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 X13,18,14,1 X3948 X9,3,10,2 X7,15,8,14 X15,11,16,10 X5,12,6,13 X11,17,12,16 X17,7,18,6
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} Topological 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} Concordance genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3} Rasmussen s-Invariant 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)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 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	Failed to parse (SVG (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} }
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^{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} }

Further Quantum Invariants

Further quantum knot invariants for 9_32.

A1 Invariants.

Weight	Invariant
1	$-q^{5}+3q^{3}-2q+3q^{-1}-q^{-3}+q^{-7}-3q^{-9}+3q^{-11}-2q^{-13}+q^{-15}$
2	$q^{16}-3q^{14}-2q^{12}+11q^{10}-4q^{8}-14q^{6}+18q^{4}+5q^{2}-23+13q^{-2}+13q^{-4}-19q^{-6}+q^{-8}+13q^{-10}-5q^{-12}-10q^{-14}+6q^{-16}+13q^{-18}-17q^{-20}-5q^{-22}+25q^{-24}-13q^{-26}-13q^{-28}+19q^{-30}-3q^{-32}-9q^{-34}+6q^{-36}-2q^{-40}+q^{-42}$
3	$-q^{33}+3q^{31}+2q^{29}-7q^{27}-10q^{25}+8q^{23}+30q^{21}-5q^{19}-48q^{17}-18q^{15}+66q^{13}+54q^{11}-67q^{9}-93q^{7}+50q^{5}+124q^{3}-14q-143q^{-1}-23q^{-3}+139q^{-5}+56q^{-7}-117q^{-9}-80q^{-11}+93q^{-13}+90q^{-15}-59q^{-17}-93q^{-19}+26q^{-21}+87q^{-23}+13q^{-25}-83q^{-27}-52q^{-29}+69q^{-31}+93q^{-33}-48q^{-35}-125q^{-37}+16q^{-39}+147q^{-41}+21q^{-43}-146q^{-45}-52q^{-47}+117q^{-49}+75q^{-51}-83q^{-53}-76q^{-55}+46q^{-57}+60q^{-59}-16q^{-61}-39q^{-63}+3q^{-65}+21q^{-67}-q^{-69}-8q^{-71}+3q^{-75}-2q^{-79}+q^{-81}$
4	$q^{56}-3q^{54}-2q^{52}+7q^{50}+6q^{48}+6q^{46}-24q^{44}-31q^{42}+11q^{40}+45q^{38}+83q^{36}-29q^{34}-139q^{32}-103q^{30}+35q^{28}+282q^{26}+170q^{24}-155q^{22}-379q^{20}-280q^{18}+334q^{16}+561q^{14}+220q^{12}-448q^{10}-794q^{8}-75q^{6}+684q^{4}+799q^{2}-33-989q^{-2}-661q^{-4}+321q^{-6}+1039q^{-8}+514q^{-10}-697q^{-12}-919q^{-14}-162q^{-16}+832q^{-18}+751q^{-20}-263q^{-22}-793q^{-24}-418q^{-26}+472q^{-28}+692q^{-30}+74q^{-32}-546q^{-34}-518q^{-36}+131q^{-38}+573q^{-40}+390q^{-42}-277q^{-44}-629q^{-46}-287q^{-48}+418q^{-50}+776q^{-52}+124q^{-54}-670q^{-56}-794q^{-58}+49q^{-60}+1005q^{-62}+652q^{-64}-376q^{-66}-1079q^{-68}-494q^{-70}+763q^{-72}+929q^{-74}+169q^{-76}-815q^{-78}-774q^{-80}+197q^{-82}+673q^{-84}+469q^{-86}-269q^{-88}-552q^{-90}-138q^{-92}+212q^{-94}+330q^{-96}+32q^{-98}-191q^{-100}-111q^{-102}-5q^{-104}+105q^{-106}+40q^{-108}-32q^{-110}-20q^{-112}-16q^{-114}+18q^{-116}+7q^{-118}-6q^{-120}+q^{-122}-3q^{-124}+3q^{-126}-2q^{-130}+q^{-132}$
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^{85}+3 q^{83}+2 q^{81}-7 q^{79}-6 q^{77}-2 q^{75}+10 q^{73}+25 q^{71}+25 q^{69}-20 q^{67}-73 q^{65}-74 q^{63}-4 q^{61}+121 q^{59}+203 q^{57}+134 q^{55}-145 q^{53}-410 q^{51}-392 q^{49}-14 q^{47}+561 q^{45}+873 q^{43}+484 q^{41}-523 q^{39}-1376 q^{37}-1291 q^{35}-44 q^{33}+1653 q^{31}+2363 q^{29}+1179 q^{27}-1344 q^{25}-3271 q^{23}-2809 q^{21}+185 q^{19}+3609 q^{17}+4532 q^{15}+1734 q^{13}-3011 q^{11}-5824 q^9-4017 q^7+1422 q^5+6227 q^3+6163 q+822 q^{-1} -5611 q^{-3} -7593 q^{-5} -3196 q^{-7} +4095 q^{-9} +8075 q^{-11} +5192 q^{-13} -2147 q^{-15} -7620 q^{-17} -6426 q^{-19} +244 q^{-21} +6455 q^{-23} +6841 q^{-25} +1297 q^{-27} -5029 q^{-29} -6557 q^{-31} -2264 q^{-33} +3612 q^{-35} +5830 q^{-37} +2780 q^{-39} -2434 q^{-41} -5011 q^{-43} -2947 q^{-45} +1501 q^{-47} +4259 q^{-49} +3079 q^{-51} -734 q^{-53} -3704 q^{-55} -3331 q^{-57} -62 q^{-59} +3284 q^{-61} +3846 q^{-63} +1053 q^{-65} -2839 q^{-67} -4548 q^{-69} -2406 q^{-71} +2149 q^{-73} +5300 q^{-75} +4045 q^{-77} -1004 q^{-79} -5743 q^{-81} -5840 q^{-83} -687 q^{-85} +5612 q^{-87} +7407 q^{-89} +2741 q^{-91} -4635 q^{-93} -8311 q^{-95} -4861 q^{-97} +2869 q^{-99} +8250 q^{-101} +6537 q^{-103} -679 q^{-105} -7065 q^{-107} -7315 q^{-109} -1521 q^{-111} +5098 q^{-113} +7029 q^{-115} +3096 q^{-117} -2832 q^{-119} -5760 q^{-121} -3789 q^{-123} +819 q^{-125} +4014 q^{-127} +3582 q^{-129} +506 q^{-131} -2296 q^{-133} -2751 q^{-135} -1078 q^{-137} +971 q^{-139} +1769 q^{-141} +1069 q^{-143} -215 q^{-145} -942 q^{-147} -753 q^{-149} -98 q^{-151} +396 q^{-153} +432 q^{-155} +150 q^{-157} -139 q^{-159} -200 q^{-161} -91 q^{-163} +29 q^{-165} +73 q^{-167} +46 q^{-169} -29 q^{-173} -16 q^{-175} +5 q^{-177} +4 q^{-179} +2 q^{-181} +3 q^{-183} -2 q^{-185} -3 q^{-187} +3 q^{-189} -2 q^{-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	Failed to parse (SVG (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}-6 q^{18}+20 q^{16}-50 q^{14}+97 q^{12}-170 q^{10}+268 q^8-368 q^6+464 q^4-522 q^2+534-462 q^{-2} +323 q^{-4} -120 q^{-6} -132 q^{-8} +394 q^{-10} -641 q^{-12} +832 q^{-14} -960 q^{-16} +998 q^{-18} -942 q^{-20} +802 q^{-22} -590 q^{-24} +342 q^{-26} -79 q^{-28} -154 q^{-30} +334 q^{-32} -448 q^{-34} +491 q^{-36} -470 q^{-38} +408 q^{-40} -330 q^{-42} +247 q^{-44} -170 q^{-46} +110 q^{-48} -70 q^{-50} +40 q^{-52} -20 q^{-54} +10 q^{-56} -4 q^{-58} + q^{-60} }
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^{18}-2 q^{16}-3 q^{14}+4 q^{12}+4 q^{10}-3 q^8-5 q^6+7 q^4+10 q^2-7-5 q^{-2} +10 q^{-4} + q^{-6} -8 q^{-8} -2 q^{-10} +6 q^{-12} -4 q^{-14} -4 q^{-16} +5 q^{-18} -6 q^{-22} +5 q^{-24} +8 q^{-26} -10 q^{-28} -2 q^{-30} +9 q^{-32} +3 q^{-34} -8 q^{-36} -2 q^{-38} +9 q^{-40} -6 q^{-44} - q^{-46} +2 q^{-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	$-q^{7}+2q^{5}-q^{3}+3q+3q^{-3}-q^{-5}+q^{-7}-q^{-11}-2q^{-15}+q^{-17}-3q^{-19}+2q^{-21}-q^{-23}+q^{-25}+q^{-29}$

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	$-q^{14}+3q^{12}-7q^{10}+11q^{8}-14q^{6}+19q^{4}-18q^{2}+18-13q^{-2}+9q^{-4}+q^{-6}-10q^{-8}+20q^{-10}-28q^{-12}+33q^{-14}-36q^{-16}+33q^{-18}-29q^{-20}+21q^{-22}-12q^{-24}+2q^{-26}+6q^{-28}-13q^{-30}+17q^{-32}-19q^{-34}+18q^{-36}-14q^{-38}+11q^{-40}-7q^{-42}+4q^{-44}-2q^{-46}+q^{-48}$
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^{24}-3 q^{20}-3 q^{18}+4 q^{16}+8 q^{14}-2 q^{12}-12 q^{10}-5 q^8+16 q^6+14 q^4-9 q^2-18+2 q^{-2} +21 q^{-4} +10 q^{-6} -16 q^{-8} -14 q^{-10} +8 q^{-12} +14 q^{-14} -4 q^{-16} -14 q^{-18} - q^{-20} +12 q^{-22} +3 q^{-24} -13 q^{-26} -6 q^{-28} +11 q^{-30} +10 q^{-32} -9 q^{-34} -13 q^{-36} +7 q^{-38} +17 q^{-40} -2 q^{-42} -19 q^{-44} -5 q^{-46} +18 q^{-48} +13 q^{-50} -11 q^{-52} -18 q^{-54} +2 q^{-56} +16 q^{-58} +6 q^{-60} -8 q^{-62} -9 q^{-64} + q^{-66} +6 q^{-68} +2 q^{-70} -2 q^{-72} -2 q^{-74} + q^{-78} }

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^{18}-3 q^{16}+4 q^{14}-6 q^{12}+9 q^{10}-12 q^8+14 q^6-13 q^4+18 q^2-12+13 q^{-2} -7 q^{-4} +9 q^{-6} - q^{-8} -7 q^{-10} +10 q^{-12} -14 q^{-14} +19 q^{-16} -26 q^{-18} +24 q^{-20} -26 q^{-22} +28 q^{-24} -26 q^{-26} +19 q^{-28} -19 q^{-30} +15 q^{-32} -7 q^{-34} + q^{-38} -6 q^{-40} +14 q^{-42} -12 q^{-44} +13 q^{-46} -15 q^{-48} +16 q^{-50} -10 q^{-52} +8 q^{-54} -9 q^{-56} +6 q^{-58} -3 q^{-60} +2 q^{-62} -2 q^{-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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

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]=

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

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 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)

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} $-16$ $8$ 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{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} Failed to parse (SVG (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{224}{3}} ${\frac {128}{3}}$ $16$ 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}} ${\frac {12689}{30}}$ 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}} ${\frac {79}{18}}$ 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{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 $j-2r=s+1$, 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	1
13									2		-2
11								4	1		3
9							5	2			-3
7						5	4				1
5					5	5					0
3				4	5						-1
1			3	6							3
-1		1	3								-2
-3		3									3
-5	1										-1

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