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]

Minimum Braid Representative:

Length is 9, width is 4.

Braid index is 4.

A Morse Link Presentation:

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	$z^{6}-z^{2}+1$
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	Failed to parse (SVG (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^7-3 q^6+6 q^5-9 q^4+10 q^3-10 q^2+9 q-6+4 q^{-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	$q^{32}-3q^{30}+7q^{28}-13q^{26}+13q^{24}-9q^{22}-6q^{20}+30q^{18}-50q^{16}+66q^{14}-56q^{12}+17q^{10}+39q^{8}-93q^{6}+126q^{4}-112q^{2}+58+22q^{-2}-92q^{-4}+126q^{-6}-106q^{-8}+48q^{-10}+29q^{-12}-83q^{-14}+89q^{-16}-47q^{-18}-23q^{-20}+92q^{-22}-122q^{-24}+101q^{-26}-35q^{-28}-53q^{-30}+131q^{-32}-173q^{-34}+158q^{-36}-91q^{-38}-6q^{-40}+98q^{-42}-157q^{-44}+157q^{-46}-103q^{-48}+19q^{-50}+58q^{-52}-102q^{-54}+89q^{-56}-33q^{-58}-39q^{-60}+90q^{-62}-94q^{-64}+49q^{-66}+22q^{-68}-90q^{-70}+125q^{-72}-111q^{-74}+63q^{-76}+q^{-78}-59q^{-80}+88q^{-82}-86q^{-84}+64q^{-86}-26q^{-88}-5q^{-90}+25q^{-92}-33q^{-94}+30q^{-96}-21q^{-98}+12q^{-100}-2q^{-102}-4q^{-104}+5q^{-106}-6q^{-108}+4q^{-110}-2q^{-112}+q^{-114}$

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	Failed to parse (SVG (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	$q^{18}-2q^{16}-3q^{14}+4q^{12}+4q^{10}-3q^{8}-5q^{6}+7q^{4}+10q^{2}-7-5q^{-2}+10q^{-4}+q^{-6}-8q^{-8}-2q^{-10}+6q^{-12}-4q^{-14}-4q^{-16}+5q^{-18}-6q^{-22}+5q^{-24}+8q^{-26}-10q^{-28}-2q^{-30}+9q^{-32}+3q^{-34}-8q^{-36}-2q^{-38}+9q^{-40}-6q^{-44}-q^{-46}+2q^{-48}-q^{-52}+q^{-56}$

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

-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

Failed to parse (SVG (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

Failed to parse (SVG (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

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}

.

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

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]= Failed to parse (SVG (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^7-3 q^6+6 q^5-9 q^4+10 q^3-10 q^2+9 q-6+4 q^{-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}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n52, K11n124, ...}

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}} ): {...}

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}

Failed to parse (SVG (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}}

Failed to parse (SVG (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}}

{\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}}

Failed to parse (SVG (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 Failed to parse (SVG (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}
Failed to parse (SVG (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}}
Failed to parse (SVG (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}	Failed to parse (SVG (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}}

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^{20}-3 q^{19}+2 q^{18}+7 q^{17}-18 q^{16}+8 q^{15}+29 q^{14}-50 q^{13}+8 q^{12}+67 q^{11}-80 q^{10}-4 q^9+97 q^8-87 q^7-20 q^6+102 q^5-69 q^4-32 q^3+82 q^2-37 q-32+46 q^{-1} -9 q^{-2} -19 q^{-3} +14 q^{-4} + q^{-5} -4 q^{-6} + q^{-7} }
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^{39}-3 q^{38}+2 q^{37}+3 q^{36}-2 q^{35}-11 q^{34}+9 q^{33}+25 q^{32}-20 q^{31}-53 q^{30}+32 q^{29}+101 q^{28}-34 q^{27}-175 q^{26}+25 q^{25}+259 q^{24}+8 q^{23}-344 q^{22}-69 q^{21}+426 q^{20}+134 q^{19}-475 q^{18}-210 q^{17}+503 q^{16}+275 q^{15}-499 q^{14}-331 q^{13}+472 q^{12}+371 q^{11}-425 q^{10}-392 q^9+353 q^8+405 q^7-276 q^6-389 q^5+180 q^4+368 q^3-103 q^2-306 q+18+248 q^{-1} +26 q^{-2} -168 q^{-3} -56 q^{-4} +105 q^{-5} +52 q^{-6} -47 q^{-7} -44 q^{-8} +21 q^{-9} +22 q^{-10} -4 q^{-11} -9 q^{-12} - q^{-13} +4 q^{-14} - q^{-15} }
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^{64}-3 q^{63}+2 q^{62}+3 q^{61}-6 q^{60}+5 q^{59}-10 q^{58}+15 q^{57}+14 q^{56}-40 q^{55}+q^{54}-22 q^{53}+87 q^{52}+79 q^{51}-150 q^{50}-105 q^{49}-102 q^{48}+310 q^{47}+377 q^{46}-268 q^{45}-455 q^{44}-516 q^{43}+593 q^{42}+1115 q^{41}-64 q^{40}-931 q^{39}-1487 q^{38}+552 q^{37}+2099 q^{36}+696 q^{35}-1097 q^{34}-2744 q^{33}-33 q^{32}+2802 q^{31}+1724 q^{30}-744 q^{29}-3700 q^{28}-876 q^{27}+2926 q^{26}+2518 q^{25}-92 q^{24}-4058 q^{23}-1581 q^{22}+2584 q^{21}+2870 q^{20}+575 q^{19}-3875 q^{18}-2023 q^{17}+1935 q^{16}+2842 q^{15}+1195 q^{14}-3257 q^{13}-2243 q^{12}+1045 q^{11}+2467 q^{10}+1725 q^9-2243 q^8-2162 q^7+51 q^6+1710 q^5+1947 q^4-1032 q^3-1637 q^2-667 q+728+1619 q^{-1} -76 q^{-2} -805 q^{-3} -782 q^{-4} -31 q^{-5} +900 q^{-6} +270 q^{-7} -137 q^{-8} -441 q^{-9} -258 q^{-10} +286 q^{-11} +171 q^{-12} +87 q^{-13} -116 q^{-14} -146 q^{-15} +39 q^{-16} +33 q^{-17} +51 q^{-18} -6 q^{-19} -34 q^{-20} + q^{-21} - q^{-22} +9 q^{-23} + q^{-24} -4 q^{-25} + q^{-26} }
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^{95}-3 q^{94}+2 q^{93}+3 q^{92}-6 q^{91}+q^{90}+6 q^{89}-4 q^{88}+4 q^{87}+4 q^{86}-27 q^{85}-12 q^{84}+35 q^{83}+42 q^{82}+31 q^{81}-40 q^{80}-147 q^{79}-121 q^{78}+96 q^{77}+331 q^{76}+313 q^{75}-76 q^{74}-641 q^{73}-776 q^{72}-93 q^{71}+1058 q^{70}+1597 q^{69}+624 q^{68}-1439 q^{67}-2825 q^{66}-1766 q^{65}+1513 q^{64}+4399 q^{63}+3700 q^{62}-1007 q^{61}-6020 q^{60}-6374 q^{59}-458 q^{58}+7327 q^{57}+9628 q^{56}+2926 q^{55}-7951 q^{54}-12993 q^{53}-6252 q^{52}+7577 q^{51}+16014 q^{50}+10142 q^{49}-6238 q^{48}-18374 q^{47}-13982 q^{46}+4127 q^{45}+19690 q^{44}+17518 q^{43}-1572 q^{42}-20169 q^{41}-20281 q^{40}-1026 q^{39}+19787 q^{38}+22257 q^{37}+3477 q^{36}-18914 q^{35}-23432 q^{34}-5581 q^{33}+17645 q^{32}+23966 q^{31}+7369 q^{30}-16121 q^{29}-23994 q^{28}-8927 q^{27}+14376 q^{26}+23593 q^{25}+10339 q^{24}-12308 q^{23}-22814 q^{22}-11685 q^{21}+9928 q^{20}+21529 q^{19}+12916 q^{18}-7094 q^{17}-19764 q^{16}-13903 q^{15}+4052 q^{14}+17236 q^{13}+14444 q^{12}-768 q^{11}-14220 q^{10}-14289 q^9-2159 q^8+10566 q^7+13250 q^6+4705 q^5-6881 q^4-11406 q^3-6139 q^2+3275 q+8853+6687 q^{-1} -448 q^{-2} -6065 q^{-3} -6075 q^{-4} -1530 q^{-5} +3414 q^{-6} +4880 q^{-7} +2365 q^{-8} -1320 q^{-9} -3277 q^{-10} -2453 q^{-11} -10 q^{-12} +1886 q^{-13} +1903 q^{-14} +607 q^{-15} -754 q^{-16} -1269 q^{-17} -720 q^{-18} +189 q^{-19} +656 q^{-20} +522 q^{-21} +99 q^{-22} -262 q^{-23} -331 q^{-24} -123 q^{-25} +81 q^{-26} +144 q^{-27} +81 q^{-28} +3 q^{-29} -52 q^{-30} -54 q^{-31} - q^{-32} +19 q^{-33} +11 q^{-34} +4 q^{-35} + q^{-36} -9 q^{-37} - q^{-38} +4 q^{-39} - q^{-40} }
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^{132}-3 q^{131}+2 q^{130}+3 q^{129}-6 q^{128}+q^{127}+2 q^{126}+12 q^{125}-15 q^{124}-6 q^{123}+17 q^{122}-30 q^{121}+4 q^{120}+31 q^{119}+65 q^{118}-36 q^{117}-75 q^{116}-17 q^{115}-136 q^{114}+21 q^{113}+227 q^{112}+387 q^{111}+45 q^{110}-325 q^{109}-451 q^{108}-838 q^{107}-198 q^{106}+889 q^{105}+1910 q^{104}+1288 q^{103}-320 q^{102}-1953 q^{101}-4001 q^{100}-2719 q^{99}+1130 q^{98}+5982 q^{97}+7019 q^{96}+3609 q^{95}-2908 q^{94}-11755 q^{93}-12929 q^{92}-5042 q^{91}+9892 q^{90}+20073 q^{89}+19301 q^{88}+5530 q^{87}-19619 q^{86}-34059 q^{85}-27884 q^{84}+1648 q^{83}+33577 q^{82}+49646 q^{81}+35307 q^{80}-12472 q^{79}-56251 q^{78}-68728 q^{77}-31404 q^{76}+29748 q^{75}+81352 q^{74}+84964 q^{73}+21832 q^{72}-59742 q^{71}-110550 q^{70}-84404 q^{69}-1980 q^{68}+93434 q^{67}+134097 q^{66}+74956 q^{65}-35893 q^{64}-131885 q^{63}-134970 q^{62}-50512 q^{61}+79231 q^{60}+161902 q^{59}+124029 q^{58}+2940 q^{57}-127781 q^{56}-164451 q^{55}-94293 q^{54}+51048 q^{53}+165275 q^{52}+153502 q^{51}+38125 q^{50}-109645 q^{49}-171876 q^{48}-121540 q^{47}+23969 q^{46}+154579 q^{45}+164354 q^{44}+61828 q^{43}-89099 q^{42}-166670 q^{41}-135004 q^{40}+2412 q^{39}+138590 q^{38}+165053 q^{37}+78198 q^{36}-67952 q^{35}-155281 q^{34}-142225 q^{33}-18612 q^{32}+117220 q^{31}+159998 q^{30}+93712 q^{29}-40914 q^{28}-135979 q^{27}-145520 q^{26}-44187 q^{25}+85033 q^{24}+145596 q^{23}+108204 q^{22}-4583 q^{21}-102796 q^{20}-138786 q^{19}-71117 q^{18}+40159 q^{17}+114625 q^{16}+112292 q^{15}+34544 q^{14}-54804 q^{13}-112907 q^{12}-86365 q^{11}-7813 q^{10}+66676 q^9+94702 q^8+59951 q^7-4087 q^6-68014 q^5-77020 q^4-39655 q^3+16095 q^2+56492 q+58074+28574 q^{-1} -20644 q^{-2} -45896 q^{-3} -42500 q^{-4} -15598 q^{-5} +16165 q^{-6} +34152 q^{-7} +32323 q^{-8} +7842 q^{-9} -13086 q^{-10} -24431 q^{-11} -20201 q^{-12} -6222 q^{-13} +9095 q^{-14} +17942 q^{-15} +12261 q^{-16} +3608 q^{-17} -6096 q^{-18} -10190 q^{-19} -8666 q^{-20} -2333 q^{-21} +4520 q^{-22} +5656 q^{-23} +4916 q^{-24} +1259 q^{-25} -1819 q^{-26} -3733 q^{-27} -2765 q^{-28} -291 q^{-29} +799 q^{-30} +1798 q^{-31} +1315 q^{-32} +512 q^{-33} -660 q^{-34} -888 q^{-35} -437 q^{-36} -245 q^{-37} +233 q^{-38} +334 q^{-39} +324 q^{-40} -11 q^{-41} -118 q^{-42} -77 q^{-43} -110 q^{-44} -10 q^{-45} +27 q^{-46} +73 q^{-47} +4 q^{-48} -12 q^{-49} +4 q^{-50} -16 q^{-51} -4 q^{-52} - q^{-53} +9 q^{-54} + q^{-55} -4 q^{-56} + q^{-57} }
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^{175}-3 q^{174}+2 q^{173}+3 q^{172}-6 q^{171}+q^{170}+2 q^{169}+8 q^{168}+q^{167}-25 q^{166}+7 q^{165}+14 q^{164}-14 q^{163}+10 q^{162}+13 q^{161}+39 q^{160}-122 q^{158}-44 q^{157}+18 q^{156}+25 q^{155}+151 q^{154}+148 q^{153}+179 q^{152}-44 q^{151}-557 q^{150}-553 q^{149}-334 q^{148}+145 q^{147}+1000 q^{146}+1347 q^{145}+1318 q^{144}+214 q^{143}-2076 q^{142}-3397 q^{141}-3455 q^{140}-1292 q^{139}+3115 q^{138}+6743 q^{137}+8395 q^{136}+5285 q^{135}-3214 q^{134}-12095 q^{133}-17673 q^{132}-14436 q^{131}-342 q^{130}+17614 q^{129}+32371 q^{128}+32798 q^{127}+12676 q^{126}-19664 q^{125}-51986 q^{124}-63534 q^{123}-40070 q^{122}+11067 q^{121}+71544 q^{120}+107217 q^{119}+89212 q^{118}+18600 q^{117}-81855 q^{116}-159932 q^{115}-163410 q^{114}-79204 q^{113}+69749 q^{112}+210456 q^{111}+259411 q^{110}+177698 q^{109}-20488 q^{108}-243093 q^{107}-366766 q^{106}-313090 q^{105}-75419 q^{104}+240410 q^{103}+466810 q^{102}+474676 q^{101}+220283 q^{100}-188080 q^{99}-539814 q^{98}-644240 q^{97}-404663 q^{96}+81887 q^{95}+568344 q^{94}+798380 q^{93}+610234 q^{92}+72472 q^{91}-543085 q^{90}-917388 q^{89}-814225 q^{88}-258583 q^{87}+466948 q^{86}+988216 q^{85}+993376 q^{84}+454830 q^{83}-350667 q^{82}-1007602 q^{81}-1133393 q^{80}-640204 q^{79}+213254 q^{78}+983011 q^{77}+1226367 q^{76}+797405 q^{75}-72075 q^{74}-926425 q^{73}-1275408 q^{72}-919017 q^{71}-56996 q^{70}+853575 q^{69}+1288481 q^{68}+1003430 q^{67}+165485 q^{66}-776310 q^{65}-1277125 q^{64}-1056639 q^{63}-251190 q^{62}+703612 q^{61}+1252009 q^{60}+1086812 q^{59}+317114 q^{58}-638364 q^{57}-1220714 q^{56}-1103145 q^{55}-369907 q^{54}+579062 q^{53}+1187340 q^{52}+1113068 q^{51}+417185 q^{50}-521047 q^{49}-1151735 q^{48}-1120943 q^{47}-466067 q^{46}+457494 q^{45}+1110493 q^{44}+1128133 q^{43}+521580 q^{42}-381833 q^{41}-1057824 q^{40}-1131910 q^{39}-585697 q^{38}+288357 q^{37}+986871 q^{36}+1126789 q^{35}+656067 q^{34}-174596 q^{33}-890498 q^{32}-1104431 q^{31}-726912 q^{30}+41884 q^{29}+765105 q^{28}+1055734 q^{27}+786917 q^{26}+102804 q^{25}-609132 q^{24}-972385 q^{23}-824806 q^{22}-247638 q^{21}+429692 q^{20}+850811 q^{19}+826008 q^{18}+376048 q^{17}-237132 q^{16}-692579 q^{15}-783220 q^{14}-471908 q^{13}+51082 q^{12}+508938 q^{11}+692548 q^{10}+519701 q^9+110089 q^8-315866 q^7-562486 q^6-514084 q^5-227070 q^4+136576 q^3+407418 q^2+456726 q+289913+10048 q^{-1} -249838 q^{-2} -362267 q^{-3} -297100 q^{-4} -108674 q^{-5} +110713 q^{-6} +249742 q^{-7} +259807 q^{-8} +155997 q^{-9} -7231 q^{-10} -141472 q^{-11} -194768 q^{-12} -158264 q^{-13} -54934 q^{-14} +54368 q^{-15} +123208 q^{-16} +130419 q^{-17} +77577 q^{-18} +2657 q^{-19} -60293 q^{-20} -89229 q^{-21} -73055 q^{-22} -30554 q^{-23} +16850 q^{-24} +49846 q^{-25} +53589 q^{-26} +35681 q^{-27} +7106 q^{-28} -20034 q^{-29} -31954 q^{-30} -29028 q^{-31} -15014 q^{-32} +3251 q^{-33} +14636 q^{-34} +18046 q^{-35} +13731 q^{-36} +4079 q^{-37} -4078 q^{-38} -9107 q^{-39} -9313 q^{-40} -4946 q^{-41} -389 q^{-42} +3232 q^{-43} +4792 q^{-44} +3582 q^{-45} +1738 q^{-46} -545 q^{-47} -2123 q^{-48} -1942 q^{-49} -1286 q^{-50} -242 q^{-51} +602 q^{-52} +753 q^{-53} +771 q^{-54} +403 q^{-55} -146 q^{-56} -297 q^{-57} -325 q^{-58} -175 q^{-59} +9 q^{-60} +17 q^{-61} +115 q^{-62} +115 q^{-63} +22 q^{-64} -20 q^{-65} -48 q^{-66} -23 q^{-67} +9 q^{-68} -11 q^{-69} + q^{-70} +16 q^{-71} +4 q^{-72} + q^{-73} -9 q^{-74} - q^{-75} +4 q^{-76} - q^{-77} }

Computer Talk

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[9, 32]]
Out[2]=	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[3]:=	GaussCode[Knot[9, 32]]
Out[3]=	GaussCode[-1, 4, -3, 1, -7, 9, -5, 3, -4, 6, -8, 7, -2, 5, -6, 8, -9, 2]
In[4]:=	DTCode[Knot[9, 32]]
Out[4]=	DTCode[4, 8, 12, 14, 2, 16, 18, 10, 6]
In[5]:=	br = BR[Knot[9, 32]]
Out[5]=	BR[4, {1, 1, -2, 1, -2, 1, 3, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 32]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 32]]]

`Out[8]=`	`-Graphics-`
In[9]:=	(#[Knot[9, 32]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 2, 3, 3, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 32]][t]
Out[10]=	-3 6 14 2 3 -17 + t - -- + -- + 14 t - 6 t + t 2 t t
In[11]:=	Conway[Knot[9, 32]][z]
Out[11]=	2 6 1 - z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 32], Knot[11, NonAlternating, 52], Knot[11, NonAlternating, 124]}
In[13]:=	{KnotDet[Knot[9, 32]], KnotSignature[Knot[9, 32]]}
Out[13]=	{59, 2}
In[14]:=	Jones[Knot[9, 32]][q]
Out[14]=	-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[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 32]}
In[16]:=	A2Invariant[Knot[9, 32]][q]
Out[16]=	-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[17]:=	HOMFLYPT[Knot[9, 32]][a, z]
Out[17]=	2 2 2 4 4 6 -6 2 -2 2 z 4 z 3 z 4 2 z 3 z z 1 + a - -- + a - z + -- - ---- + ---- - z - ---- + ---- + -- 4 6 4 2 4 2 2 a a a a a a a
In[18]:=	Kauffman[Knot[9, 32]][a, z]
Out[18]=	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[19]:=	{Vassiliev[2][Knot[9, 32]], Vassiliev[3][Knot[9, 32]]}
Out[19]=	{-1, -2}
In[20]:=	Kh[Knot[9, 32]][q, t]
Out[20]=	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
In[21]:=	ColouredJones[Knot[9, 32], 2][q]
Out[21]=	-7 4 -5 14 19 9 46 2 3 -32 + q - -- + q + -- - -- - -- + -- - 37 q + 82 q - 32 q - 6 4 3 2 q q q q q 4 5 6 7 8 9 10 11 69 q + 102 q - 20 q - 87 q + 97 q - 4 q - 80 q + 67 q + 12 13 14 15 16 17 18 19 20 8 q - 50 q + 29 q + 8 q - 18 q + 7 q + 2 q - 3 q + q

See/edit the Rolfsen_Splice_Template.

9 32

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

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