10 90

10_89

10_91

(KnotPlot image)

See the full Rolfsen Knot Table.

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[edit 10 90 Quick Notes]

[edit 10 90 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₆₂₇₁ X₂₈₃₇ X_18,14,19,13 X_14,5,15,6 X_20,12,1,11 X_12,20,13,19 X_8,15,9,16 X_10,4,11,3 X_16,9,17,10 X_4,17,5,18
Gauss code	1, -2, 8, -10, 4, -1, 2, -7, 9, -8, 5, -6, 3, -4, 7, -9, 10, -3, 6, -5
Dowker-Thistlethwaite code	6 10 14 2 16 20 18 8 4 12
Conway Notation	[.3.2.2]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 11, width is 4,

Braid index is 4

[{7, 12}, {2, 11}, {12, 8}, {4, 1}, {3, 5}, {6, 4}, {5, 7}, {9, 6}, {8, 2}, {10, 3}, {11, 9}, {1, 10}]

[edit Notes on presentations of 10 90]

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["10 90"];

In[4]:= PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= X₆₂₇₁ X₂₈₃₇ X_18,14,19,13 X_14,5,15,6 X_20,12,1,11 X_12,20,13,19 X_8,15,9,16 X_10,4,11,3 X_16,9,17,10 X_4,17,5,18

In[5]:= GaussCode[K]

Out[5]= 1, -2, 8, -10, 4, -1, 2, -7, 9, -8, 5, -6, 3, -4, 7, -9, 10, -3, 6, -5

In[6]:= DTCode[K]

Out[6]= 6 10 14 2 16 20 18 8 4 12

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:= ConwayNotation[K]

Out[8]= [.3.2.2]

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,-1,2,3,-2,-1,3,2,2\})}

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, 11, 4 }

In[11]:= Show[BraidPlot[br]]

Out[11]= -Graphics-

In[12]:= Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]= -Graphics-

In[13]:= ap = ArcPresentation[K]

Out[13]= ArcPresentation[{7, 12}, {2, 11}, {12, 8}, {4, 1}, {3, 5}, {6, 4}, {5, 7}, {9, 6}, {8, 2}, {10, 3}, {11, 9}, {1, 10}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Chiral
Unknotting number	2
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-5][-7]
Hyperbolic Volume	13.8661
A-Polynomial	See Data:10 90/A-polynomial

[edit Notes for 10 90'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 10 90's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+8t^{2}-17t+23-17t^{-1}+8t^{-2}-2t^{-3}$
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 -2 z^6-4 z^4-3 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	{ 77, 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^6-3 q^5+6 q^4-9 q^3+12 q^2-13 q+12-10 q^{-1} +7 q^{-2} -3 q^{-3} + q^{-4} }
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 a^{-2} -z^6+a^2 z^4-3 z^4 a^{-2} +z^4 a^{-4} -3 z^4+2 a^2 z^2-3 z^2 a^{-2} +2 z^2 a^{-4} -4 z^2+2 a^2+ a^{-4} -2}
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 2 z^9 a^{-1} +2 z^9 a^{-3} +9 z^8 a^{-2} +4 z^8 a^{-4} +5 z^8+7 a z^7+5 z^7 a^{-1} +z^7 a^{-3} +3 z^7 a^{-5} +6 a^2 z^6-21 z^6 a^{-2} -11 z^6 a^{-4} +z^6 a^{-6} -3 z^6+3 a^3 z^5-10 a z^5-15 z^5 a^{-1} -11 z^5 a^{-3} -9 z^5 a^{-5} +a^4 z^4-8 a^2 z^4+15 z^4 a^{-2} +9 z^4 a^{-4} -3 z^4 a^{-6} -6 z^4-2 a^3 z^3+7 a z^3+9 z^3 a^{-1} +7 z^3 a^{-3} +7 z^3 a^{-5} -a^4 z^2+6 a^2 z^2-5 z^2 a^{-2} -4 z^2 a^{-4} +2 z^2 a^{-6} +8 z^2-2 a z-2 z a^{-1} -z a^{-3} -z a^{-5} -2 a^2+ a^{-4} -2}
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^{12}-q^{10}+2 q^8+2 q^6-2 q^4+2 q^2-3- q^{-6} +3 q^{-8} -2 q^{-10} + q^{-12} + q^{-14} - q^{-16} + q^{-18} }
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^{66}-2 q^{64}+4 q^{62}-6 q^{60}+6 q^{58}-5 q^{56}+11 q^{52}-22 q^{50}+33 q^{48}-39 q^{46}+30 q^{44}-11 q^{42}-20 q^{40}+62 q^{38}-90 q^{36}+104 q^{34}-85 q^{32}+36 q^{30}+31 q^{28}-103 q^{26}+157 q^{24}-161 q^{22}+114 q^{20}-23 q^{18}-74 q^{16}+146 q^{14}-151 q^{12}+96 q^{10}-2 q^8-96 q^6+137 q^4-108 q^2+9+113 q^{-2} -199 q^{-4} +206 q^{-6} -126 q^{-8} -18 q^{-10} +157 q^{-12} -255 q^{-14} +265 q^{-16} -189 q^{-18} +54 q^{-20} +93 q^{-22} -195 q^{-24} +229 q^{-26} -176 q^{-28} +68 q^{-30} +46 q^{-32} -131 q^{-34} +148 q^{-36} -93 q^{-38} -4 q^{-40} +110 q^{-42} -162 q^{-44} +139 q^{-46} -48 q^{-48} -75 q^{-50} +172 q^{-52} -204 q^{-54} +161 q^{-56} -64 q^{-58} -48 q^{-60} +133 q^{-62} -159 q^{-64} +134 q^{-66} -69 q^{-68} + q^{-70} +45 q^{-72} -66 q^{-74} +59 q^{-76} -37 q^{-78} +16 q^{-80} +2 q^{-82} -10 q^{-84} +10 q^{-86} -9 q^{-88} +5 q^{-90} -2 q^{-92} + q^{-94} }

Further Quantum Invariants

Further quantum knot invariants for 10_90.

A1 Invariants.

Weight	Invariant
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^9-2 q^7+4 q^5-3 q^3+2 q- q^{-1} - q^{-3} +3 q^{-5} -3 q^{-7} +3 q^{-9} -2 q^{-11} + q^{-13} }
2	$q^{26}-2q^{24}+q^{22}+6q^{20}-10q^{18}+18q^{14}-20q^{12}-7q^{10}+30q^{8}-17q^{6}-16q^{4}+26q^{2}-17q^{-2}+5q^{-4}+16q^{-6}-9q^{-8}-17q^{-10}+22q^{-12}+5q^{-14}-29q^{-16}+16q^{-18}+17q^{-20}-26q^{-22}+3q^{-24}+18q^{-26}-12q^{-28}-5q^{-30}+8q^{-32}-q^{-34}-2q^{-36}+q^{-38}$
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 q^{51}-2 q^{49}+q^{47}+3 q^{45}-q^{43}-7 q^{41}+q^{39}+15 q^{37}-5 q^{35}-28 q^{33}+11 q^{31}+48 q^{29}-7 q^{27}-86 q^{25}-3 q^{23}+124 q^{21}+31 q^{19}-151 q^{17}-78 q^{15}+160 q^{13}+125 q^{11}-133 q^9-160 q^7+85 q^5+169 q^3-21 q-155 q^{-1} -40 q^{-3} +123 q^{-5} +91 q^{-7} -83 q^{-9} -127 q^{-11} +42 q^{-13} +152 q^{-15} -170 q^{-19} -37 q^{-21} +171 q^{-23} +80 q^{-25} -159 q^{-27} -123 q^{-29} +131 q^{-31} +155 q^{-33} -82 q^{-35} -171 q^{-37} +25 q^{-39} +162 q^{-41} +28 q^{-43} -126 q^{-45} -64 q^{-47} +78 q^{-49} +73 q^{-51} -34 q^{-53} -59 q^{-55} +2 q^{-57} +37 q^{-59} +10 q^{-61} -17 q^{-63} -8 q^{-65} +5 q^{-67} +4 q^{-69} - q^{-71} -2 q^{-73} + q^{-75} }
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^{84}-2 q^{82}+q^{80}+3 q^{78}-4 q^{76}+2 q^{74}-6 q^{72}+4 q^{70}+10 q^{68}-16 q^{66}+7 q^{64}-6 q^{62}+22 q^{60}+21 q^{58}-76 q^{56}-29 q^{54}+15 q^{52}+152 q^{50}+122 q^{48}-214 q^{46}-269 q^{44}-102 q^{42}+421 q^{40}+560 q^{38}-170 q^{36}-731 q^{34}-672 q^{32}+451 q^{30}+1245 q^{28}+442 q^{26}-844 q^{24}-1467 q^{22}-195 q^{20}+1414 q^{18}+1269 q^{16}-178 q^{14}-1617 q^{12}-1035 q^{10}+686 q^8+1430 q^6+703 q^4-882 q^2-1258-279 q^{-2} +843 q^{-4} +1071 q^{-6} +40 q^{-8} -903 q^{-10} -852 q^{-12} +175 q^{-14} +1020 q^{-16} +633 q^{-18} -525 q^{-20} -1116 q^{-22} -252 q^{-24} +928 q^{-26} +1047 q^{-28} -235 q^{-30} -1333 q^{-32} -676 q^{-34} +758 q^{-36} +1448 q^{-38} +259 q^{-40} -1309 q^{-42} -1202 q^{-44} +169 q^{-46} +1532 q^{-48} +976 q^{-50} -687 q^{-52} -1398 q^{-54} -715 q^{-56} +911 q^{-58} +1307 q^{-60} +290 q^{-62} -832 q^{-64} -1143 q^{-66} -63 q^{-68} +817 q^{-70} +769 q^{-72} +45 q^{-74} -733 q^{-76} -515 q^{-78} +59 q^{-80} +471 q^{-82} +393 q^{-84} -121 q^{-86} -292 q^{-88} -209 q^{-90} +50 q^{-92} +208 q^{-94} +78 q^{-96} -24 q^{-98} -96 q^{-100} -48 q^{-102} +33 q^{-104} +26 q^{-106} +19 q^{-108} -11 q^{-110} -14 q^{-112} +2 q^{-114} + q^{-116} +4 q^{-118} - q^{-120} -2 q^{-122} + q^{-124} }
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^{125}-2 q^{123}+q^{121}+3 q^{119}-4 q^{117}-q^{115}+3 q^{113}-3 q^{111}-q^{109}+5 q^{107}-q^{105}+15 q^{101}-q^{99}-27 q^{97}-39 q^{95}-16 q^{93}+48 q^{91}+121 q^{89}+114 q^{87}-61 q^{85}-286 q^{83}-338 q^{81}-35 q^{79}+498 q^{77}+802 q^{75}+409 q^{73}-667 q^{71}-1577 q^{69}-1237 q^{67}+547 q^{65}+2514 q^{63}+2730 q^{61}+284 q^{59}-3369 q^{57}-4870 q^{55}-2063 q^{53}+3520 q^{51}+7226 q^{49}+4996 q^{47}-2436 q^{45}-9142 q^{43}-8627 q^{41}-136 q^{39}+9702 q^{37}+12097 q^{35}+4020 q^{33}-8385 q^{31}-14428 q^{29}-8283 q^{27}+5247 q^{25}+14680 q^{23}+11867 q^{21}-914 q^{19}-12781 q^{17}-13779 q^{15}-3448 q^{13}+9153 q^{11}+13615 q^9+6881 q^7-4783 q^5-11677 q^3-8780 q+692 q^{-1} +8712 q^{-3} +9167 q^{-5} +2435 q^{-7} -5583 q^{-9} -8546 q^{-11} -4429 q^{-13} +3030 q^{-15} +7576 q^{-17} +5501 q^{-19} -1302 q^{-21} -6829 q^{-23} -6153 q^{-25} +275 q^{-27} +6647 q^{-29} +6865 q^{-31} +376 q^{-33} -6928 q^{-35} -7942 q^{-37} -1212 q^{-39} +7358 q^{-41} +9501 q^{-43} +2569 q^{-45} -7451 q^{-47} -11203 q^{-49} -4686 q^{-51} +6652 q^{-53} +12622 q^{-55} +7432 q^{-57} -4693 q^{-59} -13095 q^{-61} -10270 q^{-63} +1496 q^{-65} +12104 q^{-67} +12524 q^{-69} +2473 q^{-71} -9478 q^{-73} -13383 q^{-75} -6370 q^{-77} +5424 q^{-79} +12375 q^{-81} +9281 q^{-83} -798 q^{-85} -9546 q^{-87} -10368 q^{-89} -3345 q^{-91} +5490 q^{-93} +9409 q^{-95} +6061 q^{-97} -1303 q^{-99} -6839 q^{-101} -6794 q^{-103} -1963 q^{-105} +3538 q^{-107} +5770 q^{-109} +3679 q^{-111} -605 q^{-113} -3737 q^{-115} -3768 q^{-117} -1258 q^{-119} +1580 q^{-121} +2800 q^{-123} +1913 q^{-125} -51 q^{-127} -1513 q^{-129} -1629 q^{-131} -664 q^{-133} +462 q^{-135} +993 q^{-137} +739 q^{-139} +80 q^{-141} -418 q^{-143} -488 q^{-145} -226 q^{-147} +76 q^{-149} +230 q^{-151} +181 q^{-153} +28 q^{-155} -74 q^{-157} -82 q^{-159} -37 q^{-161} +6 q^{-163} +33 q^{-165} +22 q^{-167} -3 q^{-169} -8 q^{-171} -4 q^{-173} -2 q^{-175} + q^{-177} +4 q^{-179} - q^{-181} -2 q^{-183} + q^{-185} }

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^{12}-q^{10}+2 q^8+2 q^6-2 q^4+2 q^2-3- q^{-6} +3 q^{-8} -2 q^{-10} + q^{-12} + q^{-14} - q^{-16} + q^{-18} }
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^{36}-4 q^{34}+10 q^{32}-20 q^{30}+42 q^{28}-74 q^{26}+120 q^{24}-180 q^{22}+264 q^{20}-358 q^{18}+454 q^{16}-556 q^{14}+643 q^{12}-692 q^{10}+672 q^8-584 q^6+404 q^4-130 q^2-206+588 q^{-2} -941 q^{-4} +1248 q^{-6} -1450 q^{-8} +1526 q^{-10} -1481 q^{-12} +1298 q^{-14} -1014 q^{-16} +658 q^{-18} -275 q^{-20} -94 q^{-22} +424 q^{-24} -660 q^{-26} +789 q^{-28} -818 q^{-30} +760 q^{-32} -640 q^{-34} +489 q^{-36} -344 q^{-38} +220 q^{-40} -126 q^{-42} +66 q^{-44} -30 q^{-46} +12 q^{-48} -4 q^{-50} + q^{-52} }
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^{32}-q^{30}+5 q^{26}-6 q^{22}+2 q^{20}+9 q^{18}-2 q^{16}-15 q^{14}+2 q^{12}+12 q^{10}-10 q^8-10 q^6+11 q^4+8 q^2-4+9 q^{-4} -2 q^{-6} -6 q^{-8} +7 q^{-10} - q^{-12} -11 q^{-14} +5 q^{-16} +11 q^{-18} -8 q^{-20} -8 q^{-22} +7 q^{-24} +8 q^{-26} -7 q^{-28} -8 q^{-30} +8 q^{-32} +4 q^{-34} -5 q^{-36} -3 q^{-38} +2 q^{-40} +3 q^{-42} - q^{-44} - q^{-46} + q^{-48} }

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^{28}-2 q^{26}+6 q^{22}-7 q^{20}-3 q^{18}+17 q^{16}-11 q^{14}-8 q^{12}+24 q^{10}-11 q^8-13 q^6+21 q^4-6 q^2-11+8 q^{-2} +2 q^{-4} -4 q^{-6} -9 q^{-8} +10 q^{-10} +7 q^{-12} -19 q^{-14} +10 q^{-16} +14 q^{-18} -21 q^{-20} +7 q^{-22} +13 q^{-24} -16 q^{-26} +6 q^{-28} +6 q^{-30} -8 q^{-32} +3 q^{-34} + q^{-36} -2 q^{-38} + q^{-40} }

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^{15}-q^{13}+3 q^{11}+3 q^7-2 q^5+2 q^3-3 q- q^{-1} - q^{-3} - q^{-5} + q^{-7} - q^{-9} +3 q^{-11} -2 q^{-13} +2 q^{-15} - q^{-17} +2 q^{-19} - q^{-21} + q^{-23} }

A4 Invariants.

Weight

Invariant

0,1,0,0

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^{34}-q^{32}-q^{30}+4 q^{28}-5 q^{24}+3 q^{22}+10 q^{20}-10 q^{16}+6 q^{14}+13 q^{12}-12 q^{10}-15 q^8+15 q^6+4 q^4-17 q^2+3+16 q^{-2} -5 q^{-4} -11 q^{-6} +13 q^{-8} +5 q^{-10} -19 q^{-12} +2 q^{-14} +19 q^{-16} -14 q^{-18} -13 q^{-20} +19 q^{-22} +6 q^{-24} -16 q^{-26} - q^{-28} +15 q^{-30} - q^{-32} -10 q^{-34} +5 q^{-36} +7 q^{-38} -6 q^{-40} -2 q^{-42} +4 q^{-44} - q^{-46} - q^{-48} + q^{-50} }

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^{18}-q^{16}+3 q^{14}+q^{12}+q^{10}+3 q^8-2 q^6+2 q^4-3 q^2-1-2 q^{-2} - q^{-4} - q^{-6} + q^{-10} - q^{-12} +3 q^{-14} -2 q^{-16} +2 q^{-18} +2 q^{-24} - q^{-26} + q^{-28} }

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^{28}-2 q^{26}+4 q^{24}-8 q^{22}+13 q^{20}-17 q^{18}+23 q^{16}-25 q^{14}+28 q^{12}-24 q^{10}+19 q^8-9 q^6-3 q^4+16 q^2-31+40 q^{-2} -50 q^{-4} +52 q^{-6} -51 q^{-8} +44 q^{-10} -33 q^{-12} +21 q^{-14} -6 q^{-16} -6 q^{-18} +17 q^{-20} -23 q^{-22} +27 q^{-24} -26 q^{-26} +24 q^{-28} -20 q^{-30} +14 q^{-32} -9 q^{-34} +5 q^{-36} -2 q^{-38} + q^{-40} }

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^{46}-2 q^{42}-2 q^{40}+2 q^{38}+7 q^{36}+2 q^{34}-10 q^{32}-10 q^{30}+6 q^{28}+20 q^{26}+5 q^{24}-20 q^{22}-18 q^{20}+13 q^{18}+27 q^{16}+q^{14}-27 q^{12}-13 q^{10}+20 q^8+20 q^6-12 q^4-22 q^2+5+21 q^{-2} +2 q^{-4} -20 q^{-6} -6 q^{-8} +16 q^{-10} +9 q^{-12} -15 q^{-14} -12 q^{-16} +14 q^{-18} +17 q^{-20} -10 q^{-22} -24 q^{-24} +3 q^{-26} +28 q^{-28} +11 q^{-30} -24 q^{-32} -23 q^{-34} +13 q^{-36} +28 q^{-38} + q^{-40} -23 q^{-42} -12 q^{-44} +14 q^{-46} +15 q^{-48} -4 q^{-50} -11 q^{-52} -2 q^{-54} +6 q^{-56} +3 q^{-58} -2 q^{-60} -2 q^{-62} + q^{-66} }

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^{38}-2 q^{36}+2 q^{34}-3 q^{32}+7 q^{30}-10 q^{28}+10 q^{26}-12 q^{24}+20 q^{22}-19 q^{20}+19 q^{18}-19 q^{16}+24 q^{14}-15 q^{12}+11 q^{10}-9 q^8+2 q^6+9 q^4-16 q^2+19-31 q^{-2} +35 q^{-4} -38 q^{-6} +38 q^{-8} -43 q^{-10} +38 q^{-12} -32 q^{-14} +29 q^{-16} -23 q^{-18} +14 q^{-20} -4 q^{-22} +7 q^{-26} -13 q^{-28} +20 q^{-30} -19 q^{-32} +21 q^{-34} -21 q^{-36} +21 q^{-38} -17 q^{-40} +14 q^{-42} -12 q^{-44} +8 q^{-46} -5 q^{-48} +3 q^{-50} -2 q^{-52} + q^{-54} }

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^{66}-2 q^{64}+4 q^{62}-6 q^{60}+6 q^{58}-5 q^{56}+11 q^{52}-22 q^{50}+33 q^{48}-39 q^{46}+30 q^{44}-11 q^{42}-20 q^{40}+62 q^{38}-90 q^{36}+104 q^{34}-85 q^{32}+36 q^{30}+31 q^{28}-103 q^{26}+157 q^{24}-161 q^{22}+114 q^{20}-23 q^{18}-74 q^{16}+146 q^{14}-151 q^{12}+96 q^{10}-2 q^8-96 q^6+137 q^4-108 q^2+9+113 q^{-2} -199 q^{-4} +206 q^{-6} -126 q^{-8} -18 q^{-10} +157 q^{-12} -255 q^{-14} +265 q^{-16} -189 q^{-18} +54 q^{-20} +93 q^{-22} -195 q^{-24} +229 q^{-26} -176 q^{-28} +68 q^{-30} +46 q^{-32} -131 q^{-34} +148 q^{-36} -93 q^{-38} -4 q^{-40} +110 q^{-42} -162 q^{-44} +139 q^{-46} -48 q^{-48} -75 q^{-50} +172 q^{-52} -204 q^{-54} +161 q^{-56} -64 q^{-58} -48 q^{-60} +133 q^{-62} -159 q^{-64} +134 q^{-66} -69 q^{-68} + q^{-70} +45 q^{-72} -66 q^{-74} +59 q^{-76} -37 q^{-78} +16 q^{-80} +2 q^{-82} -10 q^{-84} +10 q^{-86} -9 q^{-88} +5 q^{-90} -2 q^{-92} + q^{-94} }

.

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.

In[3]:= K = Knot["10 90"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= $-2t^{3}+8t^{2}-17t+23-17t^{-1}+8t^{-2}-2t^{-3}$

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 -2 z^6-4 z^4-3 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]= { 77, 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^6-3 q^5+6 q^4-9 q^3+12 q^2-13 q+12-10 q^{-1} +7 q^{-2} -3 q^{-3} + q^{-4} }

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 a^{-2} -z^6+a^2 z^4-3 z^4 a^{-2} +z^4 a^{-4} -3 z^4+2 a^2 z^2-3 z^2 a^{-2} +2 z^2 a^{-4} -4 z^2+2 a^2+ a^{-4} -2}

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 2 z^9 a^{-1} +2 z^9 a^{-3} +9 z^8 a^{-2} +4 z^8 a^{-4} +5 z^8+7 a z^7+5 z^7 a^{-1} +z^7 a^{-3} +3 z^7 a^{-5} +6 a^2 z^6-21 z^6 a^{-2} -11 z^6 a^{-4} +z^6 a^{-6} -3 z^6+3 a^3 z^5-10 a z^5-15 z^5 a^{-1} -11 z^5 a^{-3} -9 z^5 a^{-5} +a^4 z^4-8 a^2 z^4+15 z^4 a^{-2} +9 z^4 a^{-4} -3 z^4 a^{-6} -6 z^4-2 a^3 z^3+7 a z^3+9 z^3 a^{-1} +7 z^3 a^{-3} +7 z^3 a^{-5} -a^4 z^2+6 a^2 z^2-5 z^2 a^{-2} -4 z^2 a^{-4} +2 z^2 a^{-6} +8 z^2-2 a z-2 z a^{-1} -z a^{-3} -z a^{-5} -2 a^2+ a^{-4} -2}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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}} ): {}

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.

In[3]:= K = Knot["10 90"];

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 -2 t^3+8 t^2-17 t+23-17 t^{-1} +8 t^{-2} -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^6-3 q^5+6 q^4-9 q^3+12 q^2-13 q+12-10 q^{-1} +7 q^{-2} -3 q^{-3} + q^{-4} } }

In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]= {}

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:

(-3, -1)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

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 -12}

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 72}

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 130}

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 70}

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 96}

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{592}{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{160}{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 56}

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 -288}

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 -1560}

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 -840}

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{13391}{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 \frac{9826}{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{9474}{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 \frac{613}{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 -\frac{4751}{10}}

V_2,1 through V_6,9 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

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 10 90. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

13

1

11

2

-2

9

4

1

3

7

5

2

-3

5

7

4

3

6

5

-1

1

6

7

-1

5

7

2

-3

2

5

-3

-5

1

5

4

-7

2

-2

-9

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=-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}}
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}^{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=-2}	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	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=-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}^{5}\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}^{7}\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}^{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 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}^{7}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 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^{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 {\mathbb Z}^{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 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}^{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=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^{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=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}\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=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 {\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^{18}-3 q^{17}+q^{16}+10 q^{15}-16 q^{14}-6 q^{13}+40 q^{12}-31 q^{11}-35 q^{10}+83 q^9-32 q^8-80 q^7+117 q^6-15 q^5-119 q^4+125 q^3+10 q^2-130 q+103+27 q^{-1} -104 q^{-2} +61 q^{-3} +26 q^{-4} -57 q^{-5} +24 q^{-6} +13 q^{-7} -19 q^{-8} +6 q^{-9} +3 q^{-10} -3 q^{-11} + q^{-12} }
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^{36}-3 q^{35}+q^{34}+5 q^{33}+2 q^{32}-16 q^{31}-8 q^{30}+32 q^{29}+29 q^{28}-51 q^{27}-69 q^{26}+57 q^{25}+136 q^{24}-46 q^{23}-211 q^{22}-5 q^{21}+290 q^{20}+88 q^{19}-348 q^{18}-201 q^{17}+379 q^{16}+325 q^{15}-372 q^{14}-455 q^{13}+343 q^{12}+564 q^{11}-281 q^{10}-663 q^9+210 q^8+734 q^7-129 q^6-773 q^5+41 q^4+778 q^3+45 q^2-741 q-122+663 q^{-1} +179 q^{-2} -551 q^{-3} -206 q^{-4} +418 q^{-5} +206 q^{-6} -293 q^{-7} -171 q^{-8} +180 q^{-9} +133 q^{-10} -111 q^{-11} -78 q^{-12} +53 q^{-13} +50 q^{-14} -32 q^{-15} -23 q^{-16} +16 q^{-17} +11 q^{-18} -9 q^{-19} -3 q^{-20} +2 q^{-21} +3 q^{-22} -3 q^{-23} + q^{-24} }
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^{60}-3 q^{59}+q^{58}+5 q^{57}-3 q^{56}+2 q^{55}-19 q^{54}+4 q^{53}+35 q^{52}+4 q^{51}+9 q^{50}-100 q^{49}-44 q^{48}+107 q^{47}+106 q^{46}+139 q^{45}-258 q^{44}-303 q^{43}+24 q^{42}+277 q^{41}+653 q^{40}-180 q^{39}-715 q^{38}-550 q^{37}+59 q^{36}+1431 q^{35}+544 q^{34}-667 q^{33}-1430 q^{32}-1021 q^{31}+1742 q^{30}+1666 q^{29}+350 q^{28}-1826 q^{27}-2647 q^{26}+1059 q^{25}+2377 q^{24}+2013 q^{23}-1270 q^{22}-4010 q^{21}-312 q^{20}+2270 q^{19}+3581 q^{18}-81 q^{17}-4700 q^{16}-1746 q^{15}+1613 q^{14}+4679 q^{13}+1201 q^{12}-4819 q^{11}-2926 q^{10}+749 q^9+5270 q^8+2359 q^7-4432 q^6-3771 q^5-278 q^4+5219 q^3+3302 q^2-3401 q-3999-1400 q^{-1} +4240 q^{-2} +3678 q^{-3} -1816 q^{-4} -3272 q^{-5} -2144 q^{-6} +2519 q^{-7} +3096 q^{-8} -377 q^{-9} -1825 q^{-10} -1999 q^{-11} +910 q^{-12} +1824 q^{-13} +246 q^{-14} -539 q^{-15} -1196 q^{-16} +116 q^{-17} +701 q^{-18} +187 q^{-19} +22 q^{-20} -466 q^{-21} -23 q^{-22} +178 q^{-23} +20 q^{-24} +77 q^{-25} -130 q^{-26} +7 q^{-27} +41 q^{-28} -24 q^{-29} +30 q^{-30} -33 q^{-31} +8 q^{-32} +13 q^{-33} -11 q^{-34} +7 q^{-35} -7 q^{-36} +2 q^{-37} +3 q^{-38} -3 q^{-39} + q^{-40} }
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^{90}-3 q^{89}+q^{88}+5 q^{87}-3 q^{86}-3 q^{85}-q^{84}-7 q^{83}+6 q^{82}+30 q^{81}+8 q^{80}-30 q^{79}-44 q^{78}-52 q^{77}+14 q^{76}+132 q^{75}+161 q^{74}+19 q^{73}-198 q^{72}-354 q^{71}-248 q^{70}+202 q^{69}+659 q^{68}+678 q^{67}+56 q^{66}-885 q^{65}-1374 q^{64}-763 q^{63}+775 q^{62}+2140 q^{61}+2020 q^{60}+2 q^{59}-2594 q^{58}-3601 q^{57}-1735 q^{56}+2171 q^{55}+5152 q^{54}+4286 q^{53}-503 q^{52}-5833 q^{51}-7236 q^{50}-2660 q^{49}+5107 q^{48}+9822 q^{47}+6861 q^{46}-2485 q^{45}-11155 q^{44}-11495 q^{43}-1916 q^{42}+10644 q^{41}+15609 q^{40}+7594 q^{39}-8061 q^{38}-18446 q^{37}-13710 q^{36}+3631 q^{35}+19514 q^{34}+19545 q^{33}+1990 q^{32}-18866 q^{31}-24318 q^{30}-8135 q^{29}+16689 q^{28}+27947 q^{27}+14115 q^{26}-13676 q^{25}-30288 q^{24}-19473 q^{23}+10172 q^{22}+31699 q^{21}+24135 q^{20}-6744 q^{19}-32431 q^{18}-28043 q^{17}+3442 q^{16}+32713 q^{15}+31439 q^{14}-255 q^{13}-32649 q^{12}-34415 q^{11}-2986 q^{10}+32037 q^9+36966 q^8+6548 q^7-30574 q^6-38961 q^5-10445 q^4+27920 q^3+39929 q^2+14566 q-23842-39416 q^{-1} -18465 q^{-2} +18448 q^{-3} +37032 q^{-4} +21460 q^{-5} -12178 q^{-6} -32682 q^{-7} -22927 q^{-8} +5847 q^{-9} +26701 q^{-10} +22458 q^{-11} -311 q^{-12} -19901 q^{-13} -20114 q^{-14} -3586 q^{-15} +13171 q^{-16} +16313 q^{-17} +5732 q^{-18} -7496 q^{-19} -12037 q^{-20} -5981 q^{-21} +3333 q^{-22} +7822 q^{-23} +5217 q^{-24} -790 q^{-25} -4605 q^{-26} -3751 q^{-27} -373 q^{-28} +2239 q^{-29} +2410 q^{-30} +711 q^{-31} -952 q^{-32} -1305 q^{-33} -589 q^{-34} +272 q^{-35} +626 q^{-36} +371 q^{-37} -42 q^{-38} -229 q^{-39} -196 q^{-40} -32 q^{-41} +93 q^{-42} +68 q^{-43} +10 q^{-44} -4 q^{-45} -21 q^{-46} -25 q^{-47} +20 q^{-48} +4 q^{-49} -13 q^{-50} +8 q^{-51} +5 q^{-52} -9 q^{-53} +5 q^{-54} +3 q^{-55} -7 q^{-56} +2 q^{-57} +3 q^{-58} -3 q^{-59} + q^{-60} }
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^{126}-3 q^{125}+q^{124}+5 q^{123}-3 q^{122}-3 q^{121}-6 q^{120}+11 q^{119}-5 q^{118}+q^{117}+33 q^{116}-11 q^{115}-30 q^{114}-59 q^{113}+13 q^{112}+5 q^{111}+48 q^{110}+188 q^{109}+66 q^{108}-78 q^{107}-326 q^{106}-227 q^{105}-236 q^{104}+61 q^{103}+752 q^{102}+815 q^{101}+530 q^{100}-525 q^{99}-1047 q^{98}-1842 q^{97}-1431 q^{96}+699 q^{95}+2489 q^{94}+3599 q^{93}+2096 q^{92}+28 q^{91}-4346 q^{90}-6743 q^{89}-4655 q^{88}+237 q^{87}+6841 q^{86}+9782 q^{85}+9804 q^{84}+662 q^{83}-10201 q^{82}-16362 q^{81}-14479 q^{80}-2649 q^{79}+11757 q^{78}+26139 q^{77}+22534 q^{76}+5933 q^{75}-16988 q^{74}-34124 q^{73}-33295 q^{72}-13902 q^{71}+23809 q^{70}+46402 q^{69}+46852 q^{68}+18206 q^{67}-25773 q^{66}-61525 q^{65}-66867 q^{64}-22705 q^{63}+32950 q^{62}+79896 q^{61}+81819 q^{60}+34263 q^{59}-43783 q^{58}-106330 q^{57}-98185 q^{56}-37368 q^{55}+60004 q^{54}+127146 q^{53}+123013 q^{52}+33696 q^{51}-88642 q^{50}-152642 q^{49}-135055 q^{48}-19963 q^{47}+114550 q^{46}+189733 q^{45}+136731 q^{44}-13264 q^{43}-151878 q^{42}-211104 q^{41}-123129 q^{40}+49348 q^{39}+205989 q^{38}+219738 q^{37}+82307 q^{36}-105328 q^{35}-242719 q^{34}-208320 q^{33}-32203 q^{32}+183269 q^{31}+264744 q^{30}+162109 q^{29}-46006 q^{28}-241959 q^{27}-261671 q^{26}-99959 q^{25}+149827 q^{24}+283267 q^{23}+216416 q^{22}+2714 q^{21}-231908 q^{20}-294186 q^{19}-148925 q^{18}+121756 q^{17}+293522 q^{16}+256837 q^{15}+41686 q^{14}-221716 q^{13}-320428 q^{12}-192793 q^{11}+91814 q^{10}+298416 q^9+295685 q^8+87821 q^7-197948 q^6-337805 q^5-242412 q^4+39952 q^3+278740 q^2+324994 q+150023-138342 q^{-1} -321569 q^{-2} -283807 q^{-3} -39051 q^{-4} +211280 q^{-5} +314459 q^{-6} +206845 q^{-7} -43312 q^{-8} -248797 q^{-9} -281236 q^{-10} -114816 q^{-11} +103697 q^{-12} +242732 q^{-13} +217374 q^{-14} +47526 q^{-15} -135261 q^{-16} -216162 q^{-17} -142272 q^{-18} +3646 q^{-19} +133411 q^{-20} +166740 q^{-21} +87514 q^{-22} -34037 q^{-23} -118629 q^{-24} -111009 q^{-25} -43989 q^{-26} +41095 q^{-27} +89014 q^{-28} +71913 q^{-29} +15195 q^{-30} -40685 q^{-31} -56681 q^{-32} -40723 q^{-33} -2598 q^{-34} +30328 q^{-35} +36299 q^{-36} +19757 q^{-37} -4920 q^{-38} -17781 q^{-39} -20186 q^{-40} -9482 q^{-41} +4922 q^{-42} +11596 q^{-43} +9742 q^{-44} +2510 q^{-45} -2360 q^{-46} -6198 q^{-47} -4929 q^{-48} -613 q^{-49} +2172 q^{-50} +2798 q^{-51} +1427 q^{-52} +626 q^{-53} -1162 q^{-54} -1498 q^{-55} -514 q^{-56} +149 q^{-57} +474 q^{-58} +285 q^{-59} +442 q^{-60} -102 q^{-61} -326 q^{-62} -103 q^{-63} -15 q^{-64} +38 q^{-65} -18 q^{-66} +135 q^{-67} +3 q^{-68} -64 q^{-69} +3 q^{-70} +2 q^{-71} +6 q^{-72} -29 q^{-73} +28 q^{-74} +5 q^{-75} -17 q^{-76} +7 q^{-77} + q^{-78} +3 q^{-79} -7 q^{-80} +2 q^{-81} +3 q^{-82} -3 q^{-83} + q^{-84} }
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^{168}-3 q^{167}+q^{166}+5 q^{165}-3 q^{164}-3 q^{163}-6 q^{162}+6 q^{161}+13 q^{160}-10 q^{159}+4 q^{158}+14 q^{157}-12 q^{156}-25 q^{155}-50 q^{154}-3 q^{153}+80 q^{152}+41 q^{151}+73 q^{150}+75 q^{149}-39 q^{148}-137 q^{147}-349 q^{146}-280 q^{145}+62 q^{144}+262 q^{143}+611 q^{142}+777 q^{141}+482 q^{140}-33 q^{139}-1197 q^{138}-1954 q^{137}-1637 q^{136}-841 q^{135}+1141 q^{134}+3156 q^{133}+4160 q^{132}+3903 q^{131}+627 q^{130}-3929 q^{129}-7439 q^{128}-9165 q^{127}-6025 q^{126}+921 q^{125}+9122 q^{124}+16600 q^{123}+16643 q^{122}+8562 q^{121}-5163 q^{120}-21616 q^{119}-30455 q^{118}-27134 q^{117}-10817 q^{116}+17023 q^{115}+41700 q^{114}+52255 q^{113}+41646 q^{112}+6061 q^{111}-38093 q^{110}-73835 q^{109}-84140 q^{108}-53476 q^{107}+6259 q^{106}+75098 q^{105}+124316 q^{104}+120307 q^{103}+61803 q^{102}-36089 q^{101}-137700 q^{100}-188091 q^{99}-162167 q^{98}-55908 q^{97}+98297 q^{96}+225794 q^{95}+270991 q^{94}+196626 q^{93}+13720 q^{92}-197282 q^{91}-350689 q^{90}-361566 q^{89}-196908 q^{88}+77351 q^{87}+356667 q^{86}+505217 q^{85}+425739 q^{84}+139974 q^{83}-253986 q^{82}-576901 q^{81}-653121 q^{80}-430915 q^{79}+30934 q^{78}+532543 q^{77}+821293 q^{76}+748904 q^{75}+296493 q^{74}-351335 q^{73}-880619 q^{72}-1035087 q^{71}-684996 q^{70}+41884 q^{69}+801119 q^{68}+1234285 q^{67}+1077796 q^{66}+361126 q^{65}-581315 q^{64}-1310135 q^{63}-1419250 q^{62}-806151 q^{61}+247201 q^{60}+1251602 q^{59}+1666695 q^{58}+1238912 q^{57}+159293 q^{56}-1072241 q^{55}-1801501 q^{54}-1616053 q^{53}-588314 q^{52}+805598 q^{51}+1825437 q^{50}+1909470 q^{49}+997097 q^{48}-490836 q^{47}-1759253 q^{46}-2113116 q^{45}-1354592 q^{44}+168796 q^{43}+1633267 q^{42}+2233769 q^{41}+1644804 q^{40}+131037 q^{39}-1478630 q^{38}-2291644 q^{37}-1867561 q^{36}-388246 q^{35}+1323942 q^{34}+2309291 q^{33}+2031422 q^{32}+596649 q^{31}-1187234 q^{30}-2309954 q^{29}-2154151 q^{28}-760431 q^{27}+1078614 q^{26}+2311809 q^{25}+2254664 q^{24}+892576 q^{23}-995948 q^{22}-2324969 q^{21}-2351935 q^{20}-1012902 q^{19}+927171 q^{18}+2350868 q^{17}+2459920 q^{16}+1142223 q^{15}-852161 q^{14}-2378137 q^{13}-2582437 q^{12}-1300059 q^{11}+744547 q^{10}+2385478 q^9+2712137 q^8+1495817 q^7-580254 q^6-2342637 q^5-2825972 q^4-1724804 q^3+342533 q^2+2218128 q+2890038+1964744 q^{-1} -32001 q^{-2} -1988955 q^{-3} -2865538 q^{-4} -2176687 q^{-5} -329195 q^{-6} +1649485 q^{-7} +2720281 q^{-8} +2314102 q^{-9} +699005 q^{-10} -1219352 q^{-11} -2441560 q^{-12} -2335101 q^{-13} -1022762 q^{-14} +742845 q^{-15} +2043268 q^{-16} +2215929 q^{-17} +1248347 q^{-18} -278917 q^{-19} -1566594 q^{-20} -1962250 q^{-21} -1341777 q^{-22} -112090 q^{-23} +1072105 q^{-24} +1607740 q^{-25} +1295785 q^{-26} +386653 q^{-27} -620831 q^{-28} -1206390 q^{-29} -1135216 q^{-30} -528988 q^{-31} +262448 q^{-32} +818064 q^{-33} +903270 q^{-34} +550088 q^{-35} -18644 q^{-36} -488230 q^{-37} -652137 q^{-38} -486290 q^{-39} -113670 q^{-40} +245319 q^{-41} +424422 q^{-42} +376722 q^{-43} +158567 q^{-44} -88826 q^{-45} -245886 q^{-46} -261194 q^{-47} -148954 q^{-48} +6301 q^{-49} +124257 q^{-50} +161838 q^{-51} +114092 q^{-52} +26878 q^{-53} -51605 q^{-54} -90157 q^{-55} -75935 q^{-56} -31887 q^{-57} +14988 q^{-58} +44849 q^{-59} +44695 q^{-60} +25310 q^{-61} +90 q^{-62} -19681 q^{-63} -23602 q^{-64} -16554 q^{-65} -4353 q^{-66} +7448 q^{-67} +11403 q^{-68} +9515 q^{-69} +4041 q^{-70} -2372 q^{-71} -4825 q^{-72} -4802 q^{-73} -2914 q^{-74} +356 q^{-75} +1946 q^{-76} +2429 q^{-77} +1637 q^{-78} -66 q^{-79} -605 q^{-80} -899 q^{-81} -889 q^{-82} -210 q^{-83} +145 q^{-84} +496 q^{-85} +444 q^{-86} -26 q^{-87} -28 q^{-88} -73 q^{-89} -168 q^{-90} -64 q^{-91} -51 q^{-92} +87 q^{-93} +105 q^{-94} -41 q^{-95} - q^{-96} +8 q^{-97} -12 q^{-98} -2 q^{-99} -27 q^{-100} +12 q^{-101} +25 q^{-102} -17 q^{-103} - q^{-104} +3 q^{-105} + q^{-106} +3 q^{-107} -7 q^{-108} +2 q^{-109} +3 q^{-110} -3 q^{-111} + q^{-112} }

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.

10_89

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10 90

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

Modifying This Page

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