7 7

7_6

8_1

Visit 7 7's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 7 7's page at Knotilus! Visit 7 7's page at the original Knot Atlas! This is the Chinese crown loop of practical knot tying.

Ornamental knot

Mongolian ornament ; sum of two 7.7

Depiction with three loops

Sum of 4.1 and 7.7

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 X11,14,12,1 X7,13,8,12 X13,7,14,6
Gauss code	-1, 4, -3, 1, -2, 7, -6, 3, -4, 2, -5, 6, -7, 5
Dowker-Thistlethwaite code	4 8 10 12 2 14 6
Conway Notation	[21112]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{$\$$Failed}} Hyperbolic Volume 7.64338 A-Polynomial See Data:7 7/A-polynomial

[edit Notes for 7 7'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 2} Rasmussen s-Invariant 0

[edit Notes for 7 7's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{2}+t^{-2}-5t-5t^{-1}+9$
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^4-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	{ 21, 0 }
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^4-2 q^3- q^{-3} +3 q^2+3 q^{-2} -4 q-3 q^{-1} +4}
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-4} -a^2 z^2-2 z^2 a^{-2} -2 a^{-2} +z^4+2 z^2+2}
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^4 a^{-4} -2 z^2 a^{-4} + a^{-4} +2 z^5 a^{-3} +a^3 z^3-4 z^3 a^{-3} +2 z a^{-3} +z^6 a^{-2} +3 a^2 z^4+2 z^4 a^{-2} -3 a^2 z^2-6 z^2 a^{-2} +2 a^{-2} +3 a z^5+5 z^5 a^{-1} -3 a z^3-8 z^3 a^{-1} +a z+3 z a^{-1} +z^6+4 z^4-7 z^2+2}
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{10}+q^8+q^6+2 q^2+ q^{-2} - q^{-4} - q^{-6} - q^{-10} + q^{-12} + q^{-14} }
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^{52}-2 q^{50}+3 q^{48}-4 q^{46}+q^{42}-4 q^{40}+9 q^{38}-9 q^{36}+9 q^{34}-3 q^{32}-4 q^{30}+9 q^{28}-10 q^{26}+9 q^{24}-5 q^{22}-q^{20}+5 q^{18}-4 q^{16}+4 q^{14}+2 q^{12}-7 q^{10}+10 q^8-5 q^6-2 q^4+8 q^2-12+17 q^{-2} -11 q^{-4} +5 q^{-6} +3 q^{-8} -9 q^{-10} +15 q^{-12} -14 q^{-14} +6 q^{-16} - q^{-18} -4 q^{-20} +6 q^{-22} -6 q^{-24} + q^{-26} +3 q^{-28} -7 q^{-30} +5 q^{-32} -4 q^{-34} -5 q^{-36} +10 q^{-38} -11 q^{-40} +9 q^{-42} -4 q^{-44} - q^{-46} +7 q^{-48} -8 q^{-50} +9 q^{-52} -4 q^{-54} + q^{-56} + q^{-58} -3 q^{-60} +3 q^{-62} - q^{-64} + q^{-66} }

Further Quantum Invariants

Further quantum knot invariants for 7_7.

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^7+2 q^5+q- q^{-3} + q^{-5} - q^{-7} + q^{-9} }
2	$q^{20}-2q^{18}-2q^{16}+5q^{14}-q^{12}-3q^{10}+5q^{8}-3q^{4}+2q^{2}+1-q^{-2}-2q^{-4}+2q^{-6}+2q^{-8}-4q^{-10}+2q^{-12}+4q^{-14}-4q^{-16}+3q^{-20}-2q^{-22}-q^{-24}+q^{-26}$
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{39}+2 q^{37}+2 q^{35}-3 q^{33}-5 q^{31}+q^{29}+10 q^{27}-2 q^{25}-11 q^{23}+14 q^{19}+3 q^{17}-13 q^{15}-5 q^{13}+12 q^{11}+6 q^9-9 q^7-5 q^5+4 q^3+6 q- q^{-1} -5 q^{-3} -4 q^{-5} +5 q^{-7} +7 q^{-9} -2 q^{-11} -9 q^{-13} +3 q^{-15} +13 q^{-17} - q^{-19} -13 q^{-21} -3 q^{-23} +12 q^{-25} +4 q^{-27} -11 q^{-29} -6 q^{-31} +8 q^{-33} +7 q^{-35} -4 q^{-37} -6 q^{-39} +2 q^{-41} +4 q^{-43} -2 q^{-47} - q^{-49} + q^{-51} }
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}-2 q^{62}-2 q^{60}+3 q^{58}+3 q^{56}+5 q^{54}-8 q^{52}-10 q^{50}+2 q^{48}+8 q^{46}+20 q^{44}-11 q^{42}-26 q^{40}-7 q^{38}+15 q^{36}+40 q^{34}-3 q^{32}-36 q^{30}-25 q^{28}+8 q^{26}+50 q^{24}+13 q^{22}-30 q^{20}-33 q^{18}-4 q^{16}+39 q^{14}+19 q^{12}-13 q^{10}-26 q^8-13 q^6+18 q^4+16 q^2+5-10 q^{-2} -15 q^{-4} -4 q^{-6} +14 q^{-8} +21 q^{-10} + q^{-12} -18 q^{-14} -23 q^{-16} +10 q^{-18} +30 q^{-20} +10 q^{-22} -19 q^{-24} -40 q^{-26} +4 q^{-28} +37 q^{-30} +22 q^{-32} -11 q^{-34} -46 q^{-36} -8 q^{-38} +29 q^{-40} +30 q^{-42} +6 q^{-44} -38 q^{-46} -19 q^{-48} +10 q^{-50} +25 q^{-52} +18 q^{-54} -19 q^{-56} -18 q^{-58} -6 q^{-60} +10 q^{-62} +16 q^{-64} -3 q^{-66} -6 q^{-68} -6 q^{-70} +6 q^{-74} + q^{-76} -2 q^{-80} - q^{-82} + q^{-84} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{95}+2 q^{93}+2 q^{91}-3 q^{89}-3 q^{87}-3 q^{85}+2 q^{83}+8 q^{81}+10 q^{79}-2 q^{77}-17 q^{75}-17 q^{73}+23 q^{69}+30 q^{67}+14 q^{65}-34 q^{63}-58 q^{61}-23 q^{59}+36 q^{57}+77 q^{55}+55 q^{53}-31 q^{51}-102 q^{49}-82 q^{47}+17 q^{45}+108 q^{43}+113 q^{41}+9 q^{39}-107 q^{37}-128 q^{35}-35 q^{33}+93 q^{31}+132 q^{29}+55 q^{27}-69 q^{25}-125 q^{23}-67 q^{21}+42 q^{19}+102 q^{17}+69 q^{15}-16 q^{13}-77 q^{11}-64 q^9-q^7+48 q^5+58 q^3+23 q-26 q^{-1} -45 q^{-3} -34 q^{-5} +42 q^{-9} +51 q^{-11} +15 q^{-13} -37 q^{-15} -62 q^{-17} -35 q^{-19} +34 q^{-21} +80 q^{-23} +47 q^{-25} -34 q^{-27} -93 q^{-29} -65 q^{-31} +28 q^{-33} +104 q^{-35} +86 q^{-37} -20 q^{-39} -110 q^{-41} -100 q^{-43} +4 q^{-45} +107 q^{-47} +116 q^{-49} +20 q^{-51} -96 q^{-53} -120 q^{-55} -42 q^{-57} +69 q^{-59} +117 q^{-61} +62 q^{-63} -39 q^{-65} -103 q^{-67} -76 q^{-69} +8 q^{-71} +77 q^{-73} +76 q^{-75} +18 q^{-77} -46 q^{-79} -67 q^{-81} -32 q^{-83} +21 q^{-85} +48 q^{-87} +34 q^{-89} -27 q^{-93} -29 q^{-95} -8 q^{-97} +12 q^{-99} +17 q^{-101} +8 q^{-103} -2 q^{-105} -8 q^{-107} -8 q^{-109} +4 q^{-113} +3 q^{-115} + q^{-117} -2 q^{-121} - q^{-123} + q^{-125} }
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}-2 q^{130}-2 q^{128}+3 q^{126}+3 q^{124}+3 q^{122}-4 q^{120}-2 q^{118}-8 q^{116}-10 q^{114}+11 q^{112}+17 q^{110}+17 q^{108}-3 q^{106}-12 q^{104}-38 q^{102}-37 q^{100}+16 q^{98}+54 q^{96}+68 q^{94}+21 q^{92}-17 q^{90}-106 q^{88}-127 q^{86}-23 q^{84}+94 q^{82}+180 q^{80}+132 q^{78}+31 q^{76}-176 q^{74}-281 q^{72}-165 q^{70}+57 q^{68}+278 q^{66}+310 q^{64}+191 q^{62}-145 q^{60}-391 q^{58}-355 q^{56}-98 q^{54}+250 q^{52}+416 q^{50}+368 q^{48}+q^{46}-350 q^{44}-444 q^{42}-259 q^{40}+103 q^{38}+359 q^{36}+422 q^{34}+145 q^{32}-189 q^{30}-364 q^{28}-299 q^{26}-42 q^{24}+198 q^{22}+329 q^{20}+191 q^{18}-30 q^{16}-198 q^{14}-226 q^{12}-114 q^{10}+43 q^8+179 q^6+161 q^4+69 q^2-54-129 q^{-2} -134 q^{-4} -59 q^{-6} +56 q^{-8} +125 q^{-10} +134 q^{-12} +44 q^{-14} -69 q^{-16} -159 q^{-18} -135 q^{-20} -25 q^{-22} +128 q^{-24} +209 q^{-26} +120 q^{-28} -47 q^{-30} -215 q^{-32} -220 q^{-34} -90 q^{-36} +155 q^{-38} +306 q^{-40} +212 q^{-42} -20 q^{-44} -271 q^{-46} -319 q^{-48} -181 q^{-50} +147 q^{-52} +378 q^{-54} +320 q^{-56} +61 q^{-58} -264 q^{-60} -389 q^{-62} -305 q^{-64} +49 q^{-66} +357 q^{-68} +396 q^{-70} +196 q^{-72} -141 q^{-74} -356 q^{-76} -391 q^{-78} -111 q^{-80} +209 q^{-82} +362 q^{-84} +297 q^{-86} +50 q^{-88} -193 q^{-90} -355 q^{-92} -227 q^{-94} +3 q^{-96} +197 q^{-98} +267 q^{-100} +174 q^{-102} +4 q^{-104} -194 q^{-106} -205 q^{-108} -117 q^{-110} +17 q^{-112} +126 q^{-114} +154 q^{-116} +97 q^{-118} -36 q^{-120} -88 q^{-122} -98 q^{-124} -54 q^{-126} +8 q^{-128} +61 q^{-130} +72 q^{-132} +20 q^{-134} -6 q^{-136} -31 q^{-138} -33 q^{-140} -20 q^{-142} +6 q^{-144} +22 q^{-146} +11 q^{-148} +8 q^{-150} -2 q^{-152} -7 q^{-154} -10 q^{-156} -2 q^{-158} +4 q^{-160} + q^{-162} +3 q^{-164} + q^{-166} -2 q^{-170} - q^{-172} + q^{-174} }

A2 Invariants.

Weight	Invariant
1,0	$-q^{10}+q^{8}+q^{6}+2q^{2}+q^{-2}-q^{-4}-q^{-6}-q^{-10}+q^{-12}+q^{-14}$
1,1	$q^{28}-4q^{26}+8q^{24}-12q^{22}+18q^{20}-28q^{18}+30q^{16}-30q^{14}+30q^{12}-18q^{10}+12q^{8}+10q^{6}-23q^{4}+38q^{2}-52+54q^{-2}-60q^{-4}+52q^{-6}-44q^{-8}+30q^{-10}-11q^{-12}+2q^{-14}+16q^{-16}-24q^{-18}+31q^{-20}-30q^{-22}+26q^{-24}-24q^{-26}+15q^{-28}-10q^{-30}+6q^{-32}-2q^{-34}+q^{-36}$
2,0	$q^{26}-q^{24}-2q^{22}+2q^{18}+q^{16}-3q^{14}+2q^{12}+4q^{10}+q^{8}-q^{6}+2q^{4}+q^{2}-1-q^{-2}-q^{-4}-2q^{-6}-2q^{-8}+2q^{-10}-q^{-14}+3q^{-16}+4q^{-18}-q^{-22}+q^{-24}+q^{-26}-2q^{-28}-3q^{-30}+q^{-34}+q^{-36}$
3,0	$-q^{48}+q^{46}+2q^{44}+q^{42}-2q^{40}-6q^{38}+q^{36}+4q^{34}+5q^{32}-4q^{30}-11q^{28}+10q^{24}+12q^{22}-4q^{20}-11q^{18}+11q^{14}+8q^{12}-4q^{10}-8q^{8}+q^{6}+4q^{4}+q^{2}-5-4q^{-2}-q^{-4}-2q^{-6}-3q^{-8}+8q^{-12}+5q^{-14}-q^{-18}+8q^{-20}+11q^{-22}-2q^{-24}-11q^{-26}-7q^{-28}+5q^{-30}+7q^{-32}-6q^{-34}-11q^{-36}-5q^{-38}+6q^{-40}+10q^{-42}-4q^{-46}-4q^{-48}+4q^{-50}+7q^{-52}+2q^{-54}-2q^{-56}-5q^{-58}-2q^{-60}+q^{-64}+q^{-66}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{22}-2q^{20}-q^{18}+3q^{16}-3q^{14}+q^{12}+5q^{10}-q^{8}+3q^{4}-q^{2}-1+q^{-4}-2q^{-8}+3q^{-10}+q^{-12}-4q^{-14}+2q^{-16}+q^{-18}-3q^{-20}+2q^{-22}+q^{-24}-q^{-26}+q^{-28}$
1,0,0	$-q^{13}+q^{11}+q^{7}+2q^{3}+q+q^{-1}+q^{-3}-q^{-5}-q^{-7}-2q^{-9}-q^{-13}+q^{-15}+q^{-17}+q^{-19}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{28}-q^{26}-2q^{24}+q^{22}+q^{20}-2q^{18}-q^{16}+4q^{14}+3q^{12}+2q^{8}+6q^{6}-2q^{2}+1-2q^{-2}-5q^{-4}-q^{-6}+q^{-8}-q^{-10}+q^{-12}+5q^{-14}+3q^{-16}-2q^{-18}+2q^{-22}-2q^{-24}-3q^{-26}+q^{-30}+q^{-36}+q^{-38}$
1,0,0,0	$-q^{16}+q^{14}+q^{8}+2q^{4}+q^{2}+2+q^{-2}+q^{-4}-q^{-6}-q^{-8}-2q^{-10}-2q^{-12}-q^{-16}+q^{-18}+q^{-20}+q^{-22}+q^{-24}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{22}+2q^{20}-3q^{18}+3q^{16}-3q^{14}+3q^{12}-q^{10}+q^{8}+2q^{6}-q^{4}+5q^{2}-5+6q^{-2}-5q^{-4}+4q^{-6}-4q^{-8}+q^{-10}-q^{-12}-2q^{-14}+2q^{-16}-3q^{-18}+3q^{-20}-2q^{-22}+3q^{-24}-q^{-26}+q^{-28}$
1,0	$q^{36}-2q^{32}-2q^{30}+q^{28}+3q^{26}-3q^{22}-q^{20}+4q^{18}+3q^{16}-2q^{12}+q^{10}+2q^{8}+q^{6}-3q^{4}-q^{2}+2+q^{-2}-2q^{-4}-2q^{-6}+q^{-8}+2q^{-10}-2q^{-14}+q^{-16}+3q^{-18}+q^{-20}-3q^{-22}-2q^{-24}+2q^{-26}+3q^{-28}-q^{-30}-3q^{-32}-q^{-34}+2q^{-36}+2q^{-38}-q^{-40}-q^{-42}+q^{-46}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{30}-2q^{28}+q^{26}-2q^{24}+3q^{22}-3q^{20}+2q^{18}-q^{16}+3q^{14}+q^{12}+q^{8}+4q^{4}-3q^{2}+4-4q^{-2}+5q^{-4}-3q^{-6}+3q^{-8}-4q^{-10}+2q^{-12}-q^{-14}-q^{-18}-2q^{-20}+2q^{-22}-2q^{-24}+2q^{-26}-2q^{-28}+3q^{-30}-q^{-32}+2q^{-34}-q^{-36}+q^{-38}$

G2 Invariants.

Weight

Invariant

1,0

$q^{52}-2q^{50}+3q^{48}-4q^{46}+q^{42}-4q^{40}+9q^{38}-9q^{36}+9q^{34}-3q^{32}-4q^{30}+9q^{28}-10q^{26}+9q^{24}-5q^{22}-q^{20}+5q^{18}-4q^{16}+4q^{14}+2q^{12}-7q^{10}+10q^{8}-5q^{6}-2q^{4}+8q^{2}-12+17q^{-2}-11q^{-4}+5q^{-6}+3q^{-8}-9q^{-10}+15q^{-12}-14q^{-14}+6q^{-16}-q^{-18}-4q^{-20}+6q^{-22}-6q^{-24}+q^{-26}+3q^{-28}-7q^{-30}+5q^{-32}-4q^{-34}-5q^{-36}+10q^{-38}-11q^{-40}+9q^{-42}-4q^{-44}-q^{-46}+7q^{-48}-8q^{-50}+9q^{-52}-4q^{-54}+q^{-56}+q^{-58}-3q^{-60}+3q^{-62}-q^{-64}+q^{-66}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["7 7"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{2}+t^{-2}-5t-5t^{-1}+9$

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^4-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]=

{ 21, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-4} -a^2 z^2-2 z^2 a^{-2} -2 a^{-2} +z^4+2 z^2+2}

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^4 a^{-4} -2 z^2 a^{-4} + a^{-4} +2 z^5 a^{-3} +a^3 z^3-4 z^3 a^{-3} +2 z a^{-3} +z^6 a^{-2} +3 a^2 z^4+2 z^4 a^{-2} -3 a^2 z^2-6 z^2 a^{-2} +2 a^{-2} +3 a z^5+5 z^5 a^{-1} -3 a z^3-8 z^3 a^{-1} +a z+3 z a^{-1} +z^6+4 z^4-7 z^2+2}

Vassiliev invariants

V2 and V3:

(-1, -1)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-4$ $-8$ $8$ $-{\frac {14}{3}}$ $-{\frac {10}{3}}$ $32$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{112}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{64}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -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{32}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 32} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{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{40}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2849}{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{218}{15}} ${\frac {3418}{45}}$ 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{161}{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{449}{30}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 0 is the signature of 7 7. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 χ 9               1 1 7             1   -1 5           2 1   1 3         2 1     -1 1       2 2       0 -1     2 3         1 -3   1 1           0 -5   2             2 -7 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} $i=1$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-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}^{2}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-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}\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=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}^{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}^{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=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_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[7, 7]]
Out[2]=	7
In[3]:=	PD[Knot[7, 7]]
Out[3]=	PD[X[1, 4, 2, 5], X[5, 10, 6, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[11, 14, 12, 1], X[7, 13, 8, 12], X[13, 7, 14, 6]]
In[4]:=	GaussCode[Knot[7, 7]]
Out[4]=	GaussCode[-1, 4, -3, 1, -2, 7, -6, 3, -4, 2, -5, 6, -7, 5]
In[5]:=	BR[Knot[7, 7]]
Out[5]=	BR[4, {1, -2, 1, -2, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[7, 7]][t]
Out[6]=	-2 5 2 9 + t - - - 5 t + t t
In[7]:=	Conway[Knot[7, 7]][z]
Out[7]=	2 4 1 - z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[7, 7], Knot[11, NonAlternating, 28]}
In[9]:=	{KnotDet[Knot[7, 7]], KnotSignature[Knot[7, 7]]}
Out[9]=	{21, 0}
In[10]:=	J=Jones[Knot[7, 7]][q]
Out[10]=	-3 3 3 2 3 4 4 - q + -- - - - 4 q + 3 q - 2 q + q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[7, 7]}
In[12]:=	A2Invariant[Knot[7, 7]][q]
Out[12]=	-10 -8 -6 2 2 4 6 10 12 14 -q + q + q + -- + q - q - q - q + q + q 2 q
In[13]:=	Kauffman[Knot[7, 7]][a, z]
Out[13]=	2 2 3 -4 2 2 z 3 z 2 2 z 6 z 2 2 4 z 2 + a + -- + --- + --- + a z - 7 z - ---- - ---- - 3 a z - ---- - 2 3 a 4 2 3 a a a a a 3 4 4 5 5 8 z 3 3 3 4 z 2 z 2 4 2 z 5 z ---- - 3 a z + a z + 4 z + -- + ---- + 3 a z + ---- + ---- + a 4 2 3 a a a a 6 5 6 z 3 a z + z + -- 2 a
In[14]:=	{Vassiliev[2][Knot[7, 7]], Vassiliev[3][Knot[7, 7]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[7, 7]][q, t]
Out[15]=	3 1 2 1 1 2 3 3 2 - + 2 q + ----- + ----- + ----- + ---- + --- + 2 q t + 2 q t + q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 5 3 7 3 9 4 2 q t + q t + q t + q t