7 7

7_6

8_1

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 7 7's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 7 7 at Knotilus!
	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]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 7, width is 4, Braid index is 4		[{9, 3}, {2, 7}, {8, 4}, {3, 5}, {7, 9}, {4, 1}, {6, 2}, {5, 8}, {1, 6}]

[edit Notes on presentations of 7 7]

Knot 7_7.

A graph, knot 7_7.

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["7 7"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X5,10,6,11 X3948 X9,3,10,2 X11,14,12,1 X7,13,8,12 X13,7,14,6

In[5]:=

GaussCode[K]

Out[5]=

-1, 4, -3, 1, -2, 7, -6, 3, -4, 2, -5, 6, -7, 5

In[6]:=

DTCode[K]

Out[6]=

4 8 10 12 2 14 6

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[21112]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

${\textrm {BR}}(4,\{1,-2,1,-2,3,-2,3\})$

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

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

Out[10]=

{ 4, 7, 4 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{9, 3}, {2, 7}, {8, 4}, {3, 5}, {7, 9}, {4, 1}, {6, 2}, {5, 8}, {1, 6}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

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 (syntax error): {\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 $1$ Topological 4 genus $1$ Concordance genus $2$ 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	$z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
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	Failed to parse (SVG (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	$q^{28}-q^{26}-2q^{24}+q^{22}+q^{20}-2q^{18}-q^{16}+4q^{14}+3q^{12}+2q^{8}+6q^{6}-2q^{2}+1-2q^{-2}-5q^{-4}-q^{-6}+q^{-8}-q^{-10}+q^{-12}+5q^{-14}+3q^{-16}-2q^{-18}+2q^{-22}-2q^{-24}-3q^{-26}+q^{-30}+q^{-36}+q^{-38}$
1,0,0,0	Failed to parse (SVG (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]=

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

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]=

$\{1\}$

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}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n28,}

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}} ): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["7 7"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

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

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

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} , Failed to parse (SVG (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[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{K11n28,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

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 Failed to parse (SVG (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} $8$ 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}} $-{\frac {10}{3}}$ $32$ ${\frac {112}{3}}$ ${\frac {64}{3}}$ $-8$ $-{\frac {32}{3}}$ $32$ ${\frac {56}{3}}$ ${\frac {40}{3}}$ ${\frac {2849}{30}}$ $-{\frac {218}{15}}$ ${\frac {3418}{45}}$ $-{\frac {161}{18}}$ ${\frac {449}{30}}$

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} Failed to parse (SVG (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 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}} $r=4$ 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}}

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^{12}-2 q^{11}-q^{10}+6 q^9-5 q^8-5 q^7+14 q^6-7 q^5-11 q^4+20 q^3-7 q^2-15 q+21-5 q^{-1} -14 q^{-2} +16 q^{-3} -2 q^{-4} -9 q^{-5} +8 q^{-6} -3 q^{-8} + q^{-9} }
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^{24}-2 q^{23}-q^{22}+2 q^{21}+5 q^{20}-4 q^{19}-9 q^{18}+4 q^{17}+16 q^{16}-3 q^{15}-23 q^{14}-q^{13}+31 q^{12}+5 q^{11}-38 q^{10}-11 q^9+43 q^8+19 q^7-48 q^6-23 q^5+50 q^4+28 q^3-50 q^2-32 q+49+32 q^{-1} -43 q^{-2} -34 q^{-3} +40 q^{-4} +28 q^{-5} -28 q^{-6} -28 q^{-7} +23 q^{-8} +20 q^{-9} -12 q^{-10} -17 q^{-11} +9 q^{-12} +9 q^{-13} -3 q^{-14} -5 q^{-15} +3 q^{-17} - q^{-18} }
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^{40}-2 q^{39}-q^{38}+2 q^{37}+q^{36}+6 q^{35}-8 q^{34}-7 q^{33}+2 q^{32}+4 q^{31}+25 q^{30}-14 q^{29}-23 q^{28}-10 q^{27}+3 q^{26}+62 q^{25}-7 q^{24}-38 q^{23}-39 q^{22}-16 q^{21}+106 q^{20}+17 q^{19}-39 q^{18}-76 q^{17}-54 q^{16}+141 q^{15}+50 q^{14}-24 q^{13}-109 q^{12}-98 q^{11}+162 q^{10}+79 q^9-4 q^8-129 q^7-131 q^6+167 q^5+98 q^4+16 q^3-136 q^2-149 q+156+103 q^{-1} +31 q^{-2} -125 q^{-3} -147 q^{-4} +125 q^{-5} +90 q^{-6} +44 q^{-7} -93 q^{-8} -127 q^{-9} +82 q^{-10} +61 q^{-11} +47 q^{-12} -50 q^{-13} -90 q^{-14} +40 q^{-15} +28 q^{-16} +36 q^{-17} -17 q^{-18} -47 q^{-19} +15 q^{-20} +6 q^{-21} +17 q^{-22} -2 q^{-23} -16 q^{-24} +3 q^{-25} +5 q^{-27} -3 q^{-29} + q^{-30} }
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^{60}-2 q^{59}-q^{58}+2 q^{57}+q^{56}+2 q^{55}+2 q^{54}-6 q^{53}-9 q^{52}+2 q^{51}+7 q^{50}+12 q^{49}+11 q^{48}-11 q^{47}-29 q^{46}-19 q^{45}+9 q^{44}+39 q^{43}+45 q^{42}+3 q^{41}-56 q^{40}-72 q^{39}-26 q^{38}+60 q^{37}+109 q^{36}+61 q^{35}-55 q^{34}-141 q^{33}-110 q^{32}+33 q^{31}+173 q^{30}+162 q^{29}-189 q^{27}-221 q^{26}-45 q^{25}+197 q^{24}+278 q^{23}+96 q^{22}-198 q^{21}-324 q^{20}-149 q^{19}+187 q^{18}+368 q^{17}+202 q^{16}-180 q^{15}-400 q^{14}-242 q^{13}+159 q^{12}+427 q^{11}+283 q^{10}-147 q^9-446 q^8-311 q^7+132 q^6+452 q^5+335 q^4-111 q^3-455 q^2-353 q+98+441 q^{-1} +354 q^{-2} -62 q^{-3} -420 q^{-4} -363 q^{-5} +49 q^{-6} +378 q^{-7} +341 q^{-8} - q^{-9} -335 q^{-10} -330 q^{-11} -11 q^{-12} +269 q^{-13} +283 q^{-14} +55 q^{-15} -211 q^{-16} -253 q^{-17} -50 q^{-18} +141 q^{-19} +190 q^{-20} +76 q^{-21} -95 q^{-22} -149 q^{-23} -55 q^{-24} +50 q^{-25} +91 q^{-26} +56 q^{-27} -24 q^{-28} -63 q^{-29} -33 q^{-30} +9 q^{-31} +32 q^{-32} +21 q^{-33} -15 q^{-35} -17 q^{-36} +2 q^{-37} +9 q^{-38} +4 q^{-39} -5 q^{-42} +3 q^{-44} - q^{-45} }
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^{84}-2 q^{83}-q^{82}+2 q^{81}+q^{80}+2 q^{79}-2 q^{78}+4 q^{77}-8 q^{76}-9 q^{75}+5 q^{74}+6 q^{73}+12 q^{72}+q^{71}+15 q^{70}-24 q^{69}-35 q^{68}-8 q^{67}+8 q^{66}+37 q^{65}+27 q^{64}+67 q^{63}-35 q^{62}-88 q^{61}-70 q^{60}-36 q^{59}+47 q^{58}+79 q^{57}+200 q^{56}+22 q^{55}-116 q^{54}-179 q^{53}-170 q^{52}-41 q^{51}+90 q^{50}+398 q^{49}+192 q^{48}-23 q^{47}-249 q^{46}-364 q^{45}-271 q^{44}-38 q^{43}+560 q^{42}+435 q^{41}+224 q^{40}-184 q^{39}-517 q^{38}-591 q^{37}-318 q^{36}+595 q^{35}+650 q^{34}+561 q^{33}+16 q^{32}-556 q^{31}-899 q^{30}-672 q^{29}+511 q^{28}+775 q^{27}+886 q^{26}+275 q^{25}-498 q^{24}-1130 q^{23}-1000 q^{22}+373 q^{21}+823 q^{20}+1137 q^{19}+507 q^{18}-404 q^{17}-1281 q^{16}-1245 q^{15}+243 q^{14}+828 q^{13}+1305 q^{12}+674 q^{11}-315 q^{10}-1362 q^9-1398 q^8+133 q^7+804 q^6+1395 q^5+787 q^4-225 q^3-1371 q^2-1467 q+18+729 q^{-1} +1400 q^{-2} +862 q^{-3} -102 q^{-4} -1279 q^{-5} -1449 q^{-6} -118 q^{-7} +572 q^{-8} +1288 q^{-9} +890 q^{-10} +66 q^{-11} -1058 q^{-12} -1311 q^{-13} -249 q^{-14} +332 q^{-15} +1031 q^{-16} +825 q^{-17} +241 q^{-18} -724 q^{-19} -1034 q^{-20} -312 q^{-21} +76 q^{-22} +669 q^{-23} +640 q^{-24} +335 q^{-25} -373 q^{-26} -667 q^{-27} -264 q^{-28} -90 q^{-29} +321 q^{-30} +383 q^{-31} +299 q^{-32} -127 q^{-33} -331 q^{-34} -145 q^{-35} -122 q^{-36} +100 q^{-37} +161 q^{-38} +183 q^{-39} -22 q^{-40} -124 q^{-41} -44 q^{-42} -74 q^{-43} +14 q^{-44} +44 q^{-45} +79 q^{-46} - q^{-47} -35 q^{-48} -6 q^{-49} -27 q^{-50} +6 q^{-52} +26 q^{-53} -2 q^{-54} -9 q^{-55} +3 q^{-56} -7 q^{-57} +5 q^{-60} -3 q^{-62} + q^{-63} }
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^{112}-2 q^{111}-q^{110}+2 q^{109}+q^{108}+2 q^{107}-2 q^{106}+2 q^{104}-8 q^{103}-6 q^{102}+4 q^{101}+6 q^{100}+14 q^{99}+2 q^{98}-2 q^{97}+5 q^{96}-27 q^{95}-28 q^{94}-9 q^{93}+8 q^{92}+49 q^{91}+37 q^{90}+25 q^{89}+25 q^{88}-58 q^{87}-93 q^{86}-81 q^{85}-53 q^{84}+76 q^{83}+123 q^{82}+140 q^{81}+149 q^{80}-30 q^{79}-165 q^{78}-250 q^{77}-275 q^{76}-44 q^{75}+154 q^{74}+325 q^{73}+461 q^{72}+217 q^{71}-77 q^{70}-391 q^{69}-661 q^{68}-446 q^{67}-101 q^{66}+368 q^{65}+854 q^{64}+741 q^{63}+372 q^{62}-239 q^{61}-984 q^{60}-1057 q^{59}-750 q^{58}-4 q^{57}+1036 q^{56}+1344 q^{55}+1174 q^{54}+369 q^{53}-956 q^{52}-1583 q^{51}-1633 q^{50}-821 q^{49}+776 q^{48}+1739 q^{47}+2061 q^{46}+1329 q^{45}-498 q^{44}-1789 q^{43}-2451 q^{42}-1857 q^{41}+147 q^{40}+1772 q^{39}+2782 q^{38}+2352 q^{37}+228 q^{36}-1684 q^{35}-3024 q^{34}-2818 q^{33}-617 q^{32}+1557 q^{31}+3231 q^{30}+3224 q^{29}+954 q^{28}-1422 q^{27}-3361 q^{26}-3554 q^{25}-1278 q^{24}+1274 q^{23}+3479 q^{22}+3840 q^{21}+1535 q^{20}-1168 q^{19}-3540 q^{18}-4050 q^{17}-1755 q^{16}+1044 q^{15}+3592 q^{14}+4235 q^{13}+1934 q^{12}-955 q^{11}-3618 q^{10}-4354 q^9-2082 q^8+849 q^7+3599 q^6+4443 q^5+2233 q^4-730 q^3-3566 q^2-4497 q-2339+596 q^{-1} +3437 q^{-2} +4481 q^{-3} +2486 q^{-4} -398 q^{-5} -3297 q^{-6} -4431 q^{-7} -2556 q^{-8} +194 q^{-9} +3011 q^{-10} +4263 q^{-11} +2666 q^{-12} +81 q^{-13} -2708 q^{-14} -4040 q^{-15} -2651 q^{-16} -333 q^{-17} +2251 q^{-18} +3675 q^{-19} +2636 q^{-20} +618 q^{-21} -1817 q^{-22} -3250 q^{-23} -2449 q^{-24} -819 q^{-25} +1279 q^{-26} +2701 q^{-27} +2262 q^{-28} +983 q^{-29} -853 q^{-30} -2162 q^{-31} -1892 q^{-32} -1008 q^{-33} +393 q^{-34} +1587 q^{-35} +1567 q^{-36} +988 q^{-37} -138 q^{-38} -1116 q^{-39} -1131 q^{-40} -838 q^{-41} -98 q^{-42} +682 q^{-43} +818 q^{-44} +686 q^{-45} +161 q^{-46} -414 q^{-47} -495 q^{-48} -478 q^{-49} -196 q^{-50} +201 q^{-51} +287 q^{-52} +334 q^{-53} +160 q^{-54} -103 q^{-55} -149 q^{-56} -195 q^{-57} -108 q^{-58} +41 q^{-59} +57 q^{-60} +108 q^{-61} +84 q^{-62} -17 q^{-63} -34 q^{-64} -60 q^{-65} -34 q^{-66} +16 q^{-67} -2 q^{-68} +23 q^{-69} +27 q^{-70} -6 q^{-72} -17 q^{-73} -7 q^{-74} +9 q^{-75} -3 q^{-76} +7 q^{-78} -5 q^{-81} +3 q^{-83} - q^{-84} }

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

7_6 8_1