7 7: Difference between revisions

Revision as of 21:42, 27 August 2005

Visit 7 7's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit [Failed to parse (SVG (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{KnotilusURL}(\textrm{GaussCode}())} 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 X_5,10,6,11 X₃₉₄₈ X_9,3,10,2 X_11,14,12,1 X_7,13,8,12 X_13,7,14,6
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	Failed to parse (SVG (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^2+ t^{-2} -5 t-5 t^{-1} +9}
Conway polynomial	$z^{4}-z^{2}+1$
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	Failed to parse (SVG (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}-2 q^{18}-2 q^{16}+5 q^{14}-q^{12}-3 q^{10}+5 q^8-3 q^4+2 q^2+1- q^{-2} -2 q^{-4} +2 q^{-6} +2 q^{-8} -4 q^{-10} +2 q^{-12} +4 q^{-14} -4 q^{-16} +3 q^{-20} -2 q^{-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	Failed to parse (SVG (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^{28}-4 q^{26}+8 q^{24}-12 q^{22}+18 q^{20}-28 q^{18}+30 q^{16}-30 q^{14}+30 q^{12}-18 q^{10}+12 q^8+10 q^6-23 q^4+38 q^2-52+54 q^{-2} -60 q^{-4} +52 q^{-6} -44 q^{-8} +30 q^{-10} -11 q^{-12} +2 q^{-14} +16 q^{-16} -24 q^{-18} +31 q^{-20} -30 q^{-22} +26 q^{-24} -24 q^{-26} +15 q^{-28} -10 q^{-30} +6 q^{-32} -2 q^{-34} + q^{-36} }
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^{26}-q^{24}-2 q^{22}+2 q^{18}+q^{16}-3 q^{14}+2 q^{12}+4 q^{10}+q^8-q^6+2 q^4+q^2-1- q^{-2} - q^{-4} -2 q^{-6} -2 q^{-8} +2 q^{-10} - q^{-14} +3 q^{-16} +4 q^{-18} - q^{-22} + q^{-24} + q^{-26} -2 q^{-28} -3 q^{-30} + q^{-34} + q^{-36} }
3,0	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^{48}+q^{46}+2 q^{44}+q^{42}-2 q^{40}-6 q^{38}+q^{36}+4 q^{34}+5 q^{32}-4 q^{30}-11 q^{28}+10 q^{24}+12 q^{22}-4 q^{20}-11 q^{18}+11 q^{14}+8 q^{12}-4 q^{10}-8 q^8+q^6+4 q^4+q^2-5-4 q^{-2} - q^{-4} -2 q^{-6} -3 q^{-8} +8 q^{-12} +5 q^{-14} - q^{-18} +8 q^{-20} +11 q^{-22} -2 q^{-24} -11 q^{-26} -7 q^{-28} +5 q^{-30} +7 q^{-32} -6 q^{-34} -11 q^{-36} -5 q^{-38} +6 q^{-40} +10 q^{-42} -4 q^{-46} -4 q^{-48} +4 q^{-50} +7 q^{-52} +2 q^{-54} -2 q^{-56} -5 q^{-58} -2 q^{-60} + q^{-64} + q^{-66} }

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

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

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^{16}+q^{14}+q^8+2 q^4+q^2+2+ q^{-2} + q^{-4} - q^{-6} - q^{-8} -2 q^{-10} -2 q^{-12} - q^{-16} + q^{-18} + q^{-20} + q^{-22} + q^{-24} }

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^{22}+2 q^{20}-3 q^{18}+3 q^{16}-3 q^{14}+3 q^{12}-q^{10}+q^8+2 q^6-q^4+5 q^2-5+6 q^{-2} -5 q^{-4} +4 q^{-6} -4 q^{-8} + q^{-10} - q^{-12} -2 q^{-14} +2 q^{-16} -3 q^{-18} +3 q^{-20} -2 q^{-22} +3 q^{-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

Failed to parse (SVG (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^{30}-2 q^{28}+q^{26}-2 q^{24}+3 q^{22}-3 q^{20}+2 q^{18}-q^{16}+3 q^{14}+q^{12}+q^8+4 q^4-3 q^2+4-4 q^{-2} +5 q^{-4} -3 q^{-6} +3 q^{-8} -4 q^{-10} +2 q^{-12} - q^{-14} - q^{-18} -2 q^{-20} +2 q^{-22} -2 q^{-24} +2 q^{-26} -2 q^{-28} +3 q^{-30} - q^{-32} +2 q^{-34} - q^{-36} + q^{-38} }

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^{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} }

.

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["7 7"];

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

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

Out[5]= $z^{4}-z^{2}+1$

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)

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 -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 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{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 -\frac{10}{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}

{\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}}

Failed to parse (SVG (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{3418}{45}}

-{\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		\

0

χ

1

-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[0, 1]]
Out[2]=	0
In[3]:=	PD[Knot[0, 1]]
Out[3]=	PD[Loop[1]]
In[4]:=	GaussCode[Knot[0, 1]]
Out[4]=	GaussCode[]
In[5]:=	BR[Knot[0, 1]]
Out[5]=	BR[1, {}]
In[6]:=	alex = Alexander[Knot[0, 1]][t]
Out[6]=	1
In[7]:=	Conway[Knot[0, 1]][z]
Out[7]=	1
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[0, 1], Knot[11, NonAlternating, 34], Knot[11, NonAlternating, 42]}
In[9]:=	{KnotDet[Knot[0, 1]], KnotSignature[Knot[0, 1]]}
Out[9]=	{1, 0}
In[10]:=	J=Jones[Knot[0, 1]][q]
Out[10]=	1
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[0, 1]}
In[12]:=	A2Invariant[Knot[0, 1]][q]
Out[12]=	-2 2 1 + q + q
In[13]:=	Kauffman[Knot[0, 1]][a, z]
Out[13]=	1
In[14]:=	{Vassiliev[2][Knot[0, 1]], Vassiliev[3][Knot[0, 1]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[0, 1]][q, t]
Out[15]=	1 - + q q

@@ Line 1: / Line 1: @@
+<!--  -->
-{{Template:Basic Knot Invariants|name=7_7}}
+<!-- provide an anchor so we can return to the top of the page -->
+<span id="top"></span>
+<!-- this relies on transclusion for next and previous links -->
+{{Knot Navigation Links|ext=gif}}
+{| align=left
+|- valign=top
+|[[Image:{{PAGENAME}}.gif]]
+|{{Rolfsen Knot Site Links|n=0|k=1|KnotilusURL=<math>\textrm{KnotilusURL}(\textrm{GaussCode}())</math>}}
+|{{:{{PAGENAME}} Quick Notes}}
+|}
+<br style="clear:both" />
+{{:{{PAGENAME}} Further Notes and Views}}
+{{Knot Presentations}}
+{{3D Invariants}}
+{{4D Invariants}}
+{{Polynomial Invariants}}
+{{Vassiliev Invariants}}
+===[[Khovanov Homology]]===
+The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>{{Data:{{PAGENAME}}/Signature}} is the signature of {{PAGENAME}}. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.
+<center><table border=1>
+<tr align=center>
+<td width=40.%><table cellpadding=0 cellspacing=0>
+ <tr><td>\</td><td>&nbsp;</td><td>r</td></tr>
+<tr><td>&nbsp;</td><td>&nbsp;\&nbsp;</td><td>&nbsp;</td></tr>
+<tr><td>j</td><td>&nbsp;</td><td>\</td></tr>
+</table></td>
+ <td width=20.%>0</td  ><td width=40.%>&chi;</td></tr>
+<tr align=center><td>1</td><td bgcolor=yellow>1</td><td>1</td></tr>
+<tr align=center><td>-1</td><td bgcolor=yellow>1</td><td>1</td></tr>
+</table></center>
+{{Computer Talk Header}}
+<table>
+<tr valign=top>
+<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:=&nbsp;&nbsp;&nbsp;&nbsp;</pre></td>
+<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
+</tr>
+<tr valign=top><td colspan=2><pre style="border: 0px; padding: 0em">Loading KnotTheory` (version of August 17, 2005, 14:44:34)...</pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Crossings[Knot[0, 1]]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>0</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[3]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[Knot[0, 1]]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[Loop[1]]</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[4]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>GaussCode[Knot[0, 1]]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[]</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[5]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>BR[Knot[0, 1]]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[1, {}]</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[6]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>alex = Alexander[Knot[0, 1]][t]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>1</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[7]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Conway[Knot[0, 1]][z]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>1</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[8]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[0, 1], Knot[11, NonAlternating, 34], Knot[11, NonAlternating, 42]}</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{KnotDet[Knot[0, 1]], KnotSignature[Knot[0, 1]]}</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{1, 0}</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[10]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>J=Jones[Knot[0, 1]][q]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>1</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[11]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[0, 1]}</nowiki></pre></td></tr>
+<math>\textrm{Include}(\textrm{ColouredJonesM.mhtml})</math>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[12]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Knot[0, 1]][q]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>     -2    2
++ q   + q</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[13]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Knot[0, 1]][a, z]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>1</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[14]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{Vassiliev[2][Knot[0, 1]], Vassiliev[3][Knot[0, 1]]}</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{0, 0}</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[15]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Knot[0, 1]][q, t]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>1
+- + q
+q</nowiki></pre></td></tr>
+</table>

7 7: Difference between revisions

Revision as of 21:42, 27 August 2005

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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