+3 z^2 a^{-2} -8 z^2+2 a^2+2 a^{-2} -3 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a z^7+z^7 a^{-1} +2 a^2 z^6+2 z^6 a^{-2} +4 z^6+2 a^3 z^5+2 z^5 a^{-3} +a^4 z^4-4 a^2 z^4-4 z^4 a^{-2} +z^4 a^{-4} -10 z^4-4 a^3 z^3-a z^3-z^3 a^{-1} -4 z^3 a^{-3} -2 a^4 z^2+4 a^2 z^2+4 z^2 a^{-2} -2 z^2 a^{-4} +12 z^2+a^3 z+a z+z a^{-1} +z a^{-3} -2 a^2-2 a^{-2} -3 }[/math]
The A2 invariant [math]\displaystyle{ q^{12}+q^8-q^4+q^2-1+ q^{-2} - q^{-4} + q^{-8} + q^{-12} }[/math]
The G2 invariant [math]\displaystyle{ q^{66}-q^{64}+2 q^{62}-3 q^{60}+q^{58}-3 q^{54}+6 q^{52}-6 q^{50}+7 q^{48}-5 q^{46}+6 q^{42}-10 q^{40}+12 q^{38}-8 q^{36}+4 q^{34}+3 q^{32}-6 q^{30}+11 q^{28}-6 q^{26}+2 q^{24}+4 q^{22}-7 q^{20}+5 q^{18}-8 q^{14}+11 q^{12}-10 q^{10}+9 q^8-3 q^6-10 q^4+14 q^2-17+14 q^{-2} -10 q^{-4} -3 q^{-6} +9 q^{-8} -10 q^{-10} +11 q^{-12} -8 q^{-14} +5 q^{-18} -7 q^{-20} +4 q^{-22} +2 q^{-24} -6 q^{-26} +11 q^{-28} -6 q^{-30} +3 q^{-32} +4 q^{-34} -8 q^{-36} +12 q^{-38} -10 q^{-40} +6 q^{-42} -5 q^{-46} +7 q^{-48} -6 q^{-50} +6 q^{-52} -3 q^{-54} + q^{-58} -3 q^{-60} +2 q^{-62} - q^{-64} + q^{-66} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 8_9.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^9-q^7+q^5-q^3+q+ q^{-1} - q^{-3} + q^{-5} - q^{-7} + q^{-9} }[/math]
2 [math]\displaystyle{ q^{26}-q^{24}-q^{22}+3 q^{20}-q^{18}-3 q^{16}+4 q^{14}-5 q^{10}+3 q^8+q^6-3 q^4+2 q^2+3+2 q^{-2} -3 q^{-4} + q^{-6} +3 q^{-8} -5 q^{-10} +4 q^{-14} -3 q^{-16} - q^{-18} +3 q^{-20} - q^{-22} - q^{-24} + q^{-26} }[/math]
3 [math]\displaystyle{ q^{51}-q^{49}-q^{47}+q^{45}+2 q^{43}-q^{41}-4 q^{39}+q^{37}+6 q^{35}-8 q^{31}-3 q^{29}+9 q^{27}+7 q^{25}-9 q^{23}-9 q^{21}+8 q^{19}+11 q^{17}-6 q^{15}-11 q^{13}+4 q^{11}+9 q^9-q^7-8 q^5+5 q+5 q^{-1} -8 q^{-5} - q^{-7} +9 q^{-9} +4 q^{-11} -11 q^{-13} -6 q^{-15} +11 q^{-17} +8 q^{-19} -9 q^{-21} -9 q^{-23} +7 q^{-25} +9 q^{-27} -3 q^{-29} -8 q^{-31} +6 q^{-35} + q^{-37} -4 q^{-39} - q^{-41} +2 q^{-43} + q^{-45} - q^{-47} - q^{-49} + q^{-51} }[/math]
4 [math]\displaystyle{ q^{84}-q^{82}-q^{80}+q^{78}+2 q^{74}-3 q^{72}-2 q^{70}+3 q^{68}+q^{66}+6 q^{64}-6 q^{62}-9 q^{60}+q^{58}+5 q^{56}+17 q^{54}-3 q^{52}-17 q^{50}-13 q^{48}-2 q^{46}+31 q^{44}+16 q^{42}-12 q^{40}-29 q^{38}-21 q^{36}+31 q^{34}+32 q^{32}+4 q^{30}-31 q^{28}-36 q^{26}+18 q^{24}+31 q^{22}+14 q^{20}-19 q^{18}-33 q^{16}+5 q^{14}+20 q^{12}+16 q^{10}-6 q^8-20 q^6-4 q^4+8 q^2+15+8 q^{-2} -4 q^{-4} -20 q^{-6} -6 q^{-8} +16 q^{-10} +20 q^{-12} +5 q^{-14} -33 q^{-16} -19 q^{-18} +14 q^{-20} +31 q^{-22} +18 q^{-24} -36 q^{-26} -31 q^{-28} +4 q^{-30} +32 q^{-32} +31 q^{-34} -21 q^{-36} -29 q^{-38} -12 q^{-40} +16 q^{-42} +31 q^{-44} -2 q^{-46} -13 q^{-48} -17 q^{-50} -3 q^{-52} +17 q^{-54} +5 q^{-56} + q^{-58} -9 q^{-60} -6 q^{-62} +6 q^{-64} + q^{-66} +3 q^{-68} -2 q^{-70} -3 q^{-72} +2 q^{-74} + q^{-78} - q^{-80} - q^{-82} + q^{-84} }[/math]
5 [math]\displaystyle{ q^{125}-q^{123}-q^{121}+q^{119}-q^{111}-q^{109}+3 q^{107}+4 q^{105}-q^{103}-4 q^{101}-6 q^{99}-4 q^{97}+4 q^{95}+13 q^{93}+11 q^{91}-3 q^{89}-17 q^{87}-22 q^{85}-8 q^{83}+17 q^{81}+36 q^{79}+28 q^{77}-8 q^{75}-42 q^{73}-51 q^{71}-18 q^{69}+40 q^{67}+73 q^{65}+50 q^{63}-23 q^{61}-87 q^{59}-83 q^{57}-6 q^{55}+85 q^{53}+111 q^{51}+39 q^{49}-72 q^{47}-124 q^{45}-67 q^{43}+49 q^{41}+122 q^{39}+87 q^{37}-25 q^{35}-108 q^{33}-91 q^{31}+5 q^{29}+85 q^{27}+84 q^{25}+11 q^{23}-63 q^{21}-73 q^{19}-15 q^{17}+42 q^{15}+54 q^{13}+23 q^{11}-22 q^9-45 q^7-22 q^5+8 q^3+31 q+31 q^{-1} +8 q^{-3} -22 q^{-5} -45 q^{-7} -22 q^{-9} +23 q^{-11} +54 q^{-13} +42 q^{-15} -15 q^{-17} -73 q^{-19} -63 q^{-21} +11 q^{-23} +84 q^{-25} +85 q^{-27} +5 q^{-29} -91 q^{-31} -108 q^{-33} -25 q^{-35} +87 q^{-37} +122 q^{-39} +49 q^{-41} -67 q^{-43} -124 q^{-45} -72 q^{-47} +39 q^{-49} +111 q^{-51} +85 q^{-53} -6 q^{-55} -83 q^{-57} -87 q^{-59} -23 q^{-61} +50 q^{-63} +73 q^{-65} +40 q^{-67} -18 q^{-69} -51 q^{-71} -42 q^{-73} -8 q^{-75} +28 q^{-77} +36 q^{-79} +17 q^{-81} -8 q^{-83} -22 q^{-85} -17 q^{-87} -3 q^{-89} +11 q^{-91} +13 q^{-93} +4 q^{-95} -4 q^{-97} -6 q^{-99} -4 q^{-101} - q^{-103} +4 q^{-105} +3 q^{-107} - q^{-109} - q^{-111} + q^{-119} - q^{-121} - q^{-123} + q^{-125} }[/math]
6 [math]\displaystyle{ q^{174}-q^{172}-q^{170}+q^{168}-2 q^{162}+2 q^{160}-q^{156}+5 q^{154}+2 q^{152}-2 q^{150}-9 q^{148}-2 q^{146}-3 q^{144}+15 q^{140}+14 q^{138}+5 q^{136}-17 q^{134}-14 q^{132}-22 q^{130}-16 q^{128}+21 q^{126}+41 q^{124}+41 q^{122}+4 q^{120}-14 q^{118}-58 q^{116}-75 q^{114}-28 q^{112}+36 q^{110}+91 q^{108}+86 q^{106}+70 q^{104}-36 q^{102}-135 q^{100}-149 q^{98}-82 q^{96}+48 q^{94}+153 q^{92}+232 q^{90}+123 q^{88}-71 q^{86}-232 q^{84}-271 q^{82}-145 q^{80}+72 q^{78}+322 q^{76}+329 q^{74}+140 q^{72}-149 q^{70}-359 q^{68}-355 q^{66}-134 q^{64}+243 q^{62}+408 q^{60}+333 q^{58}+39 q^{56}-274 q^{54}-412 q^{52}-292 q^{50}+73 q^{48}+316 q^{46}+360 q^{44}+164 q^{42}-116 q^{40}-311 q^{38}-296 q^{36}-43 q^{34}+162 q^{32}+254 q^{30}+165 q^{28}-6 q^{26}-163 q^{24}-198 q^{22}-67 q^{20}+51 q^{18}+132 q^{16}+109 q^{14}+35 q^{12}-60 q^{10}-105 q^8-64 q^6-5 q^4+61 q^2+79+61 q^{-2} -5 q^{-4} -64 q^{-6} -105 q^{-8} -60 q^{-10} +35 q^{-12} +109 q^{-14} +132 q^{-16} +51 q^{-18} -67 q^{-20} -198 q^{-22} -163 q^{-24} -6 q^{-26} +165 q^{-28} +254 q^{-30} +162 q^{-32} -43 q^{-34} -296 q^{-36} -311 q^{-38} -116 q^{-40} +164 q^{-42} +360 q^{-44} +316 q^{-46} +73 q^{-48} -292 q^{-50} -412 q^{-52} -274 q^{-54} +39 q^{-56} +333 q^{-58} +408 q^{-60} +243 q^{-62} -134 q^{-64} -355 q^{-66} -359 q^{-68} -149 q^{-70} +140 q^{-72} +329 q^{-74} +322 q^{-76} +72 q^{-78} -145 q^{-80} -271 q^{-82} -232 q^{-84} -71 q^{-86} +123 q^{-88} +232 q^{-90} +153 q^{-92} +48 q^{-94} -82 q^{-96} -149 q^{-98} -135 q^{-100} -36 q^{-102} +70 q^{-104} +86 q^{-106} +91 q^{-108} +36 q^{-110} -28 q^{-112} -75 q^{-114} -58 q^{-116} -14 q^{-118} +4 q^{-120} +41 q^{-122} +41 q^{-124} +21 q^{-126} -16 q^{-128} -22 q^{-130} -14 q^{-132} -17 q^{-134} +5 q^{-136} +14 q^{-138} +15 q^{-140} -3 q^{-144} -2 q^{-146} -9 q^{-148} -2 q^{-150} +2 q^{-152} +5 q^{-154} - q^{-156} +2 q^{-160} -2 q^{-162} + q^{-168} - q^{-170} - q^{-172} + q^{-174} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{12}+q^8-q^4+q^2-1+ q^{-2} - q^{-4} + q^{-8} + q^{-12} }[/math]
1,1 [math]\displaystyle{ q^{36}-2 q^{34}+4 q^{32}-8 q^{30}+13 q^{28}-14 q^{26}+22 q^{24}-28 q^{22}+27 q^{20}-28 q^{18}+24 q^{16}-20 q^{14}+7 q^{12}+10 q^{10}-20 q^8+36 q^6-47 q^4+56 q^2-58+56 q^{-2} -47 q^{-4} +36 q^{-6} -20 q^{-8} +10 q^{-10} +7 q^{-12} -20 q^{-14} +24 q^{-16} -28 q^{-18} +27 q^{-20} -28 q^{-22} +22 q^{-24} -14 q^{-26} +13 q^{-28} -8 q^{-30} +4 q^{-32} -2 q^{-34} + q^{-36} }[/math]
2,0 [math]\displaystyle{ q^{32}+q^{26}+q^{24}-q^{22}-q^{20}+q^{18}-3 q^{14}-q^{12}+q^{10}-2 q^8-q^6+3 q^4+3 q^2+2+3 q^{-2} +3 q^{-4} - q^{-6} -2 q^{-8} + q^{-10} - q^{-12} -3 q^{-14} + q^{-18} - q^{-20} - q^{-22} + q^{-24} + q^{-26} + q^{-32} }[/math]
3,0 [math]\displaystyle{ q^{60}+q^{52}+q^{50}-2 q^{48}-2 q^{46}+4 q^{42}+q^{40}-5 q^{38}-6 q^{36}+q^{34}+7 q^{32}+2 q^{30}-7 q^{28}-5 q^{26}+5 q^{24}+11 q^{22}+3 q^{20}-6 q^{18}-q^{16}+5 q^{14}+5 q^{12}-5 q^{10}-7 q^8+q^6+4 q^4+q^2-4+ q^{-2} +4 q^{-4} + q^{-6} -7 q^{-8} -5 q^{-10} +5 q^{-12} +5 q^{-14} - q^{-16} -6 q^{-18} +3 q^{-20} +11 q^{-22} +5 q^{-24} -5 q^{-26} -7 q^{-28} +2 q^{-30} +7 q^{-32} + q^{-34} -6 q^{-36} -5 q^{-38} + q^{-40} +4 q^{-42} -2 q^{-46} -2 q^{-48} + q^{-50} + q^{-52} + q^{-60} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{28}-q^{26}+q^{22}-3 q^{20}+q^{18}+4 q^{16}-2 q^{14}+2 q^{12}+5 q^{10}-2 q^8-q^6+q^4-2 q^2-2-2 q^{-2} + q^{-4} - q^{-6} -2 q^{-8} +5 q^{-10} +2 q^{-12} -2 q^{-14} +4 q^{-16} + q^{-18} -3 q^{-20} + q^{-22} - q^{-26} + q^{-28} }[/math]
1,0,0 [math]\displaystyle{ q^{15}+2 q^{11}+q^7-q^5-q- q^{-1} - q^{-5} + q^{-7} +2 q^{-11} + q^{-15} }[/math]
1,0,1 [math]\displaystyle{ q^{46}-2 q^{44}+3 q^{42}-3 q^{40}-q^{38}+8 q^{36}-9 q^{34}+13 q^{32}-3 q^{30}-9 q^{28}+18 q^{26}-27 q^{24}+19 q^{22}-2 q^{20}-20 q^{18}+36 q^{16}-34 q^{14}+16 q^{12}+5 q^{10}-22 q^8+21 q^6-9 q^4+4 q^2+9+4 q^{-2} -9 q^{-4} +21 q^{-6} -22 q^{-8} +5 q^{-10} +16 q^{-12} -34 q^{-14} +36 q^{-16} -20 q^{-18} -2 q^{-20} +19 q^{-22} -27 q^{-24} +18 q^{-26} -9 q^{-28} -3 q^{-30} +13 q^{-32} -9 q^{-34} +8 q^{-36} - q^{-38} -3 q^{-40} +3 q^{-42} -2 q^{-44} + q^{-46} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{34}+q^{28}-q^{26}-2 q^{24}+q^{22}+2 q^{20}-q^{18}+2 q^{16}+5 q^{14}+3 q^{12}-q^{10}+q^8+2 q^6-5 q^4-3 q^2-3 q^{-2} -5 q^{-4} +2 q^{-6} + q^{-8} - q^{-10} +3 q^{-12} +5 q^{-14} +2 q^{-16} - q^{-18} +2 q^{-20} + q^{-22} -2 q^{-24} - q^{-26} + q^{-28} + q^{-34} }[/math]
1,0,0,0 [math]\displaystyle{ q^{18}+2 q^{14}+q^{12}+q^{10}+q^8-q^6-2 q^2-1-2 q^{-2} - q^{-6} + q^{-8} + q^{-10} + q^{-12} +2 q^{-14} + q^{-18} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{28}-q^{26}+2 q^{24}-3 q^{22}+3 q^{20}-3 q^{18}+4 q^{16}-2 q^{14}+2 q^{12}+q^{10}-2 q^8+3 q^6-5 q^4+6 q^2-8+6 q^{-2} -5 q^{-4} +3 q^{-6} -2 q^{-8} + q^{-10} +2 q^{-12} -2 q^{-14} +4 q^{-16} -3 q^{-18} +3 q^{-20} -3 q^{-22} +2 q^{-24} - q^{-26} + q^{-28} }[/math]
1,0 [math]\displaystyle{ q^{46}-q^{42}-q^{40}+q^{38}+2 q^{36}-q^{34}-3 q^{32}-q^{30}+3 q^{28}+4 q^{26}-q^{24}-3 q^{22}+4 q^{18}+2 q^{16}-2 q^{14}-2 q^{12}+q^{10}+2 q^8-q^6-3 q^4+3-3 q^{-4} - q^{-6} +2 q^{-8} + q^{-10} -2 q^{-12} -2 q^{-14} +2 q^{-16} +4 q^{-18} -3 q^{-22} - q^{-24} +4 q^{-26} +3 q^{-28} - q^{-30} -3 q^{-32} - q^{-34} +2 q^{-36} + q^{-38} - q^{-40} - q^{-42} + q^{-46} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{38}-q^{36}+q^{34}-2 q^{32}+2 q^{30}-3 q^{28}+2 q^{26}-2 q^{24}+4 q^{22}-q^{20}+3 q^{18}+2 q^{16}+3 q^{14}+3 q^{12}-2 q^{10}+2 q^8-4 q^6+3 q^4-8 q^2+2-8 q^{-2} +3 q^{-4} -4 q^{-6} +2 q^{-8} -2 q^{-10} +3 q^{-12} +3 q^{-14} +2 q^{-16} +3 q^{-18} - q^{-20} +4 q^{-22} -2 q^{-24} +2 q^{-26} -3 q^{-28} +2 q^{-30} -2 q^{-32} + q^{-34} - q^{-36} + q^{-38} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{66}-q^{64}+2 q^{62}-3 q^{60}+q^{58}-3 q^{54}+6 q^{52}-6 q^{50}+7 q^{48}-5 q^{46}+6 q^{42}-10 q^{40}+12 q^{38}-8 q^{36}+4 q^{34}+3 q^{32}-6 q^{30}+11 q^{28}-6 q^{26}+2 q^{24}+4 q^{22}-7 q^{20}+5 q^{18}-8 q^{14}+11 q^{12}-10 q^{10}+9 q^8-3 q^6-10 q^4+14 q^2-17+14 q^{-2} -10 q^{-4} -3 q^{-6} +9 q^{-8} -10 q^{-10} +11 q^{-12} -8 q^{-14} +5 q^{-18} -7 q^{-20} +4 q^{-22} +2 q^{-24} -6 q^{-26} +11 q^{-28} -6 q^{-30} +3 q^{-32} +4 q^{-34} -8 q^{-36} +12 q^{-38} -10 q^{-40} +6 q^{-42} -5 q^{-46} +7 q^{-48} -6 q^{-50} +6 q^{-52} -3 q^{-54} + q^{-58} -3 q^{-60} +2 q^{-62} - q^{-64} + q^{-66} }[/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["8 9"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^3+3 t^2-5 t+7-5 t^{-1} +3 t^{-2} - t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^6-3 z^4-2 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]= { 25, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^4-2 q^3+3 q^2-4 q+5-4 q^{-1} +3 q^{-2} -2 q^{-3} + q^{-4} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -z^6+a^2 z^4+z^4 a^{-2} -5 z^4+3 a^2 z^2+3 z^2 a^{-2} -8 z^2+2 a^2+2 a^{-2} -3 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a z^7+z^7 a^{-1} +2 a^2 z^6+2 z^6 a^{-2} +4 z^6+2 a^3 z^5+2 z^5 a^{-3} +a^4 z^4-4 a^2 z^4-4 z^4 a^{-2} +z^4 a^{-4} -10 z^4-4 a^3 z^3-a z^3-z^3 a^{-1} -4 z^3 a^{-3} -2 a^4 z^2+4 a^2 z^2+4 z^2 a^{-2} -2 z^2 a^{-4} +12 z^2+a^3 z+a z+z a^{-1} +z a^{-3} -2 a^2-2 a^{-2} -3 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_155, K11n37,}

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

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["8 9"];
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{ -t^3+3 t^2-5 t+7-5 t^{-1} +3 t^{-2} - t^{-3} }[/math], [math]\displaystyle{ q^4-2 q^3+3 q^2-4 q+5-4 q^{-1} +3 q^{-2} -2 q^{-3} + q^{-4} }[/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]= {10_155, K11n37,}
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: (-2, 0)
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{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{212}{3} }[/math] [math]\displaystyle{ \frac{124}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{256}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{1696}{3} }[/math] [math]\displaystyle{ -\frac{992}{3} }[/math] [math]\displaystyle{ -\frac{8071}{15} }[/math] [math]\displaystyle{ \frac{1668}{5} }[/math] [math]\displaystyle{ -\frac{37804}{45} }[/math] [math]\displaystyle{ \frac{1063}{9} }[/math] [math]\displaystyle{ -\frac{3271}{15} }[/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]0 is the signature of 8 9. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-101234χ
9        11
7       1 -1
5      21 1
3     21  -1
1    32   1
-1   23    1
-3  12     -1
-5 12      1
-7 1       -1
-91        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=1 }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=4 }[/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^{12}-2 q^{11}+5 q^9-6 q^8-2 q^7+12 q^6-10 q^5-7 q^4+20 q^3-12 q^2-11 q+25-11 q^{-1} -12 q^{-2} +20 q^{-3} -7 q^{-4} -10 q^{-5} +12 q^{-6} -2 q^{-7} -6 q^{-8} +5 q^{-9} -2 q^{-11} + q^{-12} }[/math]
3 [math]\displaystyle{ q^{24}-2 q^{23}+2 q^{21}+2 q^{20}-5 q^{19}-3 q^{18}+7 q^{17}+7 q^{16}-11 q^{15}-11 q^{14}+12 q^{13}+19 q^{12}-13 q^{11}-27 q^{10}+12 q^9+36 q^8-10 q^7-44 q^6+7 q^5+51 q^4-5 q^3-54 q^2+59-54 q^{-2} -5 q^{-3} +51 q^{-4} +7 q^{-5} -44 q^{-6} -10 q^{-7} +36 q^{-8} +12 q^{-9} -27 q^{-10} -13 q^{-11} +19 q^{-12} +12 q^{-13} -11 q^{-14} -11 q^{-15} +7 q^{-16} +7 q^{-17} -3 q^{-18} -5 q^{-19} +2 q^{-20} +2 q^{-21} -2 q^{-23} + q^{-24} }[/math]
4 [math]\displaystyle{ q^{40}-2 q^{39}+2 q^{37}-q^{36}+3 q^{35}-7 q^{34}+q^{33}+7 q^{32}-3 q^{31}+8 q^{30}-19 q^{29}-2 q^{28}+17 q^{27}+q^{26}+20 q^{25}-39 q^{24}-16 q^{23}+21 q^{22}+12 q^{21}+53 q^{20}-54 q^{19}-44 q^{18}+4 q^{17}+20 q^{16}+105 q^{15}-53 q^{14}-72 q^{13}-31 q^{12}+15 q^{11}+159 q^{10}-40 q^9-89 q^8-64 q^7+q^6+197 q^5-25 q^4-93 q^3-86 q^2-13 q+213-13 q^{-1} -86 q^{-2} -93 q^{-3} -25 q^{-4} +197 q^{-5} + q^{-6} -64 q^{-7} -89 q^{-8} -40 q^{-9} +159 q^{-10} +15 q^{-11} -31 q^{-12} -72 q^{-13} -53 q^{-14} +105 q^{-15} +20 q^{-16} +4 q^{-17} -44 q^{-18} -54 q^{-19} +53 q^{-20} +12 q^{-21} +21 q^{-22} -16 q^{-23} -39 q^{-24} +20 q^{-25} + q^{-26} +17 q^{-27} -2 q^{-28} -19 q^{-29} +8 q^{-30} -3 q^{-31} +7 q^{-32} + q^{-33} -7 q^{-34} +3 q^{-35} - q^{-36} +2 q^{-37} -2 q^{-39} + q^{-40} }[/math]
5 [math]\displaystyle{ q^{60}-2 q^{59}+2 q^{57}-q^{56}+q^{54}-3 q^{53}+6 q^{51}-5 q^{49}-2 q^{48}-5 q^{47}+2 q^{46}+14 q^{45}+9 q^{44}-7 q^{43}-16 q^{42}-19 q^{41}-3 q^{40}+28 q^{39}+34 q^{38}+12 q^{37}-24 q^{36}-55 q^{35}-37 q^{34}+19 q^{33}+67 q^{32}+70 q^{31}+9 q^{30}-78 q^{29}-110 q^{28}-45 q^{27}+71 q^{26}+147 q^{25}+100 q^{24}-52 q^{23}-182 q^{22}-156 q^{21}+19 q^{20}+204 q^{19}+216 q^{18}+21 q^{17}-217 q^{16}-268 q^{15}-64 q^{14}+221 q^{13}+312 q^{12}+101 q^{11}-218 q^{10}-341 q^9-138 q^8+211 q^7+370 q^6+158 q^5-206 q^4-372 q^3-183 q^2+188 q+393+188 q^{-1} -183 q^{-2} -372 q^{-3} -206 q^{-4} +158 q^{-5} +370 q^{-6} +211 q^{-7} -138 q^{-8} -341 q^{-9} -218 q^{-10} +101 q^{-11} +312 q^{-12} +221 q^{-13} -64 q^{-14} -268 q^{-15} -217 q^{-16} +21 q^{-17} +216 q^{-18} +204 q^{-19} +19 q^{-20} -156 q^{-21} -182 q^{-22} -52 q^{-23} +100 q^{-24} +147 q^{-25} +71 q^{-26} -45 q^{-27} -110 q^{-28} -78 q^{-29} +9 q^{-30} +70 q^{-31} +67 q^{-32} +19 q^{-33} -37 q^{-34} -55 q^{-35} -24 q^{-36} +12 q^{-37} +34 q^{-38} +28 q^{-39} -3 q^{-40} -19 q^{-41} -16 q^{-42} -7 q^{-43} +9 q^{-44} +14 q^{-45} +2 q^{-46} -5 q^{-47} -2 q^{-48} -5 q^{-49} +6 q^{-51} -3 q^{-53} + q^{-54} - q^{-56} +2 q^{-57} -2 q^{-59} + q^{-60} }[/math]
6 [math]\displaystyle{ q^{84}-2 q^{83}+2 q^{81}-q^{80}-2 q^{78}+5 q^{77}-4 q^{76}-q^{75}+8 q^{74}-4 q^{73}-4 q^{72}-9 q^{71}+12 q^{70}-5 q^{69}+2 q^{68}+23 q^{67}-5 q^{66}-13 q^{65}-31 q^{64}+15 q^{63}-13 q^{62}+8 q^{61}+60 q^{60}+15 q^{59}-13 q^{58}-68 q^{57}-3 q^{56}-57 q^{55}-9 q^{54}+107 q^{53}+79 q^{52}+42 q^{51}-73 q^{50}-19 q^{49}-163 q^{48}-108 q^{47}+93 q^{46}+146 q^{45}+172 q^{44}+32 q^{43}+60 q^{42}-272 q^{41}-302 q^{40}-68 q^{39}+107 q^{38}+298 q^{37}+249 q^{36}+310 q^{35}-265 q^{34}-491 q^{33}-357 q^{32}-103 q^{31}+302 q^{30}+470 q^{29}+687 q^{28}-100 q^{27}-566 q^{26}-651 q^{25}-416 q^{24}+164 q^{23}+590 q^{22}+1052 q^{21}+143 q^{20}-522 q^{19}-847 q^{18}-696 q^{17}-31 q^{16}+605 q^{15}+1305 q^{14}+348 q^{13}-430 q^{12}-936 q^{11}-867 q^{10}-188 q^9+570 q^8+1436 q^7+466 q^6-349 q^5-959 q^4-941 q^3-283 q^2+525 q+1477+525 q^{-1} -283 q^{-2} -941 q^{-3} -959 q^{-4} -349 q^{-5} +466 q^{-6} +1436 q^{-7} +570 q^{-8} -188 q^{-9} -867 q^{-10} -936 q^{-11} -430 q^{-12} +348 q^{-13} +1305 q^{-14} +605 q^{-15} -31 q^{-16} -696 q^{-17} -847 q^{-18} -522 q^{-19} +143 q^{-20} +1052 q^{-21} +590 q^{-22} +164 q^{-23} -416 q^{-24} -651 q^{-25} -566 q^{-26} -100 q^{-27} +687 q^{-28} +470 q^{-29} +302 q^{-30} -103 q^{-31} -357 q^{-32} -491 q^{-33} -265 q^{-34} +310 q^{-35} +249 q^{-36} +298 q^{-37} +107 q^{-38} -68 q^{-39} -302 q^{-40} -272 q^{-41} +60 q^{-42} +32 q^{-43} +172 q^{-44} +146 q^{-45} +93 q^{-46} -108 q^{-47} -163 q^{-48} -19 q^{-49} -73 q^{-50} +42 q^{-51} +79 q^{-52} +107 q^{-53} -9 q^{-54} -57 q^{-55} -3 q^{-56} -68 q^{-57} -13 q^{-58} +15 q^{-59} +60 q^{-60} +8 q^{-61} -13 q^{-62} +15 q^{-63} -31 q^{-64} -13 q^{-65} -5 q^{-66} +23 q^{-67} +2 q^{-68} -5 q^{-69} +12 q^{-70} -9 q^{-71} -4 q^{-72} -4 q^{-73} +8 q^{-74} - q^{-75} -4 q^{-76} +5 q^{-77} -2 q^{-78} - q^{-80} +2 q^{-81} -2 q^{-83} + q^{-84} }[/math]
7 [math]\displaystyle{ q^{112}-2 q^{111}+2 q^{109}-q^{108}-2 q^{106}+2 q^{105}+4 q^{104}-5 q^{103}+q^{102}+4 q^{101}-4 q^{100}-2 q^{99}-8 q^{98}+q^{97}+16 q^{96}-3 q^{95}+5 q^{94}+8 q^{93}-11 q^{92}-6 q^{91}-29 q^{90}-11 q^{89}+32 q^{88}+10 q^{87}+28 q^{86}+27 q^{85}-15 q^{84}-12 q^{83}-71 q^{82}-61 q^{81}+24 q^{80}+16 q^{79}+79 q^{78}+92 q^{77}+27 q^{76}+21 q^{75}-113 q^{74}-156 q^{73}-70 q^{72}-69 q^{71}+85 q^{70}+191 q^{69}+158 q^{68}+185 q^{67}-27 q^{66}-203 q^{65}-227 q^{64}-336 q^{63}-117 q^{62}+140 q^{61}+270 q^{60}+503 q^{59}+329 q^{58}+30 q^{57}-227 q^{56}-661 q^{55}-601 q^{54}-292 q^{53}+72 q^{52}+736 q^{51}+878 q^{50}+659 q^{49}+227 q^{48}-711 q^{47}-1129 q^{46}-1075 q^{45}-631 q^{44}+549 q^{43}+1283 q^{42}+1492 q^{41}+1138 q^{40}-257 q^{39}-1345 q^{38}-1870 q^{37}-1666 q^{36}-122 q^{35}+1287 q^{34}+2159 q^{33}+2185 q^{32}+554 q^{31}-1140 q^{30}-2362 q^{29}-2642 q^{28}-984 q^{27}+942 q^{26}+2477 q^{25}+3009 q^{24}+1367 q^{23}-721 q^{22}-2511 q^{21}-3289 q^{20}-1694 q^{19}+513 q^{18}+2513 q^{17}+3488 q^{16}+1926 q^{15}-347 q^{14}-2473 q^{13}-3596 q^{12}-2106 q^{11}+194 q^{10}+2437 q^9+3692 q^8+2217 q^7-126 q^6-2389 q^5-3694 q^4-2294 q^3+13 q^2+2343 q+3751+2343 q^{-1} +13 q^{-2} -2294 q^{-3} -3694 q^{-4} -2389 q^{-5} -126 q^{-6} +2217 q^{-7} +3692 q^{-8} +2437 q^{-9} +194 q^{-10} -2106 q^{-11} -3596 q^{-12} -2473 q^{-13} -347 q^{-14} +1926 q^{-15} +3488 q^{-16} +2513 q^{-17} +513 q^{-18} -1694 q^{-19} -3289 q^{-20} -2511 q^{-21} -721 q^{-22} +1367 q^{-23} +3009 q^{-24} +2477 q^{-25} +942 q^{-26} -984 q^{-27} -2642 q^{-28} -2362 q^{-29} -1140 q^{-30} +554 q^{-31} +2185 q^{-32} +2159 q^{-33} +1287 q^{-34} -122 q^{-35} -1666 q^{-36} -1870 q^{-37} -1345 q^{-38} -257 q^{-39} +1138 q^{-40} +1492 q^{-41} +1283 q^{-42} +549 q^{-43} -631 q^{-44} -1075 q^{-45} -1129 q^{-46} -711 q^{-47} +227 q^{-48} +659 q^{-49} +878 q^{-50} +736 q^{-51} +72 q^{-52} -292 q^{-53} -601 q^{-54} -661 q^{-55} -227 q^{-56} +30 q^{-57} +329 q^{-58} +503 q^{-59} +270 q^{-60} +140 q^{-61} -117 q^{-62} -336 q^{-63} -227 q^{-64} -203 q^{-65} -27 q^{-66} +185 q^{-67} +158 q^{-68} +191 q^{-69} +85 q^{-70} -69 q^{-71} -70 q^{-72} -156 q^{-73} -113 q^{-74} +21 q^{-75} +27 q^{-76} +92 q^{-77} +79 q^{-78} +16 q^{-79} +24 q^{-80} -61 q^{-81} -71 q^{-82} -12 q^{-83} -15 q^{-84} +27 q^{-85} +28 q^{-86} +10 q^{-87} +32 q^{-88} -11 q^{-89} -29 q^{-90} -6 q^{-91} -11 q^{-92} +8 q^{-93} +5 q^{-94} -3 q^{-95} +16 q^{-96} + q^{-97} -8 q^{-98} -2 q^{-99} -4 q^{-100} +4 q^{-101} + q^{-102} -5 q^{-103} +4 q^{-104} +2 q^{-105} -2 q^{-106} - q^{-108} +2 q^{-109} -2 q^{-111} + q^{-112} }[/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.

8 8.gif

8_8

8 10.gif

8_10

Retrieved from "https://katlas.org/index.php?title=8_9&oldid=61064"