9 30

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9 29.gif

9_29

9 31.gif

9_31

9 30.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 9 30's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 30 at Knotilus!

[edit 9 30 Quick Notes]
[edit 9 30 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X10,6,11,5 X8394 X2,9,3,10 X14,8,15,7 X18,15,1,16 X16,11,17,12 X12,17,13,18 X6,14,7,13
Gauss code 1, -4, 3, -1, 2, -9, 5, -3, 4, -2, 7, -8, 9, -5, 6, -7, 8, -6
Dowker-Thistlethwaite code 4 8 10 14 2 16 6 18 12
Conway Notation [211,21,2]


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

Length is 9, width is 4,

Braid index is 4

9 30 ML.gif 9 30 AP.gif
[{2, 12}, {1, 6}, {11, 4}, {12, 10}, {8, 11}, {7, 9}, {6, 8}, {3, 5}, {4, 7}, {5, 2}, {9, 3}, {10, 1}]

[edit Notes on presentations of 9 30]


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 30"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X4251 X10,6,11,5 X8394 X2,9,3,10 X14,8,15,7 X18,15,1,16 X16,11,17,12 X12,17,13,18 X6,14,7,13
In[5]:= GaussCode[K]
Out[5]= 1, -4, 3, -1, 2, -9, 5, -3, 4, -2, 7, -8, 9, -5, 6, -7, 8, -6
In[6]:= DTCode[K]
Out[6]= 4 8 10 14 2 16 6 18 12

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [211,21,2]
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]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{BR}(4,\{-1,-1,2,2,-1,2,-3,2,-3\})}
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]= { 4, 9, 4 }
In[11]:= Show[BraidPlot[br]]
BraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.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 30 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{2, 12}, {1, 6}, {11, 4}, {12, 10}, {8, 11}, {7, 9}, {6, 8}, {3, 5}, {4, 7}, {5, 2}, {9, 3}, {10, 1}]
In[14]:= Draw[ap]
9 30 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{4,6\}}
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-5]
Hyperbolic Volume 11.9545
A-Polynomial See Data:9 30/A-polynomial

[edit Notes for 9 30's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}
Topological 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}
Concordance genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3}
Rasmussen s-Invariant 0

[edit Notes for 9 30's four dimensional invariants]

Polynomial invariants

Alexander polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^3+5 t^2-12 t+17-12 t^{-1} +5 t^{-2} - t^{-3} }
Conway polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^6-z^4-z^2+1}
2nd Alexander ideal (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature { 53, 0 }
Jones polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^4-3 q^3+6 q^2-8 q+9-9 q^{-1} +8 q^{-2} -5 q^{-3} +3 q^{-4} - q^{-5} }
HOMFLY-PT polynomial (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^6+2 a^2 z^4+z^4 a^{-2} -4 z^4-a^4 z^2+5 a^2 z^2+2 z^2 a^{-2} -7 z^2-a^4+4 a^2+2 a^{-2} -4}
Kauffman polynomial (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^2 z^8+z^8+3 a^3 z^7+7 a z^7+4 z^7 a^{-1} +3 a^4 z^6+8 a^2 z^6+5 z^6 a^{-2} +10 z^6+a^5 z^5-3 a^3 z^5-9 a z^5-2 z^5 a^{-1} +3 z^5 a^{-3} -7 a^4 z^4-22 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} -23 z^4-2 a^5 z^3-3 a^3 z^3-2 z^3 a^{-1} -3 z^3 a^{-3} +5 a^4 z^2+16 a^2 z^2+5 z^2 a^{-2} -z^2 a^{-4} +17 z^2+a^5 z+2 a^3 z+a z+z a^{-1} +z a^{-3} -a^4-4 a^2-2 a^{-2} -4}
The A2 invariant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{16}+q^{12}-q^{10}+3 q^8+q^6+q^2-3+ q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} - q^{-10} + q^{-12} }
The G2 invariant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+7 q^{72}-5 q^{70}-5 q^{68}+19 q^{66}-31 q^{64}+39 q^{62}-34 q^{60}+11 q^{58}+19 q^{56}-55 q^{54}+79 q^{52}-79 q^{50}+49 q^{48}-2 q^{46}-48 q^{44}+83 q^{42}-84 q^{40}+60 q^{38}-9 q^{36}-36 q^{34}+61 q^{32}-52 q^{30}+14 q^{28}+39 q^{26}-69 q^{24}+75 q^{22}-40 q^{20}-15 q^{18}+77 q^{16}-118 q^{14}+120 q^{12}-84 q^{10}+16 q^8+55 q^6-109 q^4+123 q^2-97+43 q^{-2} +15 q^{-4} -64 q^{-6} +72 q^{-8} -52 q^{-10} +6 q^{-12} +39 q^{-14} -60 q^{-16} +48 q^{-18} -8 q^{-20} -41 q^{-22} +79 q^{-24} -87 q^{-26} +65 q^{-28} -21 q^{-30} -29 q^{-32} +67 q^{-34} -77 q^{-36} +67 q^{-38} -34 q^{-40} +5 q^{-42} +19 q^{-44} -33 q^{-46} +32 q^{-48} -23 q^{-50} +13 q^{-52} -2 q^{-54} -4 q^{-56} +5 q^{-58} -6 q^{-60} +4 q^{-62} -2 q^{-64} + q^{-66} }
Further Quantum Invariants
Further quantum knot invariants for 9_30.

A1 Invariants.

Weight Invariant
1
2 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{32}-2 q^{30}-2 q^{28}+7 q^{26}-3 q^{24}-9 q^{22}+14 q^{20}+q^{18}-18 q^{16}+13 q^{14}+8 q^{12}-17 q^{10}+4 q^8+10 q^6-6 q^4-6 q^2+6+9 q^{-2} -13 q^{-4} - q^{-6} +19 q^{-8} -13 q^{-10} -8 q^{-12} +17 q^{-14} -6 q^{-16} -8 q^{-18} +7 q^{-20} -2 q^{-24} + q^{-26} }
3 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{63}+2 q^{61}+2 q^{59}-3 q^{57}-7 q^{55}+3 q^{53}+16 q^{51}-2 q^{49}-26 q^{47}-7 q^{45}+39 q^{43}+23 q^{41}-47 q^{39}-45 q^{37}+48 q^{35}+68 q^{33}-36 q^{31}-91 q^{29}+19 q^{27}+98 q^{25}+4 q^{23}-96 q^{21}-26 q^{19}+87 q^{17}+40 q^{15}-65 q^{13}-50 q^{11}+41 q^9+55 q^7-12 q^5-57 q^3-15 q+55 q^{-1} +45 q^{-3} -48 q^{-5} -72 q^{-7} +37 q^{-9} +92 q^{-11} -20 q^{-13} -101 q^{-15} + q^{-17} +96 q^{-19} +20 q^{-21} -82 q^{-23} -32 q^{-25} +60 q^{-27} +33 q^{-29} -34 q^{-31} -30 q^{-33} +17 q^{-35} +21 q^{-37} -7 q^{-39} -10 q^{-41} + q^{-43} +4 q^{-45} -2 q^{-49} + q^{-51} }
4 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{104}-2 q^{102}-2 q^{100}+3 q^{98}+3 q^{96}+7 q^{94}-10 q^{92}-16 q^{90}+2 q^{88}+14 q^{86}+41 q^{84}-12 q^{82}-59 q^{80}-37 q^{78}+16 q^{76}+129 q^{74}+49 q^{72}-98 q^{70}-157 q^{68}-79 q^{66}+221 q^{64}+227 q^{62}-12 q^{60}-286 q^{58}-324 q^{56}+159 q^{54}+412 q^{52}+252 q^{50}-243 q^{48}-568 q^{46}-95 q^{44}+410 q^{42}+513 q^{40}-12 q^{38}-602 q^{36}-351 q^{34}+218 q^{32}+576 q^{30}+213 q^{28}-431 q^{26}-436 q^{24}+446 q^{20}+311 q^{18}-184 q^{16}-383 q^{14}-165 q^{12}+245 q^{10}+330 q^8+66 q^6-278 q^4-306 q^2+7+324 q^{-2} +339 q^{-4} -128 q^{-6} -432 q^{-8} -272 q^{-10} +246 q^{-12} +574 q^{-14} +101 q^{-16} -431 q^{-18} -517 q^{-20} +37 q^{-22} +623 q^{-24} +326 q^{-26} -233 q^{-28} -559 q^{-30} -203 q^{-32} +420 q^{-34} +381 q^{-36} +25 q^{-38} -367 q^{-40} -279 q^{-42} +144 q^{-44} +234 q^{-46} +133 q^{-48} -125 q^{-50} -176 q^{-52} + q^{-54} +70 q^{-56} +88 q^{-58} -14 q^{-60} -59 q^{-62} -11 q^{-64} +4 q^{-66} +26 q^{-68} +3 q^{-70} -11 q^{-72} - q^{-74} -2 q^{-76} +4 q^{-78} -2 q^{-82} + q^{-84} }
5 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{155}+2 q^{153}+2 q^{151}-3 q^{149}-3 q^{147}-3 q^{145}+10 q^{141}+16 q^{139}-2 q^{137}-23 q^{135}-29 q^{133}-13 q^{131}+32 q^{129}+71 q^{127}+52 q^{125}-43 q^{123}-132 q^{121}-124 q^{119}+8 q^{117}+199 q^{115}+275 q^{113}+97 q^{111}-254 q^{109}-473 q^{107}-316 q^{105}+194 q^{103}+700 q^{101}+693 q^{99}+26 q^{97}-853 q^{95}-1164 q^{93}-488 q^{91}+802 q^{89}+1646 q^{87}+1166 q^{85}-454 q^{83}-1975 q^{81}-1968 q^{79}-198 q^{77}+1994 q^{75}+2703 q^{73}+1111 q^{71}-1652 q^{69}-3227 q^{67}-2060 q^{65}+989 q^{63}+3361 q^{61}+2906 q^{59}-129 q^{57}-3148 q^{55}-3462 q^{53}-718 q^{51}+2634 q^{49}+3644 q^{47}+1441 q^{45}-1965 q^{43}-3518 q^{41}-1914 q^{39}+1292 q^{37}+3139 q^{35}+2125 q^{33}-659 q^{31}-2658 q^{29}-2166 q^{27}+164 q^{25}+2139 q^{23}+2091 q^{21}+265 q^{19}-1642 q^{17}-2014 q^{15}-655 q^{13}+1175 q^{11}+1966 q^9+1091 q^7-705 q^5-1948 q^3-1594 q+160 q^{-1} +1931 q^{-3} +2165 q^{-5} +487 q^{-7} -1821 q^{-9} -2732 q^{-11} -1254 q^{-13} +1532 q^{-15} +3201 q^{-17} +2084 q^{-19} -1027 q^{-21} -3429 q^{-23} -2860 q^{-25} +307 q^{-27} +3311 q^{-29} +3458 q^{-31} +531 q^{-33} -2851 q^{-35} -3692 q^{-37} -1334 q^{-39} +2073 q^{-41} +3538 q^{-43} +1939 q^{-45} -1174 q^{-47} -3018 q^{-49} -2189 q^{-51} +321 q^{-53} +2244 q^{-55} +2098 q^{-57} +317 q^{-59} -1438 q^{-61} -1736 q^{-63} -624 q^{-65} +727 q^{-67} +1234 q^{-69} +688 q^{-71} -244 q^{-73} -766 q^{-75} -564 q^{-77} +2 q^{-79} +393 q^{-81} +378 q^{-83} +89 q^{-85} -167 q^{-87} -223 q^{-89} -84 q^{-91} +63 q^{-93} +102 q^{-95} +54 q^{-97} -13 q^{-99} -41 q^{-101} -32 q^{-103} +3 q^{-105} +19 q^{-107} +8 q^{-109} - q^{-111} -2 q^{-113} -4 q^{-115} -2 q^{-117} +4 q^{-119} -2 q^{-123} + q^{-125} }

A2 Invariants.

Weight Invariant
1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{16}+q^{12}-q^{10}+3 q^8+q^6+q^2-3+ q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} - q^{-10} + q^{-12} }
1,1 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{44}-4 q^{42}+12 q^{40}-28 q^{38}+54 q^{36}-96 q^{34}+150 q^{32}-214 q^{30}+280 q^{28}-336 q^{26}+366 q^{24}-352 q^{22}+299 q^{20}-192 q^{18}+42 q^{16}+138 q^{14}-321 q^{12}+486 q^{10}-622 q^8+698 q^6-714 q^4+662 q^2-550+398 q^{-2} -213 q^{-4} +36 q^{-6} +132 q^{-8} -256 q^{-10} +334 q^{-12} -360 q^{-14} +344 q^{-16} -304 q^{-18} +239 q^{-20} -174 q^{-22} +118 q^{-24} -74 q^{-26} +42 q^{-28} -20 q^{-30} +10 q^{-32} -4 q^{-34} + q^{-36} }
2,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{42}-2 q^{38}-2 q^{36}+2 q^{34}+3 q^{32}-4 q^{30}-3 q^{28}+6 q^{26}+5 q^{24}-6 q^{22}-3 q^{20}+8 q^{18}+5 q^{16}-8 q^{14}-q^{12}+5 q^{10}-6 q^8-5 q^6+2 q^4-3+7 q^{-2} +9 q^{-4} -3 q^{-6} -2 q^{-8} +9 q^{-10} + q^{-12} -12 q^{-14} - q^{-16} +6 q^{-18} - q^{-20} -5 q^{-22} +4 q^{-26} - q^{-30} + q^{-32} }

A3 Invariants.

Weight Invariant
0,1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{34}-2 q^{32}+q^{30}+3 q^{28}-8 q^{26}+3 q^{24}+6 q^{22}-13 q^{20}+6 q^{18}+11 q^{16}-13 q^{14}+6 q^{12}+11 q^{10}-7 q^8-q^6+4 q^4+q^2-6-5 q^{-2} +9 q^{-4} -5 q^{-6} -10 q^{-8} +16 q^{-10} - q^{-12} -10 q^{-14} +13 q^{-16} - q^{-18} -7 q^{-20} +5 q^{-22} -2 q^{-26} + q^{-28} }
1,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{21}-q^{17}+q^{15}-q^{13}+4 q^{11}+q^9+3 q^7-2 q-3 q^{-1} -2 q^{-5} +2 q^{-7} +3 q^{-11} - q^{-13} + q^{-15} }

A4 Invariants.

Weight Invariant
0,1,0,0
1,0,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{26}-q^{22}-q^{20}+q^{18}-q^{16}+4 q^{14}+2 q^{12}+3 q^{10}+3 q^8-3 q^2-2-4 q^{-2} -2 q^{-6} +2 q^{-8} + q^{-10} + q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }

B2 Invariants.

Weight Invariant
0,1 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{34}+2 q^{32}-5 q^{30}+7 q^{28}-10 q^{26}+13 q^{24}-14 q^{22}+15 q^{20}-12 q^{18}+9 q^{16}-q^{14}-4 q^{12}+13 q^{10}-19 q^8+25 q^6-28 q^4+27 q^2-26+19 q^{-2} -13 q^{-4} +5 q^{-6} +2 q^{-8} -8 q^{-10} +13 q^{-12} -14 q^{-14} +15 q^{-16} -13 q^{-18} +11 q^{-20} -7 q^{-22} +4 q^{-24} -2 q^{-26} + q^{-28} }
1,0

D4 Invariants.

Weight Invariant
1,0,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{46}-2 q^{44}+3 q^{42}-4 q^{40}+6 q^{38}-9 q^{36}+8 q^{34}-11 q^{32}+11 q^{30}-13 q^{28}+9 q^{26}-8 q^{24}+9 q^{22}-q^{20}+9 q^{16}-4 q^{14}+17 q^{12}-17 q^{10}+19 q^8-20 q^6+21 q^4-24 q^2+14-20 q^{-2} +12 q^{-4} -9 q^{-6} +2 q^{-8} -2 q^{-10} - q^{-12} +11 q^{-14} -7 q^{-16} +11 q^{-18} -11 q^{-20} +14 q^{-22} -9 q^{-24} +8 q^{-26} -9 q^{-28} +6 q^{-30} -3 q^{-32} +2 q^{-34} -2 q^{-36} + q^{-38} }

G2 Invariants.

Weight Invariant
1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+7 q^{72}-5 q^{70}-5 q^{68}+19 q^{66}-31 q^{64}+39 q^{62}-34 q^{60}+11 q^{58}+19 q^{56}-55 q^{54}+79 q^{52}-79 q^{50}+49 q^{48}-2 q^{46}-48 q^{44}+83 q^{42}-84 q^{40}+60 q^{38}-9 q^{36}-36 q^{34}+61 q^{32}-52 q^{30}+14 q^{28}+39 q^{26}-69 q^{24}+75 q^{22}-40 q^{20}-15 q^{18}+77 q^{16}-118 q^{14}+120 q^{12}-84 q^{10}+16 q^8+55 q^6-109 q^4+123 q^2-97+43 q^{-2} +15 q^{-4} -64 q^{-6} +72 q^{-8} -52 q^{-10} +6 q^{-12} +39 q^{-14} -60 q^{-16} +48 q^{-18} -8 q^{-20} -41 q^{-22} +79 q^{-24} -87 q^{-26} +65 q^{-28} -21 q^{-30} -29 q^{-32} +67 q^{-34} -77 q^{-36} +67 q^{-38} -34 q^{-40} +5 q^{-42} +19 q^{-44} -33 q^{-46} +32 q^{-48} -23 q^{-50} +13 q^{-52} -2 q^{-54} -4 q^{-56} +5 q^{-58} -6 q^{-60} +4 q^{-62} -2 q^{-64} + q^{-66} }

.

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 30"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^3+5 t^2-12 t+17-12 t^{-1} +5 t^{-2} - t^{-3} }
In[5]:= Conway[K][z]
Out[5]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^6-z^4-z^2+1}
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]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 53, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^4-3 q^3+6 q^2-8 q+9-9 q^{-1} +8 q^{-2} -5 q^{-3} +3 q^{-4} - q^{-5} }
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^6+2 a^2 z^4+z^4 a^{-2} -4 z^4-a^4 z^2+5 a^2 z^2+2 z^2 a^{-2} -7 z^2-a^4+4 a^2+2 a^{-2} -4}
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^2 z^8+z^8+3 a^3 z^7+7 a z^7+4 z^7 a^{-1} +3 a^4 z^6+8 a^2 z^6+5 z^6 a^{-2} +10 z^6+a^5 z^5-3 a^3 z^5-9 a z^5-2 z^5 a^{-1} +3 z^5 a^{-3} -7 a^4 z^4-22 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} -23 z^4-2 a^5 z^3-3 a^3 z^3-2 z^3 a^{-1} -3 z^3 a^{-3} +5 a^4 z^2+16 a^2 z^2+5 z^2 a^{-2} -z^2 a^{-4} +17 z^2+a^5 z+2 a^3 z+a z+z a^{-1} +z a^{-3} -a^4-4 a^2-2 a^{-2} -4}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n130,}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {K11n114,}

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 30"];
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]= { Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^3+5 t^2-12 t+17-12 t^{-1} +5 t^{-2} - t^{-3} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^4-3 q^3+6 q^2-8 q+9-9 q^{-1} +8 q^{-2} -5 q^{-3} +3 q^{-4} - q^{-5} } }
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]= {K11n130,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11n114,}

Vassiliev invariants

V2 and V3: (-1, -1)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -8} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{38}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 32} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{208}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{64}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 24} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{32}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 32} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{328}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{152}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1222}{15}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{8342}{45}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{607}{18}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{1471}{30}}

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 0 is the signature of 9 30. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-101234χ
9         11
7        2 -2
5       41 3
3      42  -2
1     54   1
-1    55    0
-3   34     -1
-5  25      3
-7 13       -2
-9 2        2
-111         -1
Integral Khovanov Homology

(db, data source)

  
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{12}-3 q^{11}+2 q^{10}+8 q^9-18 q^8+4 q^7+31 q^6-43 q^5-q^4+63 q^3-63 q^2-13 q+85-66 q^{-1} -25 q^{-2} +85 q^{-3} -50 q^{-4} -31 q^{-5} +64 q^{-6} -25 q^{-7} -26 q^{-8} +33 q^{-9} -6 q^{-10} -13 q^{-11} +10 q^{-12} -3 q^{-14} + q^{-15} }
3 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{24}-3 q^{23}+2 q^{22}+4 q^{21}-2 q^{20}-14 q^{19}+5 q^{18}+32 q^{17}-6 q^{16}-61 q^{15}+q^{14}+99 q^{13}+21 q^{12}-153 q^{11}-49 q^{10}+201 q^9+97 q^8-248 q^7-151 q^6+282 q^5+209 q^4-303 q^3-260 q^2+306 q+302-293 q^{-1} -330 q^{-2} +264 q^{-3} +347 q^{-4} -226 q^{-5} -344 q^{-6} +173 q^{-7} +332 q^{-8} -121 q^{-9} -297 q^{-10} +60 q^{-11} +262 q^{-12} -21 q^{-13} -203 q^{-14} -19 q^{-15} +152 q^{-16} +34 q^{-17} -99 q^{-18} -39 q^{-19} +59 q^{-20} +32 q^{-21} -29 q^{-22} -23 q^{-23} +13 q^{-24} +13 q^{-25} -5 q^{-26} -5 q^{-27} +3 q^{-29} - q^{-30} }
4 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{40}-3 q^{39}+2 q^{38}+4 q^{37}-6 q^{36}+2 q^{35}-13 q^{34}+16 q^{33}+27 q^{32}-28 q^{31}-13 q^{30}-61 q^{29}+61 q^{28}+129 q^{27}-46 q^{26}-82 q^{25}-238 q^{24}+112 q^{23}+387 q^{22}+55 q^{21}-172 q^{20}-661 q^{19}+24 q^{18}+779 q^{17}+411 q^{16}-133 q^{15}-1284 q^{14}-332 q^{13}+1105 q^{12}+970 q^{11}+164 q^{10}-1870 q^9-886 q^8+1191 q^7+1502 q^6+637 q^5-2198 q^4-1404 q^3+1031 q^2+1806 q+1104-2213 q^{-1} -1721 q^{-2} +718 q^{-3} +1834 q^{-4} +1448 q^{-5} -1949 q^{-6} -1806 q^{-7} +308 q^{-8} +1616 q^{-9} +1647 q^{-10} -1454 q^{-11} -1671 q^{-12} -138 q^{-13} +1180 q^{-14} +1652 q^{-15} -810 q^{-16} -1308 q^{-17} -496 q^{-18} +611 q^{-19} +1401 q^{-20} -220 q^{-21} -783 q^{-22} -599 q^{-23} +106 q^{-24} +928 q^{-25} +105 q^{-26} -288 q^{-27} -439 q^{-28} -147 q^{-29} +445 q^{-30} +143 q^{-31} -14 q^{-32} -200 q^{-33} -153 q^{-34} +145 q^{-35} +65 q^{-36} +45 q^{-37} -53 q^{-38} -73 q^{-39} +32 q^{-40} +12 q^{-41} +23 q^{-42} -6 q^{-43} -20 q^{-44} +5 q^{-45} +5 q^{-47} -3 q^{-49} + q^{-50} }
5 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{60}-3 q^{59}+2 q^{58}+4 q^{57}-6 q^{56}-2 q^{55}+3 q^{54}-2 q^{53}+11 q^{52}+15 q^{51}-22 q^{50}-37 q^{49}-6 q^{48}+26 q^{47}+78 q^{46}+63 q^{45}-61 q^{44}-184 q^{43}-145 q^{42}+82 q^{41}+334 q^{40}+352 q^{39}-46 q^{38}-575 q^{37}-711 q^{36}-120 q^{35}+856 q^{34}+1284 q^{33}+500 q^{32}-1082 q^{31}-2062 q^{30}-1232 q^{29}+1154 q^{28}+3039 q^{27}+2281 q^{26}-936 q^{25}-3985 q^{24}-3742 q^{23}+325 q^{22}+4883 q^{21}+5394 q^{20}+663 q^{19}-5450 q^{18}-7149 q^{17}-2033 q^{16}+5724 q^{15}+8776 q^{14}+3590 q^{13}-5597 q^{12}-10153 q^{11}-5200 q^{10}+5155 q^9+11178 q^8+6701 q^7-4480 q^6-11822 q^5-7986 q^4+3677 q^3+12089 q^2+9009 q-2802-12056 q^{-1} -9757 q^{-2} +1923 q^{-3} +11735 q^{-4} +10252 q^{-5} -1006 q^{-6} -11181 q^{-7} -10548 q^{-8} +93 q^{-9} +10376 q^{-10} +10624 q^{-11} +901 q^{-12} -9355 q^{-13} -10500 q^{-14} -1882 q^{-15} +8046 q^{-16} +10132 q^{-17} +2900 q^{-18} -6571 q^{-19} -9486 q^{-20} -3729 q^{-21} +4840 q^{-22} +8528 q^{-23} +4453 q^{-24} -3165 q^{-25} -7283 q^{-26} -4739 q^{-27} +1488 q^{-28} +5784 q^{-29} +4767 q^{-30} -146 q^{-31} -4248 q^{-32} -4284 q^{-33} -884 q^{-34} +2735 q^{-35} +3600 q^{-36} +1429 q^{-37} -1485 q^{-38} -2692 q^{-39} -1593 q^{-40} +543 q^{-41} +1830 q^{-42} +1422 q^{-43} +36 q^{-44} -1072 q^{-45} -1113 q^{-46} -301 q^{-47} +540 q^{-48} +746 q^{-49} +347 q^{-50} -193 q^{-51} -446 q^{-52} -294 q^{-53} +34 q^{-54} +236 q^{-55} +190 q^{-56} +26 q^{-57} -95 q^{-58} -116 q^{-59} -42 q^{-60} +45 q^{-61} +58 q^{-62} +18 q^{-63} -6 q^{-64} -21 q^{-65} -23 q^{-66} +6 q^{-67} +13 q^{-68} +2 q^{-69} -5 q^{-72} +3 q^{-74} - q^{-75} }
6 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{84}-3 q^{83}+2 q^{82}+4 q^{81}-6 q^{80}-2 q^{79}-q^{78}+14 q^{77}-7 q^{76}-q^{75}+21 q^{74}-36 q^{73}-24 q^{72}-3 q^{71}+68 q^{70}+26 q^{69}+10 q^{68}+47 q^{67}-165 q^{66}-166 q^{65}-60 q^{64}+255 q^{63}+255 q^{62}+222 q^{61}+183 q^{60}-583 q^{59}-805 q^{58}-583 q^{57}+505 q^{56}+1060 q^{55}+1356 q^{54}+1143 q^{53}-1175 q^{52}-2662 q^{51}-2873 q^{50}-298 q^{49}+2254 q^{48}+4613 q^{47}+4970 q^{46}-202 q^{45}-5504 q^{44}-8702 q^{43}-5038 q^{42}+1365 q^{41}+9631 q^{40}+13916 q^{39}+6142 q^{38}-6085 q^{37}-17419 q^{36}-16308 q^{35}-6154 q^{34}+12392 q^{33}+26764 q^{32}+20456 q^{31}+655 q^{30}-24126 q^{29}-32162 q^{28}-22522 q^{27}+7565 q^{26}+37724 q^{25}+39809 q^{24}+16353 q^{23}-23280 q^{22}-46107 q^{21}-43764 q^{20}-5863 q^{19}+41144 q^{18}+57063 q^{17}+36348 q^{16}-14213 q^{15}-52623 q^{14}-62465 q^{13}-23006 q^{12}+36529 q^{11}+66871 q^{10}+53604 q^9-1437 q^8-51360 q^7-73801 q^6-37744 q^5+27760 q^4+68954 q^3+64272 q^2+10197 q-45553-77764 q^{-1} -47408 q^{-2} +18458 q^{-3} +65948 q^{-4} +68839 q^{-5} +19152 q^{-6} -37814 q^{-7} -76599 q^{-8} -52952 q^{-9} +9292 q^{-10} +59637 q^{-11} +69304 q^{-12} +26675 q^{-13} -28185 q^{-14} -71459 q^{-15} -56063 q^{-16} -1056 q^{-17} +49597 q^{-18} +66234 q^{-19} +33915 q^{-20} -15434 q^{-21} -61480 q^{-22} -56500 q^{-23} -13004 q^{-24} +34702 q^{-25} +58150 q^{-26} +39502 q^{-27} +65 q^{-28} -45519 q^{-29} -51820 q^{-30} -23870 q^{-31} +16060 q^{-32} +43584 q^{-33} +39740 q^{-34} +14429 q^{-35} -25239 q^{-36} -39997 q^{-37} -28734 q^{-38} -1411 q^{-39} +24455 q^{-40} +32015 q^{-41} +21942 q^{-42} -6334 q^{-43} -23166 q^{-44} -24707 q^{-45} -11495 q^{-46} +6871 q^{-47} +18726 q^{-48} +19971 q^{-49} +4875 q^{-50} -7707 q^{-51} -14685 q^{-52} -12040 q^{-53} -3182 q^{-54} +6384 q^{-55} +12097 q^{-56} +6920 q^{-57} +903 q^{-58} -5206 q^{-59} -7072 q^{-60} -5060 q^{-61} -208 q^{-62} +4665 q^{-63} +4072 q^{-64} +2696 q^{-65} -342 q^{-66} -2336 q^{-67} -3022 q^{-68} -1544 q^{-69} +917 q^{-70} +1252 q^{-71} +1561 q^{-72} +669 q^{-73} -200 q^{-74} -1062 q^{-75} -880 q^{-76} -27 q^{-77} +95 q^{-78} +487 q^{-79} +368 q^{-80} +189 q^{-81} -234 q^{-82} -281 q^{-83} -50 q^{-84} -78 q^{-85} +83 q^{-86} +97 q^{-87} +106 q^{-88} -37 q^{-89} -61 q^{-90} -3 q^{-91} -35 q^{-92} +6 q^{-93} +12 q^{-94} +32 q^{-95} -6 q^{-96} -13 q^{-97} +5 q^{-98} -7 q^{-99} +5 q^{-102} -3 q^{-104} + q^{-105} }
7 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{112}-3 q^{111}+2 q^{110}+4 q^{109}-6 q^{108}-2 q^{107}-q^{106}+10 q^{105}+9 q^{104}-19 q^{103}+5 q^{102}+7 q^{101}-23 q^{100}-11 q^{99}-3 q^{98}+54 q^{97}+66 q^{96}-43 q^{95}-26 q^{94}-48 q^{93}-120 q^{92}-41 q^{91}+10 q^{90}+256 q^{89}+360 q^{88}+57 q^{87}-118 q^{86}-439 q^{85}-697 q^{84}-416 q^{83}-11 q^{82}+938 q^{81}+1661 q^{80}+1136 q^{79}+234 q^{78}-1533 q^{77}-3073 q^{76}-2855 q^{75}-1462 q^{74}+2017 q^{73}+5545 q^{72}+6200 q^{71}+4188 q^{70}-1697 q^{69}-8598 q^{68}-11668 q^{67}-9984 q^{66}-939 q^{65}+11553 q^{64}+19699 q^{63}+20162 q^{62}+7772 q^{61}-12632 q^{60}-29561 q^{59}-35566 q^{58}-21280 q^{57}+8931 q^{56}+39285 q^{55}+56377 q^{54}+43541 q^{53}+2667 q^{52}-45848 q^{51}-80761 q^{50}-74920 q^{49}-25322 q^{48}+44770 q^{47}+105238 q^{46}+114826 q^{45}+60818 q^{44}-32714 q^{43}-125498 q^{42}-159320 q^{41}-108154 q^{40}+6363 q^{39}+136276 q^{38}+204052 q^{37}+165116 q^{36}+34010 q^{35}-134696 q^{34}-243111 q^{33}-225975 q^{32}-86496 q^{31}+118414 q^{30}+272168 q^{29}+285618 q^{28}+146556 q^{27}-89189 q^{26}-288323 q^{25}-338081 q^{24}-208618 q^{23}+49860 q^{22}+290962 q^{21}+379836 q^{20}+267301 q^{19}-5088 q^{18}-281941 q^{17}-409021 q^{16}-318196 q^{15}-40378 q^{14}+264251 q^{13}+425772 q^{12}+358976 q^{11}+82746 q^{10}-241485 q^9-432054 q^8-389088 q^7-119422 q^6+216898 q^5+430356 q^4+409333 q^3+149583 q^2-192532 q-423183-421788 q^{-1} -173788 q^{-2} +169564 q^{-3} +412620 q^{-4} +428330 q^{-5} +193300 q^{-6} -147605 q^{-7} -399497 q^{-8} -430936 q^{-9} -210137 q^{-10} +125684 q^{-11} +384105 q^{-12} +430664 q^{-13} +225747 q^{-14} -102197 q^{-15} -365409 q^{-16} -427749 q^{-17} -241544 q^{-18} +75535 q^{-19} +342270 q^{-20} +421529 q^{-21} +257631 q^{-22} -44562 q^{-23} -312837 q^{-24} -410688 q^{-25} -273437 q^{-26} +9110 q^{-27} +276036 q^{-28} +393037 q^{-29} +286912 q^{-30} +30306 q^{-31} -231134 q^{-32} -367093 q^{-33} -295693 q^{-34} -70743 q^{-35} +179153 q^{-36} +330748 q^{-37} +296342 q^{-38} +109569 q^{-39} -121869 q^{-40} -284915 q^{-41} -286634 q^{-42} -141581 q^{-43} +63812 q^{-44} +230046 q^{-45} +264303 q^{-46} +163756 q^{-47} -9038 q^{-48} -170584 q^{-49} -230685 q^{-50} -172069 q^{-51} -36434 q^{-52} +110653 q^{-53} +187209 q^{-54} +166245 q^{-55} +69551 q^{-56} -56314 q^{-57} -139368 q^{-58} -147372 q^{-59} -87038 q^{-60} +12099 q^{-61} +91670 q^{-62} +119293 q^{-63} +90197 q^{-64} +18811 q^{-65} -50124 q^{-66} -87023 q^{-67} -81246 q^{-68} -35776 q^{-69} +17916 q^{-70} +55884 q^{-71} +64964 q^{-72} +40523 q^{-73} +3154 q^{-74} -29803 q^{-75} -46006 q^{-76} -36794 q^{-77} -13917 q^{-78} +11200 q^{-79} +28515 q^{-80} +28303 q^{-81} +16730 q^{-82} +106 q^{-83} -14834 q^{-84} -18971 q^{-85} -14824 q^{-86} -5135 q^{-87} +5906 q^{-88} +10887 q^{-89} +10739 q^{-90} +6184 q^{-91} -907 q^{-92} -5177 q^{-93} -6764 q^{-94} -5221 q^{-95} -1033 q^{-96} +1882 q^{-97} +3592 q^{-98} +3494 q^{-99} +1452 q^{-100} -168 q^{-101} -1623 q^{-102} -2149 q^{-103} -1162 q^{-104} -277 q^{-105} +598 q^{-106} +1047 q^{-107} +671 q^{-108} +429 q^{-109} -79 q^{-110} -538 q^{-111} -402 q^{-112} -265 q^{-113} +9 q^{-114} +202 q^{-115} +121 q^{-116} +170 q^{-117} +90 q^{-118} -86 q^{-119} -87 q^{-120} -87 q^{-121} -16 q^{-122} +42 q^{-123} -5 q^{-124} +31 q^{-125} +35 q^{-126} -6 q^{-127} -12 q^{-128} -23 q^{-129} -3 q^{-130} +13 q^{-131} -5 q^{-132} +7 q^{-134} -5 q^{-137} +3 q^{-139} - q^{-140} }

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.

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