7 7

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	Failed to parse (SVG (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)	$a^{-4}-a^{2}z^{2}-2z^{2}a^{-2}-2a^{-2}+z^{4}+2z^{2}+2$
Kauffman polynomial (db, data sources)	$z^{4}a^{-4}-2z^{2}a^{-4}+a^{-4}+2z^{5}a^{-3}+a^{3}z^{3}-4z^{3}a^{-3}+2za^{-3}+z^{6}a^{-2}+3a^{2}z^{4}+2z^{4}a^{-2}-3a^{2}z^{2}-6z^{2}a^{-2}+2a^{-2}+3az^{5}+5z^{5}a^{-1}-3az^{3}-8z^{3}a^{-1}+az+3za^{-1}+z^{6}+4z^{4}-7z^{2}+2$
The A2 invariant	$-q^{10}+q^{8}+q^{6}+2q^{2}+q^{-2}-q^{-4}-q^{-6}-q^{-10}+q^{-12}+q^{-14}$
The G2 invariant	$q^{52}-2q^{50}+3q^{48}-4q^{46}+q^{42}-4q^{40}+9q^{38}-9q^{36}+9q^{34}-3q^{32}-4q^{30}+9q^{28}-10q^{26}+9q^{24}-5q^{22}-q^{20}+5q^{18}-4q^{16}+4q^{14}+2q^{12}-7q^{10}+10q^{8}-5q^{6}-2q^{4}+8q^{2}-12+17q^{-2}-11q^{-4}+5q^{-6}+3q^{-8}-9q^{-10}+15q^{-12}-14q^{-14}+6q^{-16}-q^{-18}-4q^{-20}+6q^{-22}-6q^{-24}+q^{-26}+3q^{-28}-7q^{-30}+5q^{-32}-4q^{-34}-5q^{-36}+10q^{-38}-11q^{-40}+9q^{-42}-4q^{-44}-q^{-46}+7q^{-48}-8q^{-50}+9q^{-52}-4q^{-54}+q^{-56}+q^{-58}-3q^{-60}+3q^{-62}-q^{-64}+q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 7_7.

A1 Invariants.

Weight	Invariant
1	$-q^{7}+2q^{5}+q-q^{-3}+q^{-5}-q^{-7}+q^{-9}$
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	$-q^{39}+2q^{37}+2q^{35}-3q^{33}-5q^{31}+q^{29}+10q^{27}-2q^{25}-11q^{23}+14q^{19}+3q^{17}-13q^{15}-5q^{13}+12q^{11}+6q^{9}-9q^{7}-5q^{5}+4q^{3}+6q-q^{-1}-5q^{-3}-4q^{-5}+5q^{-7}+7q^{-9}-2q^{-11}-9q^{-13}+3q^{-15}+13q^{-17}-q^{-19}-13q^{-21}-3q^{-23}+12q^{-25}+4q^{-27}-11q^{-29}-6q^{-31}+8q^{-33}+7q^{-35}-4q^{-37}-6q^{-39}+2q^{-41}+4q^{-43}-2q^{-47}-q^{-49}+q^{-51}$
4	$q^{64}-2q^{62}-2q^{60}+3q^{58}+3q^{56}+5q^{54}-8q^{52}-10q^{50}+2q^{48}+8q^{46}+20q^{44}-11q^{42}-26q^{40}-7q^{38}+15q^{36}+40q^{34}-3q^{32}-36q^{30}-25q^{28}+8q^{26}+50q^{24}+13q^{22}-30q^{20}-33q^{18}-4q^{16}+39q^{14}+19q^{12}-13q^{10}-26q^{8}-13q^{6}+18q^{4}+16q^{2}+5-10q^{-2}-15q^{-4}-4q^{-6}+14q^{-8}+21q^{-10}+q^{-12}-18q^{-14}-23q^{-16}+10q^{-18}+30q^{-20}+10q^{-22}-19q^{-24}-40q^{-26}+4q^{-28}+37q^{-30}+22q^{-32}-11q^{-34}-46q^{-36}-8q^{-38}+29q^{-40}+30q^{-42}+6q^{-44}-38q^{-46}-19q^{-48}+10q^{-50}+25q^{-52}+18q^{-54}-19q^{-56}-18q^{-58}-6q^{-60}+10q^{-62}+16q^{-64}-3q^{-66}-6q^{-68}-6q^{-70}+6q^{-74}+q^{-76}-2q^{-80}-q^{-82}+q^{-84}$
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	Failed to parse (SVG (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} }
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	Failed to parse (SVG (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^{36}-2 q^{32}-2 q^{30}+q^{28}+3 q^{26}-3 q^{22}-q^{20}+4 q^{18}+3 q^{16}-2 q^{12}+q^{10}+2 q^8+q^6-3 q^4-q^2+2+ q^{-2} -2 q^{-4} -2 q^{-6} + q^{-8} +2 q^{-10} -2 q^{-14} + q^{-16} +3 q^{-18} + q^{-20} -3 q^{-22} -2 q^{-24} +2 q^{-26} +3 q^{-28} - q^{-30} -3 q^{-32} - q^{-34} +2 q^{-36} +2 q^{-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. 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]=

Failed to parse (SVG (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]=

Failed to parse (SVG (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]=

$a^{-4}-a^{2}z^{2}-2z^{2}a^{-2}-2a^{-2}+z^{4}+2z^{2}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{4}a^{-4}-2z^{2}a^{-4}+a^{-4}+2z^{5}a^{-3}+a^{3}z^{3}-4z^{3}a^{-3}+2za^{-3}+z^{6}a^{-2}+3a^{2}z^{4}+2z^{4}a^{-2}-3a^{2}z^{2}-6z^{2}a^{-2}+2a^{-2}+3az^{5}+5z^{5}a^{-1}-3az^{3}-8z^{3}a^{-1}+az+3za^{-1}+z^{6}+4z^{4}-7z^{2}+2$