9 41

9_40

9_42

Visit 9 41's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 41's page at Knotilus! Visit 9 41's page at the original Knot Atlas! 9 41 Quick Notes

Three-fold symmetric decorative knot

Three-fold symmetric decorative knot in circle

Knot presentations

Planar diagram presentation	X6271 X12,8,13,7 X14,5,15,6 X10,3,11,4 X2,11,3,12 X4,15,5,16 X8,17,9,18 X16,9,17,10 X18,14,1,13
Gauss code	1, -5, 4, -6, 3, -1, 2, -7, 8, -4, 5, -2, 9, -3, 6, -8, 7, -9
Dowker-Thistlethwaite code	6 10 14 12 16 2 18 4 8
Conway Notation	[20:20:20]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index 4 Nakanishi index 2 Maximal Thurston-Bennequin number [-7][-4] Hyperbolic Volume 12.0989 A-Polynomial See Data:9 41/A-polynomial

[edit Notes for 9 41'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 0} 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 0} 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 0} Rasmussen s-Invariant 0

[edit Notes for 9 41's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 t^2-12 t+19-12 t^{-1} +3 t^{-2} }
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 3 z^4+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 \{7,t+1\}}
Determinant and Signature	{ 49, 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^3+3 q^2-5 q+8-8 q^{-1} +8 q^{-2} -7 q^{-3} +5 q^{-4} -3 q^{-5} + q^{-6} }
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 a^6-3 z^2 a^4-3 a^4+2 z^4 a^2+4 z^2 a^2+3 a^2+z^4-z^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^4 z^8+2 a^2 z^8+3 a^5 z^7+9 a^3 z^7+6 a z^7+a^6 z^6-a^4 z^6+5 a^2 z^6+7 z^6-10 a^5 z^5-26 a^3 z^5-11 a z^5+5 z^5 a^{-1} -3 a^6 z^4-12 a^4 z^4-23 a^2 z^4+3 z^4 a^{-2} -11 z^4+9 a^5 z^3+19 a^3 z^3+6 a z^3-3 z^3 a^{-1} +z^3 a^{-3} +3 a^6 z^2+13 a^4 z^2+17 a^2 z^2-z^2 a^{-2} +6 z^2-2 a^5 z-4 a^3 z-2 a z-a^6-3 a^4-3 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^{20}+q^{18}-2 q^{16}-q^{12}-2 q^{10}+2 q^8+2 q^4+q^2+2 q^{-2} -2 q^{-4} + q^{-6} + q^{-8} - q^{-10} }
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^{94}-2 q^{92}+6 q^{90}-10 q^{88}+11 q^{86}-7 q^{84}-7 q^{82}+27 q^{80}-39 q^{78}+44 q^{76}-28 q^{74}-3 q^{72}+40 q^{70}-66 q^{68}+70 q^{66}-45 q^{64}+q^{62}+37 q^{60}-64 q^{58}+54 q^{56}-25 q^{54}-16 q^{52}+42 q^{50}-49 q^{48}+27 q^{46}+7 q^{44}-43 q^{42}+63 q^{40}-60 q^{38}+37 q^{36}+6 q^{34}-46 q^{32}+79 q^{30}-82 q^{28}+64 q^{26}-19 q^{24}-28 q^{22}+65 q^{20}-77 q^{18}+58 q^{16}-17 q^{14}-25 q^{12}+51 q^{10}-48 q^8+18 q^6+19 q^4-46 q^2+51-30 q^{-2} -3 q^{-4} +35 q^{-6} -51 q^{-8} +53 q^{-10} -32 q^{-12} +8 q^{-14} +16 q^{-16} -32 q^{-18} +33 q^{-20} -27 q^{-22} +18 q^{-24} -7 q^{-26} -2 q^{-28} +9 q^{-30} -14 q^{-32} +13 q^{-34} -10 q^{-36} +7 q^{-38} -2 q^{-40} - q^{-42} +2 q^{-44} -4 q^{-46} +3 q^{-48} -2 q^{-50} + q^{-52} }

Further Quantum Invariants

Further quantum knot invariants for 9_41.

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^{13}-2 q^{11}+2 q^9-2 q^7+q^5+3 q^{-1} -2 q^{-3} +2 q^{-5} - q^{-7} }
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^{38}-2 q^{36}-3 q^{34}+7 q^{32}+q^{30}-11 q^{28}+7 q^{26}+9 q^{24}-13 q^{22}+12 q^{18}-8 q^{16}-6 q^{14}+9 q^{12}-8 q^8+2 q^6+9 q^4-5 q^2-8+14 q^{-2} + q^{-4} -13 q^{-6} +9 q^{-8} +4 q^{-10} -7 q^{-12} +3 q^{-14} -2 q^{-18} + q^{-20} }
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^{75}-2 q^{73}-3 q^{71}+2 q^{69}+10 q^{67}+4 q^{65}-18 q^{63}-16 q^{61}+16 q^{59}+34 q^{57}-4 q^{55}-47 q^{53}-17 q^{51}+46 q^{49}+41 q^{47}-34 q^{45}-60 q^{43}+15 q^{41}+66 q^{39}+9 q^{37}-62 q^{35}-28 q^{33}+55 q^{31}+39 q^{29}-43 q^{27}-46 q^{25}+32 q^{23}+50 q^{21}-19 q^{19}-53 q^{17}+7 q^{15}+49 q^{13}+11 q^{11}-47 q^9-33 q^7+32 q^5+57 q^3-11 q-66 q^{-1} -11 q^{-3} +66 q^{-5} +35 q^{-7} -56 q^{-9} -42 q^{-11} +34 q^{-13} +40 q^{-15} -15 q^{-17} -26 q^{-19} +5 q^{-21} +14 q^{-23} -4 q^{-25} -4 q^{-27} + q^{-31} - q^{-33} +2 q^{-37} - q^{-39} }
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^{124}-2 q^{122}-3 q^{120}+2 q^{118}+5 q^{116}+13 q^{114}-3 q^{112}-23 q^{110}-24 q^{108}-8 q^{106}+55 q^{104}+57 q^{102}+3 q^{100}-72 q^{98}-121 q^{96}-2 q^{94}+116 q^{92}+159 q^{90}+43 q^{88}-191 q^{86}-202 q^{84}-47 q^{82}+217 q^{80}+294 q^{78}+9 q^{76}-254 q^{74}-328 q^{72}-12 q^{70}+352 q^{68}+302 q^{66}-22 q^{64}-398 q^{62}-304 q^{60}+142 q^{58}+393 q^{56}+244 q^{54}-237 q^{52}-398 q^{50}-86 q^{48}+293 q^{46}+335 q^{44}-67 q^{42}-340 q^{40}-178 q^{38}+193 q^{36}+310 q^{34}+9 q^{32}-275 q^{30}-205 q^{28}+130 q^{26}+287 q^{24}+96 q^{22}-210 q^{20}-278 q^{18}-8 q^{16}+244 q^{14}+274 q^{12}-17 q^{10}-326 q^8-273 q^6+42 q^4+409 q^2+310-163 q^{-2} -441 q^{-4} -287 q^{-6} +269 q^{-8} +479 q^{-10} +133 q^{-12} -301 q^{-14} -416 q^{-16} -3 q^{-18} +316 q^{-20} +233 q^{-22} -46 q^{-24} -251 q^{-26} -98 q^{-28} +88 q^{-30} +116 q^{-32} +38 q^{-34} -76 q^{-36} -41 q^{-38} +9 q^{-40} +20 q^{-42} +20 q^{-44} -17 q^{-46} -6 q^{-48} +2 q^{-50} +6 q^{-54} -3 q^{-56} + q^{-58} -2 q^{-62} + q^{-64} }
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^{185}-2 q^{183}-3 q^{181}+2 q^{179}+5 q^{177}+8 q^{175}+6 q^{173}-8 q^{171}-31 q^{169}-30 q^{167}-2 q^{165}+44 q^{163}+84 q^{161}+72 q^{159}-17 q^{157}-143 q^{155}-190 q^{153}-106 q^{151}+98 q^{149}+309 q^{147}+346 q^{145}+105 q^{143}-294 q^{141}-567 q^{139}-486 q^{137}+2 q^{135}+617 q^{133}+899 q^{131}+537 q^{129}-314 q^{127}-1081 q^{125}-1177 q^{123}-371 q^{121}+857 q^{119}+1618 q^{117}+1232 q^{115}-157 q^{113}-1597 q^{111}-1983 q^{109}-858 q^{107}+1052 q^{105}+2327 q^{103}+1871 q^{101}-92 q^{99}-2114 q^{97}-2599 q^{95}-1026 q^{93}+1445 q^{91}+2851 q^{89}+1988 q^{87}-505 q^{85}-2620 q^{83}-2615 q^{81}-445 q^{79}+2075 q^{77}+2828 q^{75}+1183 q^{73}-1396 q^{71}-2689 q^{69}-1637 q^{67}+776 q^{65}+2358 q^{63}+1770 q^{61}-329 q^{59}-1961 q^{57}-1691 q^{55}+88 q^{53}+1632 q^{51}+1513 q^{49}-23 q^{47}-1421 q^{45}-1360 q^{43}+34 q^{41}+1345 q^{39}+1316 q^{37}+17 q^{35}-1317 q^{33}-1448 q^{31}-230 q^{29}+1238 q^{27}+1676 q^{25}+701 q^{23}-931 q^{21}-1927 q^{19}-1397 q^{17}+334 q^{15}+1957 q^{13}+2167 q^{11}+614 q^9-1637 q^7-2789 q^5-1726 q^3+888 q+2982 q^{-1} +2752 q^{-3} +198 q^{-5} -2647 q^{-7} -3387 q^{-9} -1306 q^{-11} +1844 q^{-13} +3402 q^{-15} +2149 q^{-17} -804 q^{-19} -2892 q^{-21} -2461 q^{-23} -109 q^{-25} +2034 q^{-27} +2259 q^{-29} +678 q^{-31} -1154 q^{-33} -1723 q^{-35} -843 q^{-37} +492 q^{-39} +1118 q^{-41} +702 q^{-43} -119 q^{-45} -603 q^{-47} -478 q^{-49} -30 q^{-51} +297 q^{-53} +265 q^{-55} +36 q^{-57} -118 q^{-59} -124 q^{-61} -33 q^{-63} +51 q^{-65} +55 q^{-67} +9 q^{-69} -23 q^{-71} -17 q^{-73} + q^{-75} +6 q^{-77} +8 q^{-79} +2 q^{-81} -6 q^{-83} -4 q^{-85} +3 q^{-87} - q^{-89} +2 q^{-93} - q^{-95} }

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^{20}+q^{18}-2 q^{16}-q^{12}-2 q^{10}+2 q^8+2 q^4+q^2+2 q^{-2} -2 q^{-4} + q^{-6} + q^{-8} - q^{-10} }
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^{52}-4 q^{50}+14 q^{48}-36 q^{46}+66 q^{44}-110 q^{42}+156 q^{40}-196 q^{38}+224 q^{36}-216 q^{34}+182 q^{32}-112 q^{30}+22 q^{28}+80 q^{26}-188 q^{24}+272 q^{22}-339 q^{20}+376 q^{18}-378 q^{16}+346 q^{14}-280 q^{12}+190 q^{10}-96 q^8-4 q^6+84 q^4-134 q^2+168-158 q^{-2} +149 q^{-4} -122 q^{-6} +98 q^{-8} -78 q^{-10} +56 q^{-12} -46 q^{-14} +38 q^{-16} -28 q^{-18} +18 q^{-20} -12 q^{-22} +8 q^{-24} -4 q^{-26} + q^{-28} }
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^{52}+q^{50}-q^{48}-5 q^{46}-3 q^{44}+4 q^{42}+5 q^{40}-2 q^{38}-3 q^{36}+6 q^{34}+8 q^{32}-3 q^{30}-8 q^{28}+2 q^{26}+2 q^{24}-4 q^{22}-7 q^{20}+3 q^{18}+4 q^{16}-3 q^{14}+q^{12}-q^8+q^6+6 q^4-3 q^2-3+7 q^{-2} +8 q^{-4} -5 q^{-6} -8 q^{-8} +7 q^{-10} +5 q^{-12} -5 q^{-14} -3 q^{-16} +2 q^{-18} +2 q^{-20} -2 q^{-22} - q^{-24} + q^{-26} }

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^{40}-2 q^{38}+2 q^{36}+2 q^{34}-6 q^{32}+6 q^{30}-8 q^{26}+8 q^{24}-q^{22}-8 q^{20}+7 q^{18}+2 q^{16}-7 q^{14}+2 q^{12}+2 q^{10}-q^8-3 q^6+2 q^4+9 q^2-5+ q^{-2} +11 q^{-4} -9 q^{-6} -2 q^{-8} +8 q^{-10} -5 q^{-12} -3 q^{-14} +5 q^{-16} - q^{-18} -2 q^{-20} + q^{-22} }
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^{27}+q^{25}+q^{23}-2 q^{21}-3 q^{17}-q^{15}-2 q^{13}+2 q^{11}+q^9+2 q^7+2 q^5+q^3+q- q^{-1} +2 q^{-3} -2 q^{-5} + q^{-7} + q^{-11} - q^{-13} }

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^{40}-2 q^{38}+6 q^{36}-8 q^{34}+10 q^{32}-12 q^{30}+12 q^{28}-12 q^{26}+8 q^{24}-5 q^{22}-2 q^{20}+7 q^{18}-14 q^{16}+19 q^{14}-22 q^{12}+24 q^{10}-21 q^8+19 q^6-12 q^4+9 q^2-1-3 q^{-2} +9 q^{-4} -11 q^{-6} +12 q^{-8} -12 q^{-10} +9 q^{-12} -7 q^{-14} +5 q^{-16} -3 q^{-18} +2 q^{-20} - q^{-22} }
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^{62}-2 q^{60}+4 q^{58}+5 q^{56}-4 q^{54}-8 q^{52}+11 q^{48}+5 q^{46}-11 q^{44}-10 q^{42}+7 q^{40}+13 q^{38}-13 q^{34}-5 q^{32}+9 q^{30}+8 q^{28}-5 q^{26}-8 q^{24}+2 q^{22}+7 q^{20}-q^{18}-9 q^{16}-q^{14}+9 q^{12}+3 q^{10}-9 q^8-5 q^6+8 q^4+11 q^2-4-10 q^{-2} +2 q^{-4} +14 q^{-6} +3 q^{-8} -10 q^{-10} -8 q^{-12} +5 q^{-14} +10 q^{-16} -7 q^{-20} -5 q^{-22} +2 q^{-24} +5 q^{-26} + q^{-28} -2 q^{-30} -2 q^{-32} + q^{-36} }

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^{94}-2 q^{92}+6 q^{90}-10 q^{88}+11 q^{86}-7 q^{84}-7 q^{82}+27 q^{80}-39 q^{78}+44 q^{76}-28 q^{74}-3 q^{72}+40 q^{70}-66 q^{68}+70 q^{66}-45 q^{64}+q^{62}+37 q^{60}-64 q^{58}+54 q^{56}-25 q^{54}-16 q^{52}+42 q^{50}-49 q^{48}+27 q^{46}+7 q^{44}-43 q^{42}+63 q^{40}-60 q^{38}+37 q^{36}+6 q^{34}-46 q^{32}+79 q^{30}-82 q^{28}+64 q^{26}-19 q^{24}-28 q^{22}+65 q^{20}-77 q^{18}+58 q^{16}-17 q^{14}-25 q^{12}+51 q^{10}-48 q^8+18 q^6+19 q^4-46 q^2+51-30 q^{-2} -3 q^{-4} +35 q^{-6} -51 q^{-8} +53 q^{-10} -32 q^{-12} +8 q^{-14} +16 q^{-16} -32 q^{-18} +33 q^{-20} -27 q^{-22} +18 q^{-24} -7 q^{-26} -2 q^{-28} +9 q^{-30} -14 q^{-32} +13 q^{-34} -10 q^{-36} +7 q^{-38} -2 q^{-40} - q^{-42} +2 q^{-44} -4 q^{-46} +3 q^{-48} -2 q^{-50} + q^{-52} }

.

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 41"];

In[4]:=

Alexander[K][t]

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

Out[4]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 t^2-12 t+19-12 t^{-1} +3 t^{-2} }

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 3 z^4+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 \{7,t+1\}}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 49, 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^3+3 q^2-5 q+8-8 q^{-1} +8 q^{-2} -7 q^{-3} +5 q^{-4} -3 q^{-5} + q^{-6} }

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 a^6-3 z^2 a^4-3 a^4+2 z^4 a^2+4 z^2 a^2+3 a^2+z^4-z^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^4 z^8+2 a^2 z^8+3 a^5 z^7+9 a^3 z^7+6 a z^7+a^6 z^6-a^4 z^6+5 a^2 z^6+7 z^6-10 a^5 z^5-26 a^3 z^5-11 a z^5+5 z^5 a^{-1} -3 a^6 z^4-12 a^4 z^4-23 a^2 z^4+3 z^4 a^{-2} -11 z^4+9 a^5 z^3+19 a^3 z^3+6 a z^3-3 z^3 a^{-1} +z^3 a^{-3} +3 a^6 z^2+13 a^4 z^2+17 a^2 z^2-z^2 a^{-2} +6 z^2-2 a^5 z-4 a^3 z-2 a z-a^6-3 a^4-3 a^2}

Vassiliev invariants

V2 and V3:

(0, 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 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 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 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 -48} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -24} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 \frac{368}{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 104} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 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 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 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 -136} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 296} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -328} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -88} $-72$

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 $s=$0 is the signature of 9 41. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 7                   1 -1 5                 2   2 3               3 1   -2 1             5 2     3 -1           4 4       0 -3         4 4         0 -5       3 4           1 -7     2 4             -2 -9   1 3               2 -11   2                 -2 -13 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=-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}} Failed to parse (SVG (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}^{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=-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}^{3}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-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^{3}} ${\mathbb {Z} }^{3}$ 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=-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}^{4}\oplus{\mathbb Z}_2^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=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}^{4}\oplus{\mathbb Z}_2^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\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=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}\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=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} Failed to parse (SVG (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}}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Failed to parse (SVG (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{Include}(\textrm{ColouredJonesM.mhtml})}

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[9, 41]]
Out[2]=	9
In[3]:=	PD[Knot[9, 41]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 8, 13, 7], X[14, 5, 15, 6], X[10, 3, 11, 4], X[2, 11, 3, 12], X[4, 15, 5, 16], X[8, 17, 9, 18], X[16, 9, 17, 10], X[18, 14, 1, 13]]
In[4]:=	GaussCode[Knot[9, 41]]
Out[4]=	GaussCode[1, -5, 4, -6, 3, -1, 2, -7, 8, -4, 5, -2, 9, -3, 6, -8, 7, -9]
In[5]:=	BR[Knot[9, 41]]
Out[5]=	BR[5, {-1, -1, -2, 1, 3, 2, 2, -4, -3, 2, -3, -4}]
In[6]:=	alex = Alexander[Knot[9, 41]][t]
Out[6]=	3 12 2 19 + -- - -- - 12 t + 3 t 2 t t
In[7]:=	Conway[Knot[9, 41]][z]
Out[7]=	4 1 + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 41], Knot[11, NonAlternating, 83]}
In[9]:=	{KnotDet[Knot[9, 41]], KnotSignature[Knot[9, 41]]}
Out[9]=	{49, 0}
In[10]:=	J=Jones[Knot[9, 41]][q]
Out[10]=	-6 3 5 7 8 8 2 3 8 + q - -- + -- - -- + -- - - - 5 q + 3 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 41], Knot[11, NonAlternating, 4], Knot[11, NonAlternating, 21]}
In[12]:=	A2Invariant[Knot[9, 41]][q]
Out[12]=	-20 -18 2 -12 2 2 2 -2 2 4 6 q + q - --- - q - --- + -- + -- + q + 2 q - 2 q + q + 16 10 8 4 q q q q 8 10 q - q
In[13]:=	Kauffman[Knot[9, 41]][a, z]
Out[13]=	2 2 4 6 3 5 2 z 2 2 -3 a - 3 a - a - 2 a z - 4 a z - 2 a z + 6 z - -- + 17 a z + 2 a 3 3 4 2 6 2 z 3 z 3 3 3 5 3 13 a z + 3 a z + -- - ---- + 6 a z + 19 a z + 9 a z - 3 a a 4 5 4 3 z 2 4 4 4 6 4 5 z 5 11 z + ---- - 23 a z - 12 a z - 3 a z + ---- - 11 a z - 2 a a 3 5 5 5 6 2 6 4 6 6 6 7 26 a z - 10 a z + 7 z + 5 a z - a z + a z + 6 a z + 3 7 5 7 2 8 4 8 9 a z + 3 a z + 2 a z + 2 a z
In[14]:=	{Vassiliev[2][Knot[9, 41]], Vassiliev[3][Knot[9, 41]]}
Out[14]=	{0, 1}
In[15]:=	Kh[Knot[9, 41]][q, t]
Out[15]=	4 1 2 1 3 2 4 3 - + 5 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 q t q t q t q t q t q t q t 4 4 4 4 3 3 2 5 2 7 3 ----- + ----- + ---- + --- + 2 q t + 3 q t + q t + 2 q t + q t 5 2 3 2 3 q t q t q t q t