9 33

9_32

9_34

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 33's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 33 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X12,8,13,7 X8394 X2,9,3,10 X18,13,1,14 X14,5,15,6 X6,17,7,18 X16,12,17,11 X10,16,11,15
Gauss code	1, -4, 3, -1, 6, -7, 2, -3, 4, -9, 8, -2, 5, -6, 9, -8, 7, -5
Dowker-Thistlethwaite code	4 8 14 12 2 16 18 10 6
Conway Notation	[.21.2]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 9, width is 4, Braid index is 4		[{3, 10}, {6, 2}, {1, 3}, {5, 8}, {7, 9}, {8, 11}, {10, 6}, {4, 7}, {2, 5}, {11, 4}, {9, 1}]

[edit Notes on presentations of 9 33]

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 33"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X12,8,13,7 X8394 X2,9,3,10 X18,13,1,14 X14,5,15,6 X6,17,7,18 X16,12,17,11 X10,16,11,15

In[5]:=

GaussCode[K]

Out[5]=

1, -4, 3, -1, 6, -7, 2, -3, 4, -9, 8, -2, 5, -6, 9, -8, 7, -5

In[6]:=

DTCode[K]

Out[6]=

4 8 14 12 2 16 18 10 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[.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,\{-1,2,-1,2,2,-1,-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]]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-5] Hyperbolic Volume 13.2805 A-Polynomial See Data:9 33/A-polynomial

[edit Notes for 9 33's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for 9 33's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+6t^{2}-14t+19-14t^{-1}+6t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 61, 0 }
Jones polynomial	$q^{4}-4q^{3}+7q^{2}-9q+11-10q^{-1}+9q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$
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} -3 z^4-a^4 z^2+4 a^2 z^2+z^2 a^{-2} -3 z^2-a^4+2 a^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 a^2 z^8+2 z^8+4 a^3 z^7+10 a z^7+6 z^7 a^{-1} +3 a^4 z^6+5 a^2 z^6+7 z^6 a^{-2} +9 z^6+a^5 z^5-6 a^3 z^5-16 a z^5-5 z^5 a^{-1} +4 z^5 a^{-3} -6 a^4 z^4-16 a^2 z^4-9 z^4 a^{-2} +z^4 a^{-4} -20 z^4-2 a^5 z^3+a^3 z^3+5 a z^3-z^3 a^{-1} -3 z^3 a^{-3} +4 a^4 z^2+10 a^2 z^2+3 z^2 a^{-2} +9 z^2+a^5 z+a^3 z-a^4-2 a^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^{16}+q^{12}-2 q^{10}+2 q^8+3 q^2-1+3 q^{-2} -2 q^{-4} + q^{-8} -2 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}+8 q^{72}-7 q^{70}-2 q^{68}+17 q^{66}-35 q^{64}+50 q^{62}-52 q^{60}+28 q^{58}+13 q^{56}-67 q^{54}+113 q^{52}-123 q^{50}+92 q^{48}-23 q^{46}-64 q^{44}+129 q^{42}-148 q^{40}+111 q^{38}-31 q^{36}-54 q^{34}+104 q^{32}-98 q^{30}+43 q^{28}+39 q^{26}-101 q^{24}+121 q^{22}-81 q^{20}-5 q^{18}+102 q^{16}-171 q^{14}+188 q^{12}-138 q^{10}+43 q^8+71 q^6-159 q^4+198 q^2-170+88 q^{-2} +17 q^{-4} -101 q^{-6} +132 q^{-8} -101 q^{-10} +29 q^{-12} +54 q^{-14} -99 q^{-16} +89 q^{-18} -33 q^{-20} -51 q^{-22} +119 q^{-24} -142 q^{-26} +110 q^{-28} -38 q^{-30} -44 q^{-32} +103 q^{-34} -124 q^{-36} +105 q^{-38} -58 q^{-40} +6 q^{-42} +31 q^{-44} -54 q^{-46} +53 q^{-48} -36 q^{-50} +20 q^{-52} -2 q^{-54} -6 q^{-56} +9 q^{-58} -10 q^{-60} +6 q^{-62} -3 q^{-64} + q^{-66} }

Further Quantum Invariants

Further quantum knot invariants for 9_33.

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^{11}+2 q^9-3 q^7+3 q^5-q^3+q+2 q^{-1} -2 q^{-3} +3 q^{-5} -3 q^{-7} + q^{-9} }
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}-q^{28}+8 q^{26}-6 q^{24}-11 q^{22}+19 q^{20}-q^{18}-24 q^{16}+19 q^{14}+10 q^{12}-23 q^{10}+7 q^8+14 q^6-9 q^4-7 q^2+10+10 q^{-2} -19 q^{-4} +24 q^{-8} -19 q^{-10} -10 q^{-12} +24 q^{-14} -8 q^{-16} -11 q^{-18} +10 q^{-20} -3 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}+q^{59}-4 q^{57}-5 q^{55}+9 q^{53}+17 q^{51}-14 q^{49}-37 q^{47}+7 q^{45}+64 q^{43}+17 q^{41}-89 q^{39}-53 q^{37}+97 q^{35}+98 q^{33}-83 q^{31}-142 q^{29}+55 q^{27}+160 q^{25}-14 q^{23}-163 q^{21}-24 q^{19}+146 q^{17}+55 q^{15}-116 q^{13}-71 q^{11}+78 q^9+88 q^7-37 q^5-96 q^3-7 q+100 q^{-1} +55 q^{-3} -99 q^{-5} -100 q^{-7} +84 q^{-9} +144 q^{-11} -59 q^{-13} -163 q^{-15} +19 q^{-17} +164 q^{-19} +19 q^{-21} -141 q^{-23} -48 q^{-25} +103 q^{-27} +55 q^{-29} -58 q^{-31} -50 q^{-33} +26 q^{-35} +35 q^{-37} -11 q^{-39} -15 q^{-41} + q^{-43} +7 q^{-45} -3 q^{-49} + q^{-51} }
4	$q^{104}-2q^{102}-q^{100}+4q^{98}+q^{96}+2q^{94}-15q^{92}-11q^{90}+22q^{88}+29q^{86}+29q^{84}-62q^{82}-100q^{80}-q^{78}+114q^{76}+209q^{74}-22q^{72}-287q^{70}-263q^{68}+41q^{66}+528q^{64}+357q^{62}-249q^{60}-678q^{58}-460q^{56}+546q^{54}+887q^{52}+285q^{50}-737q^{48}-1090q^{46}+37q^{44}+1015q^{42}+925q^{40}-283q^{38}-1272q^{36}-567q^{34}+639q^{32}+1143q^{30}+245q^{28}-957q^{26}-813q^{24}+157q^{22}+933q^{20}+515q^{18}-486q^{16}-764q^{14}-200q^{12}+594q^{10}+626q^{8}-30q^{6}-653q^{4}-533q^{2}+214+742q^{-2}+505q^{-4}-478q^{-6}-908q^{-8}-311q^{-10}+735q^{-12}+1078q^{-14}-47q^{-16}-1054q^{-18}-913q^{-20}+347q^{-22}+1318q^{-24}+534q^{-26}-683q^{-28}-1152q^{-30}-260q^{-32}+938q^{-34}+781q^{-36}-51q^{-38}-793q^{-40}-535q^{-42}+312q^{-44}+507q^{-46}+254q^{-48}-261q^{-50}-350q^{-52}-11q^{-54}+144q^{-56}+170q^{-58}-21q^{-60}-107q^{-62}-27q^{-64}+7q^{-66}+45q^{-68}+6q^{-70}-18q^{-72}-3q^{-74}-2q^{-76}+7q^{-78}-3q^{-82}+q^{-84}$
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}+q^{151}-4 q^{149}-q^{147}+2 q^{145}+4 q^{143}+9 q^{141}+3 q^{139}-25 q^{137}-35 q^{135}-5 q^{133}+46 q^{131}+94 q^{129}+70 q^{127}-58 q^{125}-221 q^{123}-235 q^{121}-7 q^{119}+348 q^{117}+555 q^{115}+310 q^{113}-364 q^{111}-1004 q^{109}-932 q^{107}+45 q^{105}+1342 q^{103}+1869 q^{101}+852 q^{99}-1269 q^{97}-2883 q^{95}-2321 q^{93}+449 q^{91}+3483 q^{89}+4137 q^{87}+1278 q^{85}-3263 q^{83}-5786 q^{81}-3654 q^{79}+1968 q^{77}+6658 q^{75}+6189 q^{73}+321 q^{71}-6439 q^{69}-8261 q^{67}-3059 q^{65}+5080 q^{63}+9299 q^{61}+5719 q^{59}-2916 q^{57}-9238 q^{55}-7670 q^{53}+518 q^{51}+8149 q^{49}+8637 q^{47}+1662 q^{45}-6495 q^{43}-8639 q^{41}-3221 q^{39}+4654 q^{37}+7933 q^{35}+4089 q^{33}-2980 q^{31}-6839 q^{29}-4415 q^{27}+1618 q^{25}+5711 q^{23}+4441 q^{21}-578 q^{19}-4685 q^{17}-4449 q^{15}-358 q^{13}+3874 q^{11}+4636 q^9+1353 q^7-3141 q^5-5037 q^3-2629 q+2274 q^{-1} +5598 q^{-3} +4247 q^{-5} -1081 q^{-7} -6023 q^{-9} -6083 q^{-11} -652 q^{-13} +5991 q^{-15} +7924 q^{-17} +2837 q^{-19} -5227 q^{-21} -9225 q^{-23} -5283 q^{-25} +3573 q^{-27} +9657 q^{-29} +7474 q^{-31} -1241 q^{-33} -8894 q^{-35} -8890 q^{-37} -1354 q^{-39} +7044 q^{-41} +9129 q^{-43} +3583 q^{-45} -4489 q^{-47} -8184 q^{-49} -4898 q^{-51} +1895 q^{-53} +6290 q^{-55} +5125 q^{-57} +182 q^{-59} -4094 q^{-61} -4423 q^{-63} -1321 q^{-65} +2102 q^{-67} +3189 q^{-69} +1648 q^{-71} -711 q^{-73} -1951 q^{-75} -1400 q^{-77} +17 q^{-79} +967 q^{-81} +917 q^{-83} +237 q^{-85} -379 q^{-87} -521 q^{-89} -207 q^{-91} +124 q^{-93} +217 q^{-95} +129 q^{-97} -16 q^{-99} -86 q^{-101} -64 q^{-103} +4 q^{-105} +32 q^{-107} +16 q^{-109} - q^{-111} -6 q^{-113} -6 q^{-115} -2 q^{-117} +7 q^{-119} -3 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}-2 q^{10}+2 q^8+3 q^2-1+3 q^{-2} -2 q^{-4} + q^{-8} -2 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}+60 q^{36}-114 q^{34}+194 q^{32}-296 q^{30}+411 q^{28}-526 q^{26}+600 q^{24}-622 q^{22}+565 q^{20}-416 q^{18}+186 q^{16}+102 q^{14}-411 q^{12}+716 q^{10}-966 q^8+1136 q^6-1197 q^4+1156 q^2-1002+762 q^{-2} -462 q^{-4} +146 q^{-6} +148 q^{-8} -384 q^{-10} +538 q^{-12} -606 q^{-14} +596 q^{-16} -528 q^{-18} +423 q^{-20} -312 q^{-22} +216 q^{-24} -134 q^{-26} +73 q^{-28} -38 q^{-30} +18 q^{-32} -6 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}-q^{36}+4 q^{34}+4 q^{32}-6 q^{30}-6 q^{28}+6 q^{26}+4 q^{24}-10 q^{22}-6 q^{20}+9 q^{18}+6 q^{16}-9 q^{14}+q^{12}+11 q^{10}-3 q^8-2 q^6+6 q^4+q^2-5+4 q^{-2} +6 q^{-4} -10 q^{-6} -5 q^{-8} +11 q^{-10} +4 q^{-12} -13 q^{-14} + q^{-16} +10 q^{-18} - q^{-20} -6 q^{-22} - q^{-24} +5 q^{-26} - q^{-28} -2 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}+4 q^{28}-9 q^{26}+3 q^{24}+9 q^{22}-18 q^{20}+6 q^{18}+13 q^{16}-19 q^{14}+6 q^{12}+13 q^{10}-10 q^8-q^6+8 q^4+3 q^2-3-3 q^{-2} +14 q^{-4} -5 q^{-6} -15 q^{-8} +18 q^{-10} -4 q^{-12} -16 q^{-14} +16 q^{-16} - q^{-18} -9 q^{-20} +8 q^{-22} -3 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}-2 q^{13}+3 q^{11}-q^9+2 q^7+2 q^3+q+2 q^{-3} -2 q^{-5} + q^{-7} -2 q^{-9} +2 q^{-11} -2 q^{-13} + q^{-15} }

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^{44}-q^{40}+q^{38}+2 q^{36}-3 q^{34}-5 q^{32}+3 q^{30}+4 q^{28}-11 q^{26}-5 q^{24}+14 q^{22}+q^{20}-14 q^{18}+6 q^{16}+15 q^{14}-8 q^{12}-9 q^{10}+13 q^8+7 q^6-13 q^4+8 q^2+17-10 q^{-2} -6 q^{-4} +17 q^{-6} -4 q^{-8} -18 q^{-10} +4 q^{-12} +11 q^{-14} -9 q^{-16} -9 q^{-18} +11 q^{-20} +6 q^{-22} -8 q^{-24} - q^{-26} +6 q^{-28} -2 q^{-30} -2 q^{-32} + q^{-34} }
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}-2 q^{16}+3 q^{14}+q^{10}+2 q^8+2 q^4+2- q^{-2} +2 q^{-4} -2 q^{-6} + q^{-8} - q^{-10} - q^{-12} +2 q^{-14} -2 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}+8 q^{28}-13 q^{26}+17 q^{24}-19 q^{22}+20 q^{20}-18 q^{18}+13 q^{16}-5 q^{14}-4 q^{12}+15 q^{10}-24 q^8+33 q^6-36 q^4+39 q^2-35+29 q^{-2} -18 q^{-4} +9 q^{-6} + q^{-8} -10 q^{-10} +16 q^{-12} -20 q^{-14} +20 q^{-16} -19 q^{-18} +15 q^{-20} -10 q^{-22} +6 q^{-24} -3 q^{-26} + q^{-28} }
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^{56}-2 q^{52}-2 q^{50}+3 q^{48}+6 q^{46}-q^{44}-11 q^{42}-6 q^{40}+12 q^{38}+14 q^{36}-7 q^{34}-21 q^{32}-4 q^{30}+21 q^{28}+13 q^{26}-15 q^{24}-18 q^{22}+6 q^{20}+19 q^{18}+q^{16}-15 q^{14}-4 q^{12}+14 q^{10}+7 q^8-10 q^6-9 q^4+10 q^2+14-5 q^{-2} -15 q^{-4} +4 q^{-6} +17 q^{-8} +2 q^{-10} -19 q^{-12} -10 q^{-14} +16 q^{-16} +17 q^{-18} -10 q^{-20} -22 q^{-22} - q^{-24} +18 q^{-26} +10 q^{-28} -10 q^{-30} -12 q^{-32} +2 q^{-34} +9 q^{-36} +3 q^{-38} -3 q^{-40} -3 q^{-42} + q^{-46} }

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}+7 q^{38}-11 q^{36}+11 q^{34}-14 q^{32}+15 q^{30}-18 q^{28}+13 q^{26}-13 q^{24}+12 q^{22}-5 q^{20}-q^{18}+7 q^{16}-7 q^{14}+19 q^{12}-23 q^{10}+25 q^8-25 q^6+32 q^4-28 q^2+26-23 q^{-2} +22 q^{-4} -11 q^{-6} +5 q^{-8} -5 q^{-10} -3 q^{-12} +10 q^{-14} -13 q^{-16} +12 q^{-18} -17 q^{-20} +18 q^{-22} -13 q^{-24} +12 q^{-26} -12 q^{-28} +9 q^{-30} -4 q^{-32} +3 q^{-34} -3 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}+8 q^{72}-7 q^{70}-2 q^{68}+17 q^{66}-35 q^{64}+50 q^{62}-52 q^{60}+28 q^{58}+13 q^{56}-67 q^{54}+113 q^{52}-123 q^{50}+92 q^{48}-23 q^{46}-64 q^{44}+129 q^{42}-148 q^{40}+111 q^{38}-31 q^{36}-54 q^{34}+104 q^{32}-98 q^{30}+43 q^{28}+39 q^{26}-101 q^{24}+121 q^{22}-81 q^{20}-5 q^{18}+102 q^{16}-171 q^{14}+188 q^{12}-138 q^{10}+43 q^8+71 q^6-159 q^4+198 q^2-170+88 q^{-2} +17 q^{-4} -101 q^{-6} +132 q^{-8} -101 q^{-10} +29 q^{-12} +54 q^{-14} -99 q^{-16} +89 q^{-18} -33 q^{-20} -51 q^{-22} +119 q^{-24} -142 q^{-26} +110 q^{-28} -38 q^{-30} -44 q^{-32} +103 q^{-34} -124 q^{-36} +105 q^{-38} -58 q^{-40} +6 q^{-42} +31 q^{-44} -54 q^{-46} +53 q^{-48} -36 q^{-50} +20 q^{-52} -2 q^{-54} -6 q^{-56} +9 q^{-58} -10 q^{-60} +6 q^{-62} -3 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 33"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+6t^{2}-14t+19-14t^{-1}+6t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{6}+z^{2}+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 61, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{4}-4q^{3}+7q^{2}-9q+11-10q^{-1}+9q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$

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} -3 z^4-a^4 z^2+4 a^2 z^2+z^2 a^{-2} -3 z^2-a^4+2 a^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 a^2 z^8+2 z^8+4 a^3 z^7+10 a z^7+6 z^7 a^{-1} +3 a^4 z^6+5 a^2 z^6+7 z^6 a^{-2} +9 z^6+a^5 z^5-6 a^3 z^5-16 a z^5-5 z^5 a^{-1} +4 z^5 a^{-3} -6 a^4 z^4-16 a^2 z^4-9 z^4 a^{-2} +z^4 a^{-4} -20 z^4-2 a^5 z^3+a^3 z^3+5 a z^3-z^3 a^{-1} -3 z^3 a^{-3} +4 a^4 z^2+10 a^2 z^2+3 z^2 a^{-2} +9 z^2+a^5 z+a^3 z-a^4-2 a^2}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n55,}

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

In[3]:=

K = Knot["9 33"];

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+6 t^2-14 t+19-14 t^{-1} +6 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^4-4 q^3+7 q^2-9 q+11-10 q^{-1} +9 q^{-2} -6 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]=

{K11n55,}

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:

(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{62}{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{10}{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{176}{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 -40} 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{248}{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{40}{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{5071}{30}} 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{422}{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{3062}{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{593}{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{271}{30}}

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

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   1 1 7                 3   -3 5               4 1   3 3             5 3     -2 1           6 4       2 -1         5 6         1 -3       4 5           -1 -5     2 5             3 -7   1 4               -3 -9   2                 2 -11 1                   -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}^{4}\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^{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=-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}} ${\mathbb {Z} }^{5}$ 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}^{6}\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}^{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=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}^{3}\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^{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=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}-4 q^{11}+3 q^{10}+11 q^9-25 q^8+6 q^7+43 q^6-59 q^5-3 q^4+86 q^3-83 q^2-22 q+115-83 q^{-1} -39 q^{-2} +113 q^{-3} -60 q^{-4} -46 q^{-5} +83 q^{-6} -27 q^{-7} -37 q^{-8} +40 q^{-9} -4 q^{-10} -17 q^{-11} +10 q^{-12} + q^{-13} -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}-4 q^{23}+3 q^{22}+7 q^{21}-5 q^{20}-20 q^{19}+7 q^{18}+53 q^{17}-14 q^{16}-96 q^{15}-q^{14}+166 q^{13}+34 q^{12}-247 q^{11}-94 q^{10}+326 q^9+179 q^8-392 q^7-276 q^6+430 q^5+382 q^4-452 q^3-460 q^2+431 q+536-407 q^{-1} -567 q^{-2} +342 q^{-3} +595 q^{-4} -282 q^{-5} -577 q^{-6} +193 q^{-7} +550 q^{-8} -111 q^{-9} -486 q^{-10} +23 q^{-11} +411 q^{-12} +38 q^{-13} -312 q^{-14} -82 q^{-15} +214 q^{-16} +97 q^{-17} -131 q^{-18} -83 q^{-19} +64 q^{-20} +61 q^{-21} -25 q^{-22} -36 q^{-23} +7 q^{-24} +17 q^{-25} -2 q^{-26} -5 q^{-27} - q^{-28} +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}-4 q^{39}+3 q^{38}+7 q^{37}-9 q^{36}-19 q^{34}+27 q^{33}+46 q^{32}-47 q^{31}-34 q^{30}-99 q^{29}+113 q^{28}+237 q^{27}-73 q^{26}-189 q^{25}-438 q^{24}+202 q^{23}+752 q^{22}+180 q^{21}-384 q^{20}-1285 q^{19}-56 q^{18}+1494 q^{17}+1012 q^{16}-227 q^{15}-2483 q^{14}-948 q^{13}+1963 q^{12}+2229 q^{11}+557 q^{10}-3454 q^9-2208 q^8+1822 q^7+3236 q^6+1682 q^5-3797 q^4-3254 q^3+1225 q^2+3666 q+2665-3560 q^{-1} -3782 q^{-2} +478 q^{-3} +3546 q^{-4} +3288 q^{-5} -2904 q^{-6} -3814 q^{-7} -316 q^{-8} +2982 q^{-9} +3566 q^{-10} -1903 q^{-11} -3396 q^{-12} -1092 q^{-13} +2012 q^{-14} +3422 q^{-15} -701 q^{-16} -2498 q^{-17} -1596 q^{-18} +806 q^{-19} +2717 q^{-20} +288 q^{-21} -1290 q^{-22} -1506 q^{-23} -172 q^{-24} +1590 q^{-25} +641 q^{-26} -268 q^{-27} -904 q^{-28} -513 q^{-29} +584 q^{-30} +423 q^{-31} +161 q^{-32} -298 q^{-33} -342 q^{-34} +97 q^{-35} +119 q^{-36} +137 q^{-37} -33 q^{-38} -111 q^{-39} +2 q^{-40} +4 q^{-41} +38 q^{-42} +5 q^{-43} -20 q^{-44} +2 q^{-45} -3 q^{-46} +5 q^{-47} + q^{-48} -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}-4 q^{59}+3 q^{58}+7 q^{57}-9 q^{56}-4 q^{55}+q^{54}+q^{53}+20 q^{52}+23 q^{51}-37 q^{50}-72 q^{49}-21 q^{48}+71 q^{47}+165 q^{46}+111 q^{45}-130 q^{44}-403 q^{43}-335 q^{42}+213 q^{41}+781 q^{40}+791 q^{39}-80 q^{38}-1353 q^{37}-1752 q^{36}-338 q^{35}+2021 q^{34}+3150 q^{33}+1461 q^{32}-2440 q^{31}-5175 q^{30}-3440 q^{29}+2350 q^{28}+7426 q^{27}+6404 q^{26}-1275 q^{25}-9570 q^{24}-10233 q^{23}-936 q^{22}+11121 q^{21}+14476 q^{20}+4271 q^{19}-11655 q^{18}-18631 q^{17}-8472 q^{16}+11117 q^{15}+22129 q^{14}+12986 q^{13}-9472 q^{12}-24715 q^{11}-17328 q^{10}+7175 q^9+26127 q^8+21050 q^7-4385 q^6-26648 q^5-23971 q^4+1744 q^3+26187 q^2+25992 q+943-25297 q^{-1} -27295 q^{-2} -3159 q^{-3} +23779 q^{-4} +27888 q^{-5} +5437 q^{-6} -22014 q^{-7} -28057 q^{-8} -7391 q^{-9} +19688 q^{-10} +27652 q^{-11} +9544 q^{-12} -16995 q^{-13} -26787 q^{-14} -11484 q^{-15} +13655 q^{-16} +25228 q^{-17} +13403 q^{-18} -9926 q^{-19} -22943 q^{-20} -14763 q^{-21} +5780 q^{-22} +19810 q^{-23} +15547 q^{-24} -1769 q^{-25} -15968 q^{-26} -15251 q^{-27} -1851 q^{-28} +11622 q^{-29} +13979 q^{-30} +4553 q^{-31} -7333 q^{-32} -11671 q^{-33} -6070 q^{-34} +3483 q^{-35} +8777 q^{-36} +6375 q^{-37} -573 q^{-38} -5803 q^{-39} -5601 q^{-40} -1207 q^{-41} +3155 q^{-42} +4243 q^{-43} +1950 q^{-44} -1262 q^{-45} -2742 q^{-46} -1861 q^{-47} +121 q^{-48} +1473 q^{-49} +1388 q^{-50} +352 q^{-51} -621 q^{-52} -844 q^{-53} -406 q^{-54} +176 q^{-55} +411 q^{-56} +280 q^{-57} +19 q^{-58} -170 q^{-59} -161 q^{-60} -31 q^{-61} +56 q^{-62} +52 q^{-63} +33 q^{-64} -7 q^{-65} -33 q^{-66} -7 q^{-67} +8 q^{-68} + q^{-69} +3 q^{-70} +3 q^{-71} -5 q^{-72} - q^{-73} +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}-4 q^{83}+3 q^{82}+7 q^{81}-9 q^{80}-4 q^{79}-3 q^{78}+21 q^{77}-6 q^{76}-3 q^{75}+33 q^{74}-65 q^{73}-46 q^{72}-3 q^{71}+130 q^{70}+86 q^{69}+26 q^{68}+45 q^{67}-372 q^{66}-385 q^{65}-122 q^{64}+585 q^{63}+755 q^{62}+634 q^{61}+286 q^{60}-1502 q^{59}-2193 q^{58}-1601 q^{57}+1143 q^{56}+3197 q^{55}+4151 q^{54}+3071 q^{53}-2969 q^{52}-7713 q^{51}-8728 q^{50}-2120 q^{49}+6490 q^{48}+14254 q^{47}+15272 q^{46}+1604 q^{45}-15319 q^{44}-26770 q^{43}-19157 q^{42}+1159 q^{41}+28074 q^{40}+43332 q^{39}+25450 q^{38}-12018 q^{37}-50779 q^{36}-56897 q^{35}-28710 q^{34}+29035 q^{33}+79230 q^{32}+74624 q^{31}+19753 q^{30}-60527 q^{29}-103623 q^{28}-86828 q^{27}-817 q^{26}+99561 q^{25}+133075 q^{24}+80365 q^{23}-38506 q^{22}-133685 q^{21}-152968 q^{20}-58309 q^{19}+88047 q^{18}+173726 q^{17}+146686 q^{16}+9390 q^{15}-132607 q^{14}-200071 q^{13}-119348 q^{12}+52288 q^{11}+184020 q^{10}+193635 q^9+60262 q^8-108722 q^7-218005 q^6-162501 q^5+12510 q^4+172178 q^3+214259 q^2+97330 q-78957-214818 q^{-1} -184164 q^{-2} -19189 q^{-3} +151725 q^{-4} +216694 q^{-5} +120052 q^{-6} -51346 q^{-7} -201224 q^{-8} -192445 q^{-9} -44618 q^{-10} +127353 q^{-11} +209410 q^{-12} +136239 q^{-13} -22434 q^{-14} -179367 q^{-15} -193704 q^{-16} -70883 q^{-17} +94486 q^{-18} +192292 q^{-19} +149869 q^{-20} +13884 q^{-21} -143716 q^{-22} -185196 q^{-23} -99105 q^{-24} +48156 q^{-25} +158360 q^{-26} +154703 q^{-27} +55581 q^{-28} -90163 q^{-29} -157590 q^{-30} -118833 q^{-31} -6329 q^{-32} +103658 q^{-33} +138130 q^{-34} +87871 q^{-35} -26967 q^{-36} -106483 q^{-37} -114093 q^{-38} -50107 q^{-39} +38832 q^{-40} +95210 q^{-41} +92289 q^{-42} +23687 q^{-43} -44665 q^{-44} -80241 q^{-45} -63304 q^{-46} -11723 q^{-47} +40900 q^{-48} +65843 q^{-49} +41954 q^{-50} +2172 q^{-51} -34698 q^{-52} -45436 q^{-53} -29796 q^{-54} +1254 q^{-55} +28461 q^{-56} +30486 q^{-57} +18440 q^{-58} -2915 q^{-59} -17972 q^{-60} -21421 q^{-61} -11473 q^{-62} +3842 q^{-63} +11205 q^{-64} +12832 q^{-65} +6416 q^{-66} -1467 q^{-67} -7641 q^{-68} -7524 q^{-69} -2882 q^{-70} +764 q^{-71} +4009 q^{-72} +3883 q^{-73} +2079 q^{-74} -915 q^{-75} -2121 q^{-76} -1606 q^{-77} -1015 q^{-78} +333 q^{-79} +916 q^{-80} +1000 q^{-81} +220 q^{-82} -212 q^{-83} -266 q^{-84} -391 q^{-85} -127 q^{-86} +60 q^{-87} +215 q^{-88} +65 q^{-89} +7 q^{-90} +12 q^{-91} -63 q^{-92} -34 q^{-93} -12 q^{-94} +36 q^{-95} +2 q^{-96} -6 q^{-97} +11 q^{-98} -6 q^{-99} -3 q^{-100} -3 q^{-101} +5 q^{-102} + q^{-103} -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}-4 q^{111}+3 q^{110}+7 q^{109}-9 q^{108}-4 q^{107}-3 q^{106}+17 q^{105}+14 q^{104}-29 q^{103}+7 q^{102}+5 q^{101}-39 q^{100}-18 q^{99}+4 q^{98}+111 q^{97}+132 q^{96}-62 q^{95}-86 q^{94}-180 q^{93}-291 q^{92}-86 q^{91}+121 q^{90}+684 q^{89}+935 q^{88}+297 q^{87}-399 q^{86}-1462 q^{85}-2136 q^{84}-1371 q^{83}+160 q^{82}+2973 q^{81}+5199 q^{80}+4209 q^{79}+933 q^{78}-5092 q^{77}-10410 q^{76}-10542 q^{75}-5481 q^{74}+6189 q^{73}+18797 q^{72}+23424 q^{71}+16832 q^{70}-3684 q^{69}-29195 q^{68}-44109 q^{67}-40107 q^{66}-9504 q^{65}+36910 q^{64}+73834 q^{63}+81192 q^{62}+41025 q^{61}-34268 q^{60}-107299 q^{59}-141435 q^{58}-101281 q^{57}+7287 q^{56}+134706 q^{55}+219360 q^{54}+196159 q^{53}+56536 q^{52}-139663 q^{51}-302390 q^{50}-324942 q^{49}-169637 q^{48}+103593 q^{47}+373033 q^{46}+477633 q^{45}+333623 q^{44}-11592 q^{43}-408448 q^{42}-633728 q^{41}-539777 q^{40}-143541 q^{39}+388993 q^{38}+768599 q^{37}+768535 q^{36}+354914 q^{35}-304353 q^{34}-858018 q^{33}-992330 q^{32}-604378 q^{31}+155533 q^{30}+886693 q^{29}+1185063 q^{28}+865421 q^{27}+41939 q^{26}-850809 q^{25}-1325913 q^{24}-1110482 q^{23}-265534 q^{22}+759123 q^{21}+1406486 q^{20}+1316820 q^{19}+489369 q^{18}-628897 q^{17}-1428674 q^{16}-1472214 q^{15}-692247 q^{14}+481739 q^{13}+1404365 q^{12}+1573547 q^{11}+859516 q^{10}-335270 q^9-1348375 q^8-1628437 q^7-987518 q^6+204023 q^5+1277395 q^4+1646488 q^3+1077731 q^2-92091 q-1201173-1641850 q^{-1} -1140218 q^{-2} +335 q^{-3} +1128107 q^{-4} +1623094 q^{-5} +1182743 q^{-6} +78632 q^{-7} -1056484 q^{-8} -1598273 q^{-9} -1217127 q^{-10} -151677 q^{-11} +984896 q^{-12} +1568187 q^{-13} +1247728 q^{-14} +228507 q^{-15} -904233 q^{-16} -1531416 q^{-17} -1279794 q^{-18} -315323 q^{-19} +808575 q^{-20} +1481362 q^{-21} +1310013 q^{-22} +416318 q^{-23} -688591 q^{-24} -1410389 q^{-25} -1334094 q^{-26} -530059 q^{-27} +541117 q^{-28} +1308914 q^{-29} +1340844 q^{-30} +650820 q^{-31} -364860 q^{-32} -1170292 q^{-33} -1319360 q^{-34} -765645 q^{-35} +167085 q^{-36} +990462 q^{-37} +1256627 q^{-38} +859423 q^{-39} +40157 q^{-40} -774451 q^{-41} -1145254 q^{-42} -913617 q^{-43} -236456 q^{-44} +533062 q^{-45} +982955 q^{-46} +914623 q^{-47} +400916 q^{-48} -286916 q^{-49} -779577 q^{-50} -854456 q^{-51} -512171 q^{-52} +60416 q^{-53} +551937 q^{-54} +736826 q^{-55} +558450 q^{-56} +122500 q^{-57} -326102 q^{-58} -577412 q^{-59} -537227 q^{-60} -243628 q^{-61} +127750 q^{-62} +399023 q^{-63} +460243 q^{-64} +297765 q^{-65} +22031 q^{-66} -229324 q^{-67} -348951 q^{-68} -290825 q^{-69} -112897 q^{-70} +90343 q^{-71} +228197 q^{-72} +241102 q^{-73} +148133 q^{-74} +4595 q^{-75} -121531 q^{-76} -171176 q^{-77} -140049 q^{-78} -54247 q^{-79} +42625 q^{-80} +101582 q^{-81} +107960 q^{-82} +67702 q^{-83} +4057 q^{-84} -46924 q^{-85} -69150 q^{-86} -58261 q^{-87} -23549 q^{-88} +12089 q^{-89} +35831 q^{-90} +39979 q^{-91} +25351 q^{-92} +4737 q^{-93} -13638 q^{-94} -22439 q^{-95} -18756 q^{-96} -9240 q^{-97} +1991 q^{-98} +9813 q^{-99} +10926 q^{-100} +7989 q^{-101} +2220 q^{-102} -3152 q^{-103} -5102 q^{-104} -4780 q^{-105} -2525 q^{-106} +206 q^{-107} +1678 q^{-108} +2427 q^{-109} +1790 q^{-110} +395 q^{-111} -420 q^{-112} -917 q^{-113} -791 q^{-114} -376 q^{-115} -118 q^{-116} +288 q^{-117} +409 q^{-118} +188 q^{-119} +54 q^{-120} -88 q^{-121} -88 q^{-122} -43 q^{-123} -85 q^{-124} +55 q^{-126} +29 q^{-127} +13 q^{-128} -17 q^{-129} -5 q^{-130} +11 q^{-131} -13 q^{-132} -6 q^{-133} +6 q^{-134} +3 q^{-135} +3 q^{-136} -5 q^{-137} - q^{-138} +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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