9 32

9_31

9_33

Visit 9 32's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 32's page at Knotilus!

Visit 9 32's page at the original Knot Atlas!

9 32 Quick Notes

9 32 Further Notes and Views

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_13,18,14,1 X₃₉₄₈ X_9,3,10,2 X_7,15,8,14 X_15,11,16,10 X_5,12,6,13 X_11,17,12,16 X_17,7,18,6
Gauss code	-1, 4, -3, 1, -7, 9, -5, 3, -4, 6, -8, 7, -2, 5, -6, 8, -9, 2
Dowker-Thistlethwaite code	4 8 12 14 2 16 18 10 6
Conway Notation	[.21.20]

Three dimensional invariants

Symmetry type	Chiral
Unknotting number	2
3-genus	3
Bridge index	3
Super bridge index	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{4,6\}}
Nakanishi index	1
Maximal Thurston-Bennequin number	[-2][-9]
Hyperbolic Volume	13.0999
A-Polynomial	See Data:9 32/A-polynomial

[edit Notes for 9 32's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	$1$
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	2

[edit Notes for 9 32's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^3-6 t^2+14 t-17+14 t^{-1} -6 t^{-2} + t^{-3} }
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6-z^2+1}
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 59, 2 }
Jones polynomial	$q^{7}-3q^{6}+6q^{5}-9q^{4}+10q^{3}-10q^{2}+9q-6+4q^{-1}-q^{-2}$
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} +3 z^4 a^{-2} -2 z^4 a^{-4} -z^4+3 z^2 a^{-2} -4 z^2 a^{-4} +z^2 a^{-6} -z^2+ a^{-2} -2 a^{-4} + a^{-6} +1}
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^8 a^{-2} +2 z^8 a^{-4} +5 z^7 a^{-1} +10 z^7 a^{-3} +5 z^7 a^{-5} +6 z^6 a^{-2} +7 z^6 a^{-4} +5 z^6 a^{-6} +4 z^6+a z^5-9 z^5 a^{-1} -18 z^5 a^{-3} -5 z^5 a^{-5} +3 z^5 a^{-7} -19 z^4 a^{-2} -18 z^4 a^{-4} -6 z^4 a^{-6} +z^4 a^{-8} -8 z^4-a z^3+3 z^3 a^{-1} +9 z^3 a^{-3} +2 z^3 a^{-5} -3 z^3 a^{-7} +10 z^2 a^{-2} +12 z^2 a^{-4} +4 z^2 a^{-6} -z^2 a^{-8} +3 z^2-z a^{-1} -2 z a^{-3} +z a^{-7} - a^{-2} -2 a^{-4} - a^{-6} +1}
The A2 invariant	$-q^{6}+2q^{4}+1+3q^{-2}-2q^{-4}+2q^{-6}-2q^{-8}-2q^{-14}+2q^{-16}-q^{-18}+q^{-22}$
The G2 invariant	$q^{32}-3q^{30}+7q^{28}-13q^{26}+13q^{24}-9q^{22}-6q^{20}+30q^{18}-50q^{16}+66q^{14}-56q^{12}+17q^{10}+39q^{8}-93q^{6}+126q^{4}-112q^{2}+58+22q^{-2}-92q^{-4}+126q^{-6}-106q^{-8}+48q^{-10}+29q^{-12}-83q^{-14}+89q^{-16}-47q^{-18}-23q^{-20}+92q^{-22}-122q^{-24}+101q^{-26}-35q^{-28}-53q^{-30}+131q^{-32}-173q^{-34}+158q^{-36}-91q^{-38}-6q^{-40}+98q^{-42}-157q^{-44}+157q^{-46}-103q^{-48}+19q^{-50}+58q^{-52}-102q^{-54}+89q^{-56}-33q^{-58}-39q^{-60}+90q^{-62}-94q^{-64}+49q^{-66}+22q^{-68}-90q^{-70}+125q^{-72}-111q^{-74}+63q^{-76}+q^{-78}-59q^{-80}+88q^{-82}-86q^{-84}+64q^{-86}-26q^{-88}-5q^{-90}+25q^{-92}-33q^{-94}+30q^{-96}-21q^{-98}+12q^{-100}-2q^{-102}-4q^{-104}+5q^{-106}-6q^{-108}+4q^{-110}-2q^{-112}+q^{-114}$

Further Quantum Invariants

Further quantum knot invariants for 9_32.

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^5+3 q^3-2 q+3 q^{-1} - q^{-3} + q^{-7} -3 q^{-9} +3 q^{-11} -2 q^{-13} + q^{-15} }
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^{16}-3 q^{14}-2 q^{12}+11 q^{10}-4 q^8-14 q^6+18 q^4+5 q^2-23+13 q^{-2} +13 q^{-4} -19 q^{-6} + q^{-8} +13 q^{-10} -5 q^{-12} -10 q^{-14} +6 q^{-16} +13 q^{-18} -17 q^{-20} -5 q^{-22} +25 q^{-24} -13 q^{-26} -13 q^{-28} +19 q^{-30} -3 q^{-32} -9 q^{-34} +6 q^{-36} -2 q^{-40} + q^{-42} }
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^{33}+3 q^{31}+2 q^{29}-7 q^{27}-10 q^{25}+8 q^{23}+30 q^{21}-5 q^{19}-48 q^{17}-18 q^{15}+66 q^{13}+54 q^{11}-67 q^9-93 q^7+50 q^5+124 q^3-14 q-143 q^{-1} -23 q^{-3} +139 q^{-5} +56 q^{-7} -117 q^{-9} -80 q^{-11} +93 q^{-13} +90 q^{-15} -59 q^{-17} -93 q^{-19} +26 q^{-21} +87 q^{-23} +13 q^{-25} -83 q^{-27} -52 q^{-29} +69 q^{-31} +93 q^{-33} -48 q^{-35} -125 q^{-37} +16 q^{-39} +147 q^{-41} +21 q^{-43} -146 q^{-45} -52 q^{-47} +117 q^{-49} +75 q^{-51} -83 q^{-53} -76 q^{-55} +46 q^{-57} +60 q^{-59} -16 q^{-61} -39 q^{-63} +3 q^{-65} +21 q^{-67} - q^{-69} -8 q^{-71} +3 q^{-75} -2 q^{-79} + q^{-81} }
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^{56}-3 q^{54}-2 q^{52}+7 q^{50}+6 q^{48}+6 q^{46}-24 q^{44}-31 q^{42}+11 q^{40}+45 q^{38}+83 q^{36}-29 q^{34}-139 q^{32}-103 q^{30}+35 q^{28}+282 q^{26}+170 q^{24}-155 q^{22}-379 q^{20}-280 q^{18}+334 q^{16}+561 q^{14}+220 q^{12}-448 q^{10}-794 q^8-75 q^6+684 q^4+799 q^2-33-989 q^{-2} -661 q^{-4} +321 q^{-6} +1039 q^{-8} +514 q^{-10} -697 q^{-12} -919 q^{-14} -162 q^{-16} +832 q^{-18} +751 q^{-20} -263 q^{-22} -793 q^{-24} -418 q^{-26} +472 q^{-28} +692 q^{-30} +74 q^{-32} -546 q^{-34} -518 q^{-36} +131 q^{-38} +573 q^{-40} +390 q^{-42} -277 q^{-44} -629 q^{-46} -287 q^{-48} +418 q^{-50} +776 q^{-52} +124 q^{-54} -670 q^{-56} -794 q^{-58} +49 q^{-60} +1005 q^{-62} +652 q^{-64} -376 q^{-66} -1079 q^{-68} -494 q^{-70} +763 q^{-72} +929 q^{-74} +169 q^{-76} -815 q^{-78} -774 q^{-80} +197 q^{-82} +673 q^{-84} +469 q^{-86} -269 q^{-88} -552 q^{-90} -138 q^{-92} +212 q^{-94} +330 q^{-96} +32 q^{-98} -191 q^{-100} -111 q^{-102} -5 q^{-104} +105 q^{-106} +40 q^{-108} -32 q^{-110} -20 q^{-112} -16 q^{-114} +18 q^{-116} +7 q^{-118} -6 q^{-120} + q^{-122} -3 q^{-124} +3 q^{-126} -2 q^{-130} + q^{-132} }
5	$-q^{85}+3q^{83}+2q^{81}-7q^{79}-6q^{77}-2q^{75}+10q^{73}+25q^{71}+25q^{69}-20q^{67}-73q^{65}-74q^{63}-4q^{61}+121q^{59}+203q^{57}+134q^{55}-145q^{53}-410q^{51}-392q^{49}-14q^{47}+561q^{45}+873q^{43}+484q^{41}-523q^{39}-1376q^{37}-1291q^{35}-44q^{33}+1653q^{31}+2363q^{29}+1179q^{27}-1344q^{25}-3271q^{23}-2809q^{21}+185q^{19}+3609q^{17}+4532q^{15}+1734q^{13}-3011q^{11}-5824q^{9}-4017q^{7}+1422q^{5}+6227q^{3}+6163q+822q^{-1}-5611q^{-3}-7593q^{-5}-3196q^{-7}+4095q^{-9}+8075q^{-11}+5192q^{-13}-2147q^{-15}-7620q^{-17}-6426q^{-19}+244q^{-21}+6455q^{-23}+6841q^{-25}+1297q^{-27}-5029q^{-29}-6557q^{-31}-2264q^{-33}+3612q^{-35}+5830q^{-37}+2780q^{-39}-2434q^{-41}-5011q^{-43}-2947q^{-45}+1501q^{-47}+4259q^{-49}+3079q^{-51}-734q^{-53}-3704q^{-55}-3331q^{-57}-62q^{-59}+3284q^{-61}+3846q^{-63}+1053q^{-65}-2839q^{-67}-4548q^{-69}-2406q^{-71}+2149q^{-73}+5300q^{-75}+4045q^{-77}-1004q^{-79}-5743q^{-81}-5840q^{-83}-687q^{-85}+5612q^{-87}+7407q^{-89}+2741q^{-91}-4635q^{-93}-8311q^{-95}-4861q^{-97}+2869q^{-99}+8250q^{-101}+6537q^{-103}-679q^{-105}-7065q^{-107}-7315q^{-109}-1521q^{-111}+5098q^{-113}+7029q^{-115}+3096q^{-117}-2832q^{-119}-5760q^{-121}-3789q^{-123}+819q^{-125}+4014q^{-127}+3582q^{-129}+506q^{-131}-2296q^{-133}-2751q^{-135}-1078q^{-137}+971q^{-139}+1769q^{-141}+1069q^{-143}-215q^{-145}-942q^{-147}-753q^{-149}-98q^{-151}+396q^{-153}+432q^{-155}+150q^{-157}-139q^{-159}-200q^{-161}-91q^{-163}+29q^{-165}+73q^{-167}+46q^{-169}-29q^{-173}-16q^{-175}+5q^{-177}+4q^{-179}+2q^{-181}+3q^{-183}-2q^{-185}-3q^{-187}+3q^{-189}-2q^{-193}+q^{-195}$

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^6+2 q^4+1+3 q^{-2} -2 q^{-4} +2 q^{-6} -2 q^{-8} -2 q^{-14} +2 q^{-16} - q^{-18} + q^{-22} }
1,1	$q^{20}-6q^{18}+20q^{16}-50q^{14}+97q^{12}-170q^{10}+268q^{8}-368q^{6}+464q^{4}-522q^{2}+534-462q^{-2}+323q^{-4}-120q^{-6}-132q^{-8}+394q^{-10}-641q^{-12}+832q^{-14}-960q^{-16}+998q^{-18}-942q^{-20}+802q^{-22}-590q^{-24}+342q^{-26}-79q^{-28}-154q^{-30}+334q^{-32}-448q^{-34}+491q^{-36}-470q^{-38}+408q^{-40}-330q^{-42}+247q^{-44}-170q^{-46}+110q^{-48}-70q^{-50}+40q^{-52}-20q^{-54}+10q^{-56}-4q^{-58}+q^{-60}$
2,0	$q^{18}-2q^{16}-3q^{14}+4q^{12}+4q^{10}-3q^{8}-5q^{6}+7q^{4}+10q^{2}-7-5q^{-2}+10q^{-4}+q^{-6}-8q^{-8}-2q^{-10}+6q^{-12}-4q^{-14}-4q^{-16}+5q^{-18}-6q^{-22}+5q^{-24}+8q^{-26}-10q^{-28}-2q^{-30}+9q^{-32}+3q^{-34}-8q^{-36}-2q^{-38}+9q^{-40}-6q^{-44}-q^{-46}+2q^{-48}-q^{-52}+q^{-56}$

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

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^7+2 q^5-q^3+3 q+3 q^{-3} - q^{-5} + q^{-7} - q^{-11} -2 q^{-15} + q^{-17} -3 q^{-19} +2 q^{-21} - q^{-23} + q^{-25} + q^{-29} }

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

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^8+2 q^6-q^4+2 q^2+2+3 q^{-4} - q^{-6} +2 q^{-8} - q^{-10} + q^{-12} - q^{-14} -2 q^{-18} - q^{-20} -3 q^{-24} +2 q^{-26} - q^{-28} + q^{-30} + q^{-32} + q^{-36} }

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

1,0

q^{24}-3q^{20}-3q^{18}+4q^{16}+8q^{14}-2q^{12}-12q^{10}-5q^{8}+16q^{6}+14q^{4}-9q^{2}-18+2q^{-2}+21q^{-4}+10q^{-6}-16q^{-8}-14q^{-10}+8q^{-12}+14q^{-14}-4q^{-16}-14q^{-18}-q^{-20}+12q^{-22}+3q^{-24}-13q^{-26}-6q^{-28}+11q^{-30}+10q^{-32}-9q^{-34}-13q^{-36}+7q^{-38}+17q^{-40}-2q^{-42}-19q^{-44}-5q^{-46}+18q^{-48}+13q^{-50}-11q^{-52}-18q^{-54}+2q^{-56}+16q^{-58}+6q^{-60}-8q^{-62}-9q^{-64}+q^{-66}+6q^{-68}+2q^{-70}-2q^{-72}-2q^{-74}+q^{-78}

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{18}-3q^{16}+4q^{14}-6q^{12}+9q^{10}-12q^{8}+14q^{6}-13q^{4}+18q^{2}-12+13q^{-2}-7q^{-4}+9q^{-6}-q^{-8}-7q^{-10}+10q^{-12}-14q^{-14}+19q^{-16}-26q^{-18}+24q^{-20}-26q^{-22}+28q^{-24}-26q^{-26}+19q^{-28}-19q^{-30}+15q^{-32}-7q^{-34}+q^{-38}-6q^{-40}+14q^{-42}-12q^{-44}+13q^{-46}-15q^{-48}+16q^{-50}-10q^{-52}+8q^{-54}-9q^{-56}+6q^{-58}-3q^{-60}+2q^{-62}-2q^{-64}+q^{-66}$

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^{32}-3 q^{30}+7 q^{28}-13 q^{26}+13 q^{24}-9 q^{22}-6 q^{20}+30 q^{18}-50 q^{16}+66 q^{14}-56 q^{12}+17 q^{10}+39 q^8-93 q^6+126 q^4-112 q^2+58+22 q^{-2} -92 q^{-4} +126 q^{-6} -106 q^{-8} +48 q^{-10} +29 q^{-12} -83 q^{-14} +89 q^{-16} -47 q^{-18} -23 q^{-20} +92 q^{-22} -122 q^{-24} +101 q^{-26} -35 q^{-28} -53 q^{-30} +131 q^{-32} -173 q^{-34} +158 q^{-36} -91 q^{-38} -6 q^{-40} +98 q^{-42} -157 q^{-44} +157 q^{-46} -103 q^{-48} +19 q^{-50} +58 q^{-52} -102 q^{-54} +89 q^{-56} -33 q^{-58} -39 q^{-60} +90 q^{-62} -94 q^{-64} +49 q^{-66} +22 q^{-68} -90 q^{-70} +125 q^{-72} -111 q^{-74} +63 q^{-76} + q^{-78} -59 q^{-80} +88 q^{-82} -86 q^{-84} +64 q^{-86} -26 q^{-88} -5 q^{-90} +25 q^{-92} -33 q^{-94} +30 q^{-96} -21 q^{-98} +12 q^{-100} -2 q^{-102} -4 q^{-104} +5 q^{-106} -6 q^{-108} +4 q^{-110} -2 q^{-112} + q^{-114} }

.

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["9 32"];

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

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

Out[4]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^3-6 t^2+14 t-17+14 t^{-1} -6 t^{-2} + t^{-3} }

In[5]:= Conway[K][z]

Out[5]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6-z^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]= { 59, 2 }

In[8]:= Jones[K][q]

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

Out[8]= $q^{7}-3q^{6}+6q^{5}-9q^{4}+10q^{3}-10q^{2}+9q-6+4q^{-1}-q^{-2}$

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} +3 z^4 a^{-2} -2 z^4 a^{-4} -z^4+3 z^2 a^{-2} -4 z^2 a^{-4} +z^2 a^{-6} -z^2+ a^{-2} -2 a^{-4} + a^{-6} +1}

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^8 a^{-2} +2 z^8 a^{-4} +5 z^7 a^{-1} +10 z^7 a^{-3} +5 z^7 a^{-5} +6 z^6 a^{-2} +7 z^6 a^{-4} +5 z^6 a^{-6} +4 z^6+a z^5-9 z^5 a^{-1} -18 z^5 a^{-3} -5 z^5 a^{-5} +3 z^5 a^{-7} -19 z^4 a^{-2} -18 z^4 a^{-4} -6 z^4 a^{-6} +z^4 a^{-8} -8 z^4-a z^3+3 z^3 a^{-1} +9 z^3 a^{-3} +2 z^3 a^{-5} -3 z^3 a^{-7} +10 z^2 a^{-2} +12 z^2 a^{-4} +4 z^2 a^{-6} -z^2 a^{-8} +3 z^2-z a^{-1} -2 z a^{-3} +z a^{-7} - a^{-2} -2 a^{-4} - a^{-6} +1}

Vassiliev invariants

V₂ and V₃:

(-1, -2)

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 -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 -16}

Failed to parse (SVG (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}}

{\frac {14}{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 64}

{\frac {224}{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{128}{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 16}

Failed to parse (SVG (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 128}

Failed to parse (SVG (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{56}{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{12689}{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{74}{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{9778}{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{79}{18}}

{\frac {689}{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=} 2 is the signature of 9 32. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-3

-2

-1

0

1

2

3

4

5

6

χ

15

1

13

2

-2

11

4

1

3

9

5

2

-3

7

5

4

1

5

0

3

4

5

-1

1

3

6

3

-1

1

3

-2

-3

3

-5

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=3}
$r=-3$	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}}
$r=-2$	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}	Failed to parse (SVG (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=-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}^{3}\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=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^{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}}
Failed to parse (SVG (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=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^{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=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}}

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, 32]]
Out[2]=	9
In[3]:=	PD[Knot[9, 32]]
Out[3]=	PD[X[1, 4, 2, 5], X[13, 18, 14, 1], X[3, 9, 4, 8], X[9, 3, 10, 2], X[7, 15, 8, 14], X[15, 11, 16, 10], X[5, 12, 6, 13], X[11, 17, 12, 16], X[17, 7, 18, 6]]
In[4]:=	GaussCode[Knot[9, 32]]
Out[4]=	GaussCode[-1, 4, -3, 1, -7, 9, -5, 3, -4, 6, -8, 7, -2, 5, -6, 8, -9, 2]
In[5]:=	BR[Knot[9, 32]]
Out[5]=	BR[4, {1, 1, -2, 1, -2, 1, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[9, 32]][t]
Out[6]=	-3 6 14 2 3 -17 + t - -- + -- + 14 t - 6 t + t 2 t t
In[7]:=	Conway[Knot[9, 32]][z]
Out[7]=	2 6 1 - z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 32], Knot[11, NonAlternating, 52], Knot[11, NonAlternating, 124]}
In[9]:=	{KnotDet[Knot[9, 32]], KnotSignature[Knot[9, 32]]}
Out[9]=	{59, 2}
In[10]:=	J=Jones[Knot[9, 32]][q]
Out[10]=	-2 4 2 3 4 5 6 7 -6 - q + - + 9 q - 10 q + 10 q - 9 q + 6 q - 3 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 32]}
In[12]:=	A2Invariant[Knot[9, 32]][q]
Out[12]=	-6 2 2 4 6 8 14 16 18 22 1 - q + -- + 3 q - 2 q + 2 q - 2 q - 2 q + 2 q - q + q 4 q
In[13]:=	Kauffman[Knot[9, 32]][a, z]
Out[13]=	2 2 2 2 -6 2 -2 z 2 z z 2 z 4 z 12 z 10 z 1 - a - -- - a + -- - --- - - + 3 z - -- + ---- + ----- + ----- - 4 7 3 a 8 6 4 2 a a a a a a a 3 3 3 3 4 4 4 4 3 z 2 z 9 z 3 z 3 4 z 6 z 18 z 19 z ---- + ---- + ---- + ---- - a z - 8 z + -- - ---- - ----- - ----- + 7 5 3 a 8 6 4 2 a a a a a a a 5 5 5 5 6 6 6 3 z 5 z 18 z 9 z 5 6 5 z 7 z 6 z ---- - ---- - ----- - ---- + a z + 4 z + ---- + ---- + ---- + 7 5 3 a 6 4 2 a a a a a a 7 7 7 8 8 5 z 10 z 5 z 2 z 2 z ---- + ----- + ---- + ---- + ---- 5 3 a 4 2 a a a a
In[14]:=	{Vassiliev[2][Knot[9, 32]], Vassiliev[3][Knot[9, 32]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[9, 32]][q, t]
Out[15]=	3 1 3 1 3 3 q 3 5 6 q + 4 q + ----- + ----- + ---- + --- + --- + 5 q t + 5 q t + 5 3 3 2 2 q t t q t q t q t 5 2 7 2 7 3 9 3 9 4 11 4 11 5 5 q t + 5 q t + 4 q t + 5 q t + 2 q t + 4 q t + q t + 13 5 15 6 2 q t + q t

9 32

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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