5 1

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 n} &id=Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} 5 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 5 1 at Knotilus!

[edit 5 1 Quick Notes]

An interlaced pentagram, this is known variously as the "Cinquefoil Knot", after certain herbs and shrubs of the rose family which have 5-lobed leaves and 5-petaled flowers (see e.g. [4]), as the "Pentafoil Knot" (visit Bert Jagers' pentafoil page), as the "Double Overhand Knot", as 5_1, or finally as the torus knot T(5,2).

When taken off the post the strangle knot (hitch) of practical knot tying deforms to 5_1

[edit 5 1 Further Notes and Views]

A kolam of a 2x3 dot array	The VISA Interlink Logo [1]	Version of the US bicentennial emblem
A pentagonal table by Bob Mackay [2]	The Utah State Parks logo	As impossible object ("Penrose" pentagram)
Folded ribbon which is single-sided (more complex version of Möbius Strip).	Non-pentagonal shape.	Pentagram of circles.
Alternate pentagram of intersecting circles.	3D-looking rendition.	Partial view of US bicentennial logo on a shirt seen in Lisboa [3]
Non-prime knot with two 5_1 configurations on a closed loop.	Knotted epitrochoid	Sum of two 5_1s, Vienna, orthodox church

This sentence was last edited by Dror. Sometime later, Scott added this sentence.

Knot presentations

Planar diagram presentation	X₁₆₂₇ X₃₈₄₉ X_5,10,6,1 X₇₂₈₃ X_9,4,10,5
Gauss code	-1, 4, -2, 5, -3, 1, -4, 2, -5, 3
Dowker-Thistlethwaite code	6 8 10 2 4
Conway Notation	[5]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 5, width is 2,

Braid index is 2

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

[edit Notes on presentations of 5 1]

Knot 5_1.

A graph, knot 5_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.org/wiki/KnotTheory.

In[3]:= K = Knot["5 1"];

In[4]:= PD[K]

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

Out[4]= X₁₆₂₇ X₃₈₄₉ X_5,10,6,1 X₇₂₈₃ X_9,4,10,5

In[5]:= GaussCode[K]

Out[5]= -1, 4, -2, 5, -3, 1, -4, 2, -5, 3

In[6]:= DTCode[K]

Out[6]= 6 8 10 2 4

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:= ConwayNotation[K]

Out[8]= [5]

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]= Failed to parse (SVG (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{BR}(2,\{-1,-1,-1,-1,-1\})}

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]= { 2, 5, 2 }

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

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	2
Bridge index	2
Super bridge index	3
Nakanishi index	1
Maximal Thurston-Bennequin number	[-10][3]
Hyperbolic Volume	Not hyperbolic
A-Polynomial	See Data:5 1/A-polynomial

[edit Notes for 5 1'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 2}
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 2}
Concordance genus	${\textrm {ConcordanceGenus}}({\textrm {Knot}}(5,1))$
Rasmussen s-Invariant	-4

[edit Notes for 5 1'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} -t- t^{-1} +1}
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+3 z^2+1}
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 5, -4 }
Jones polynomial	$-q^{-7}+q^{-6}-q^{-5}+q^{-4}+q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$a^{6}\left(-z^{2}\right)-2a^{6}+a^{4}z^{4}+4a^{4}z^{2}+3a^{4}$
Kauffman polynomial (db, data sources)	$a^{9}z+a^{8}z^{2}+a^{7}z^{3}-a^{7}z+a^{6}z^{4}-3a^{6}z^{2}+2a^{6}+a^{5}z^{3}-2a^{5}z+a^{4}z^{4}-4a^{4}z^{2}+3a^{4}$
The A2 invariant	$-q^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2q^{10}+q^{8}+q^{6}$
The G2 invariant	$q^{120}-q^{100}-q^{98}-q^{92}-q^{90}-q^{88}-q^{82}-q^{80}-q^{78}-q^{72}+q^{58}+q^{56}+q^{52}+2q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+2q^{40}+q^{38}+q^{34}+q^{32}+q^{30}$

Further Quantum Invariants

Further quantum knot invariants for 5_1.

A1 Invariants.

Weight	Invariant
1	$-q^{15}+q^{7}+q^{5}+q^{3}$
2	$q^{40}-q^{32}-q^{30}-q^{28}+q^{14}+q^{12}+q^{10}+q^{8}+q^{6}$
3	$-q^{75}+q^{67}+q^{65}+q^{63}-q^{49}-q^{47}-q^{45}-q^{43}-q^{41}+q^{21}+q^{19}+q^{17}+q^{15}+q^{13}+q^{11}+q^{9}$
4	$q^{120}-q^{112}-q^{110}-q^{108}+q^{94}+q^{92}+q^{90}+q^{88}+q^{86}-q^{66}-q^{64}-q^{62}-q^{60}-q^{58}-q^{56}-q^{54}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12}$
5	$-q^{175}+q^{167}+q^{165}+q^{163}-q^{149}-q^{147}-q^{145}-q^{143}-q^{141}+q^{121}+q^{119}+q^{117}+q^{115}+q^{113}+q^{111}+q^{109}-q^{83}-q^{81}-q^{79}-q^{77}-q^{75}-q^{73}-q^{71}-q^{69}-q^{67}+q^{35}+q^{33}+q^{31}+q^{29}+q^{27}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15}$
6	$q^{240}-q^{232}-q^{230}-q^{228}+q^{214}+q^{212}+q^{210}+q^{208}+q^{206}-q^{186}-q^{184}-q^{182}-q^{180}-q^{178}-q^{176}-q^{174}+q^{148}+q^{146}+q^{144}+q^{142}+q^{140}+q^{138}+q^{136}+q^{134}+q^{132}-q^{100}-q^{98}-q^{96}-q^{94}-q^{92}-q^{90}-q^{88}-q^{86}-q^{84}-q^{82}-q^{80}+q^{42}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}$
8	$q^{400}-q^{392}-q^{390}-q^{388}+q^{374}+q^{372}+q^{370}+q^{368}+q^{366}-q^{346}-q^{344}-q^{342}-q^{340}-q^{338}-q^{336}-q^{334}+q^{308}+q^{306}+q^{304}+q^{302}+q^{300}+q^{298}+q^{296}+q^{294}+q^{292}-q^{260}-q^{258}-q^{256}-q^{254}-q^{252}-q^{250}-q^{248}-q^{246}-q^{244}-q^{242}-q^{240}+q^{202}+q^{200}+q^{198}+q^{196}+q^{194}+q^{192}+q^{190}+q^{188}+q^{186}+q^{184}+q^{182}+q^{180}+q^{178}-q^{134}-q^{132}-q^{130}-q^{128}-q^{126}-q^{124}-q^{122}-q^{120}-q^{118}-q^{116}-q^{114}-q^{112}-q^{110}-q^{108}-q^{106}+q^{56}+q^{54}+q^{52}+q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2q^{10}+q^{8}+q^{6}$
1,1	$q^{60}-2q^{36}-2q^{34}-4q^{32}-4q^{30}-3q^{28}+2q^{24}+4q^{22}+5q^{20}+4q^{18}+4q^{16}+2q^{14}+q^{12}$
2,0	$q^{54}+q^{52}+2q^{50}+q^{48}-2q^{44}-3q^{42}-3q^{40}-3q^{38}-2q^{36}-q^{34}+q^{28}+q^{26}+2q^{24}+2q^{22}+3q^{20}+2q^{18}+2q^{16}+q^{14}+q^{12}$
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^{96}-q^{94}-2 q^{92}-2 q^{90}-q^{88}+q^{86}+3 q^{84}+4 q^{82}+5 q^{80}+4 q^{78}+4 q^{76}+2 q^{74}+q^{72}-q^{70}-2 q^{68}-3 q^{66}-4 q^{64}-5 q^{62}-5 q^{60}-5 q^{58}-4 q^{56}-3 q^{54}-2 q^{52}-q^{50}+q^{42}+q^{40}+2 q^{38}+2 q^{36}+3 q^{34}+3 q^{32}+4 q^{30}+3 q^{28}+3 q^{26}+2 q^{24}+2 q^{22}+q^{20}+q^{18}}

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^{50}-q^{36}-2 q^{34}-3 q^{32}-3 q^{30}-2 q^{28}-q^{26}+2 q^{24}+3 q^{22}+4 q^{20}+3 q^{18}+3 q^{16}+q^{14}+q^{12}}
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^{29}-q^{27}-2 q^{25}-q^{23}+q^{19}+2 q^{17}+2 q^{15}+2 q^{13}+q^{11}+q^9}
1,0,1	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^{80}+q^{58}+q^{56}+3 q^{54}+3 q^{52}+2 q^{50}-q^{48}-5 q^{46}-9 q^{44}-12 q^{42}-12 q^{40}-9 q^{38}-3 q^{36}+2 q^{34}+7 q^{32}+10 q^{30}+11 q^{28}+10 q^{26}+7 q^{24}+5 q^{22}+2 q^{20}+q^{18}}

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^{64}+q^{62}+q^{60}+q^{58}+q^{56}-q^{50}-2 q^{48}-4 q^{46}-5 q^{44}-6 q^{42}-6 q^{40}-4 q^{38}-q^{36}+2 q^{34}+4 q^{32}+7 q^{30}+6 q^{28}+6 q^{26}+4 q^{24}+3 q^{22}+q^{20}+q^{18}}
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^{36}-q^{34}-2 q^{32}-2 q^{30}-q^{28}+q^{24}+2 q^{22}+3 q^{20}+2 q^{18}+2 q^{16}+q^{14}+q^{12}}

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^{50}-q^{36}-q^{32}-q^{30}+q^{26}+q^{22}+2 q^{20}+q^{18}+q^{16}+q^{14}+q^{12}}
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^{80}-q^{58}-q^{56}-q^{54}-q^{52}-2 q^{50}-q^{48}-q^{46}-q^{44}+q^{38}+q^{36}+2 q^{34}+q^{32}+2 q^{30}+q^{28}+2 q^{26}+q^{24}+q^{22}+q^{18}}

B3 Invariants.

Weight	Invariant
1,0,0	$q^{120}-q^{86}-q^{82}-q^{80}-2q^{78}-q^{76}-2q^{74}-q^{72}-2q^{70}-q^{68}-q^{66}-q^{64}+q^{58}+q^{56}+2q^{54}+q^{52}+3q^{50}+q^{48}+3q^{46}+q^{44}+2q^{42}+q^{40}+2q^{38}+q^{34}+q^{30}$

B4 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^{160}-q^{114}-q^{110}-2 q^{106}-q^{104}-2 q^{102}-q^{100}-2 q^{98}-q^{96}-2 q^{94}-q^{92}-2 q^{90}-q^{88}-q^{86}+q^{78}+2 q^{74}+q^{72}+3 q^{70}+q^{68}+3 q^{66}+q^{64}+3 q^{62}+q^{60}+3 q^{58}+q^{56}+2 q^{54}+2 q^{50}+q^{46}+q^{42}}

C3 Invariants.

Weight	Invariant
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^{70}-q^{50}-q^{48}-q^{46}-q^{44}-q^{42}-q^{40}+q^{34}+q^{32}+2 q^{30}+2 q^{28}+2 q^{26}+q^{24}+2 q^{22}+q^{20}+q^{18}}

C4 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^{90}-q^{64}-q^{62}-2 q^{60}-q^{58}-q^{56}-q^{54}-q^{52}-q^{50}-q^{48}+q^{44}+2 q^{42}+2 q^{40}+2 q^{38}+2 q^{36}+2 q^{34}+2 q^{32}+2 q^{30}+2 q^{28}+q^{26}+q^{24}}

D4 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^{120}-q^{100}-q^{98}-3 q^{96}-3 q^{94}-q^{92}-q^{90}+2 q^{88}+6 q^{86}+9 q^{84}+11 q^{82}+14 q^{80}+11 q^{78}+9 q^{76}+3 q^{74}-4 q^{72}-12 q^{70}-18 q^{68}-24 q^{66}-27 q^{64}-27 q^{62}-24 q^{60}-17 q^{58}-11 q^{56}+7 q^{52}+14 q^{50}+19 q^{48}+22 q^{46}+19 q^{44}+19 q^{42}+14 q^{40}+10 q^{38}+6 q^{36}+4 q^{34}+q^{32}+q^{30}}

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^{70}-q^{50}-q^{48}-3 q^{46}-3 q^{44}-3 q^{42}-3 q^{40}-2 q^{38}+q^{34}+3 q^{32}+4 q^{30}+4 q^{28}+4 q^{26}+3 q^{24}+2 q^{22}+q^{20}+q^{18}}

G2 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^{240}-q^{198}-q^{192}+q^{178}+q^{176}+q^{172}+2 q^{170}+q^{168}+q^{166}+q^{164}+q^{162}+2 q^{160}+q^{158}+q^{154}+q^{152}-q^{148}-q^{146}-q^{144}-q^{142}-2 q^{140}-3 q^{138}-2 q^{136}-2 q^{134}-3 q^{132}-4 q^{130}-4 q^{128}-3 q^{126}-3 q^{124}-4 q^{122}-4 q^{120}-3 q^{118}-2 q^{116}-2 q^{114}-3 q^{112}-2 q^{110}+q^{102}+q^{100}+2 q^{98}+3 q^{96}+2 q^{94}+2 q^{92}+4 q^{90}+3 q^{88}+3 q^{86}+4 q^{84}+3 q^{82}+3 q^{80}+4 q^{78}+2 q^{76}+2 q^{74}+3 q^{72}+2 q^{70}+q^{68}+2 q^{66}+q^{64}+q^{62}+q^{60}+q^{54}}

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^{120}-q^{100}-q^{98}-q^{92}-q^{90}-q^{88}-q^{82}-q^{80}-q^{78}-q^{72}+q^{58}+q^{56}+q^{52}+2 q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+2 q^{40}+q^{38}+q^{34}+q^{32}+q^{30}}

.

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["5 1"];

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} -t- t^{-1} +1}

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+3 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]= { 5, -4 }

In[8]:= Jones[K][q]

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

Out[8]= $-q^{-7}+q^{-6}-q^{-5}+q^{-4}+q^{-2}$

In[9]:= HOMFLYPT[K][a, z]

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

Out[9]= $a^{6}\left(-z^{2}\right)-2a^{6}+a^{4}z^{4}+4a^{4}z^{2}+3a^{4}$

In[10]:= Kauffman[K][a, z]

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

Out[10]= $a^{9}z+a^{8}z^{2}+a^{7}z^{3}-a^{7}z+a^{6}z^{4}-3a^{6}z^{2}+2a^{6}+a^{5}z^{3}-2a^{5}z+a^{4}z^{4}-4a^{4}z^{2}+3a^{4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {[[0_1]], [[K11n34]], [[K11n42]], }

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.

In[3]:= K = Knot["5 1"];

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} -t- t^{-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^{-7} + q^{-6} - q^{-5} + q^{-4} + q^{-2} } }

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]= {[[0_1]], [[K11n34]], [[K11n42]], }

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:

(3, -5)

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 12}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -40}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 72}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 174}

26

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -480}

Failed to parse (SVG (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{2512}{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{448}{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 -104}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 288}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 800}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2088}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 312}

Failed to parse (SVG (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{41151}{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 \frac{2494}{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{7634}{5}}

{\frac {43}{2}}

{\frac {1951}{10}}

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=} -4 is the signature of 5 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

0

χ

-3

1

-5

1

-7

1

-9

0

-11

1

0

-13

0

-15

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=-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 i=-3}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-5}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (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}}
Failed to parse (SVG (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}	Failed to parse (SVG (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 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}}	Failed to parse (SVG (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 \textrm{Apart}\left[\frac{\textrm{Hold}\left[\textrm{REngine}\left(\textrm{MorseLink}(\textrm{MorseLink::Error: bad input}),\left( \begin{array}{ccccccccc} q & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & q^2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & q^3 & 0 & 0 \\ 0 & q^2 & 0 & q-q^3 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & q^2 & 0 & -(q-1) \left(q^{5/4}+\sqrt[4]{q}\right)^2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^2 & 0 \\ 0 & 0 & q^3 & 0 & q^{5/2}-q^{7/2} & 0 & (q-1)^2 q (q+1) & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & q^2 & 0 & q-q^3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q \end{array} \right),\left( \begin{array}{ccccccccc} \frac{1}{q} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{q^2-1}{q^3} & 0 & \frac{1}{q^2} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{(q-1)^2 (q+1)}{q^4} & 0 & \frac{q-1}{q^{5/2}} & 0 & \frac{1}{q^3} & 0 & 0 \\ 0 & \frac{1}{q^2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{(q-1) (q+1)^2}{q^{9/2}} & 0 & \frac{1}{q^2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{q^2-1}{q^3} & 0 & \frac{1}{q^2} & 0 \\ 0 & 0 & \frac{1}{q^3} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{q^2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{q} \end{array} \right),\left( \begin{array}{ccc} 0 & 0 & \frac{1}{\sqrt{q}} \\ 0 & 1 & 0 \\ \sqrt{q} & 0 & 0 \end{array} \right),\left( \begin{array}{ccc} 0 & 0 & \frac{1}{\sqrt{q}} \\ 0 & 1 & 0 \\ \sqrt{q} & 0 & 0 \end{array} \right),\left( \begin{array}{ccc} 0 & 0 & \frac{1}{\sqrt{q}} \\ 0 & 1 & 0 \\ \sqrt{q} & 0 & 0 \end{array} \right),\left( \begin{array}{ccc} 0 & 0 & \frac{1}{\sqrt{q}} \\ 0 & 1 & 0 \\ \sqrt{q} & 0 & 0 \end{array} \right)\right)\right]}{q+\frac{1}{q}+1}\right]}
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 \textrm{Apart}\left[\frac{\textrm{Hold}\left[\textrm{REngine}\left(\textrm{MorseLink}(\textrm{MorseLink::Error: bad input}),\left( \begin{array}{cccccccccccccccc} q^{3/2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & q^3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{9/2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^6 & 0 & 0 & 0 \\ 0 & q^3 & 0 & 0 & q^{3/2}-q^{9/2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & q^{7/2} & 0 & 0 & -q^{3/2} (q+1) \left(q^3-1\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^4 & 0 & 0 & -(q-1) q^{3/2} \left(q^2+q+1\right)^2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{9/2} & 0 & 0 \\ 0 & 0 & q^{9/2} & 0 & 0 & q^{7/2}-q^{11/2} & 0 & 0 & q^{13/2}-q^{9/2}-q^{7/2}+q^{3/2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & q^4 & 0 & 0 & -(q-1) q^{5/2} (q+1)^2 & 0 & 0 & (q+1) \left(q^3-1\right)^2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{7/2} & 0 & 0 & -q^{3/2} (q+1) \left(q^3-1\right) & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^3 & 0 \\ 0 & 0 & 0 & q^6 & 0 & 0 & q^{11/2}-q^{13/2} & 0 & 0 & (q-1)^2 q^4 (q+1) & 0 & 0 & -(q-1)^3 q^{3/2} (q+1) \left(q^2+q+1\right) & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{9/2} & 0 & 0 & q^{7/2}-q^{11/2} & 0 & 0 & q^{13/2}-q^{9/2}-q^{7/2}+q^{3/2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^3 & 0 & 0 & q^{3/2}-q^{9/2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{3/2} \end{array} \right),\left( \begin{array}{cccccccccccccccc} \frac{1}{q^{3/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{q^3-1}{q^{9/2}} & 0 & 0 & \frac{1}{q^3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{q^5-q^3-q^2+1}{q^{13/2}} & 0 & 0 & \frac{q^2-1}{q^{9/2}} & 0 & 0 & \frac{1}{q^{9/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{(q-1)^3 (q+1) \left(q^2+q+1\right)}{q^{15/2}} & 0 & 0 & \frac{(q-1)^2 (q+1)}{q^5} & 0 & 0 & \frac{q-1}{q^{9/2}} & 0 & 0 & \frac{1}{q^6} & 0 & 0 & 0 \\ 0 & \frac{1}{q^3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{(q+1) \left(q^3-1\right)}{q^{13/2}} & 0 & 0 & \frac{1}{q^{7/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{(q+1) \left(q^3-1\right)^2}{q^9} & 0 & 0 & \frac{(q-1) (q+1)^2}{q^{11/2}} & 0 & 0 & \frac{1}{q^4} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{q^5-q^3-q^2+1}{q^{13/2}} & 0 & 0 & \frac{q^2-1}{q^{9/2}} & 0 & 0 & \frac{1}{q^{9/2}} & 0 & 0 \\ 0 & 0 & \frac{1}{q^{9/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{(q-1) \left(q^2+q+1\right)^2}{q^{17/2}} & 0 & 0 & \frac{1}{q^4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{(q+1) \left(q^3-1\right)}{q^{13/2}} & 0 & 0 & \frac{1}{q^{7/2}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{q^3-1}{q^{9/2}} & 0 & 0 & \frac{1}{q^3} & 0 \\ 0 & 0 & 0 & \frac{1}{q^6} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{q^{9/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{q^3} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{q^{3/2}} \end{array} \right),\left( \begin{array}{cccc} 0 & 0 & 0 & \frac{1}{q^{3/4}} \\ 0 & 0 & \frac{1}{\sqrt[4]{q}} & 0 \\ 0 & \sqrt[4]{q} & 0 & 0 \\ q^{3/4} & 0 & 0 & 0 \end{array} \right),\left( \begin{array}{cccc} 0 & 0 & 0 & \frac{1}{q^{3/4}} \\ 0 & 0 & \frac{1}{\sqrt[4]{q}} & 0 \\ 0 & \sqrt[4]{q} & 0 & 0 \\ q^{3/4} & 0 & 0 & 0 \end{array} \right),\left( \begin{array}{cccc} 0 & 0 & 0 & \frac{1}{q^{3/4}} \\ 0 & 0 & \frac{1}{\sqrt[4]{q}} & 0 \\ 0 & \sqrt[4]{q} & 0 & 0 \\ q^{3/4} & 0 & 0 & 0 \end{array} \right),\left( \begin{array}{cccc} 0 & 0 & 0 & \frac{1}{q^{3/4}} \\ 0 & 0 & \frac{1}{\sqrt[4]{q}} & 0 \\ 0 & \sqrt[4]{q} & 0 & 0 \\ q^{3/4} & 0 & 0 & 0 \end{array} \right)\right)\right]}{q^{3/2}+\sqrt{q}+\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}}\right]}
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 \textrm{Apart}\left[\frac{\textrm{Hold}\left[\textrm{REngine}\left(\textrm{MorseLink}(\textrm{MorseLink::Error: bad input}),\left( \begin{array}{ccccccccccccccccccccccccc} q^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & q^4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{10} & 0 & 0 & 0 & 0 \\ 0 & q^4 & 0 & 0 & 0 & q^2-q^6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & q^5 & 0 & 0 & 0 & -q^{15/2}-q^{13/2}+q^{7/2}+q^{5/2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^6 & 0 & 0 & 0 & -q^3 \left(q^2+q+1\right) \left(q^4-1\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^7 & 0 & 0 & 0 & -(q-1) q^{7/2} \left(q^3+q^2+q+1\right)^2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^8 & 0 & 0 & 0 \\ 0 & 0 & q^6 & 0 & 0 & 0 & q^{9/2}-q^{15/2} & 0 & 0 & 0 & q^9-q^6-q^5+q^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^6 & 0 & 0 & 0 & -q^4 (q+1) \left(q^3-1\right) & 0 & 0 & 0 & q \left(q^3-1\right)^2 \left(q^3+q^2+q+1\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^6 & 0 & 0 & 0 & -(q-1) q^{7/2} \left(q^2+q+1\right)^2 & 0 & 0 & 0 & (q+1) \left(q^5+q^3-q^2-1\right)^2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^6 & 0 & 0 & 0 & -q^3 \left(q^2+q+1\right) \left(q^4-1\right) & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^6 & 0 & 0 \\ 0 & 0 & 0 & q^8 & 0 & 0 & 0 & q^7-q^9 & 0 & 0 & 0 & q^{10}-q^8-q^7+q^5 & 0 & 0 & 0 & -(q-1)^3 q^2 (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^7 & 0 & 0 & 0 & -(q-1) q^{11/2} (q+1)^2 & 0 & 0 & 0 & q^3 (q+1) \left(q^3-1\right)^2 & 0 & 0 & 0 & -\frac{\left(q^2-1\right)^3 \left(q^2+1\right)^2 \left(q^2+q+1\right)}{\sqrt{q}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^6 & 0 & 0 & 0 & -q^4 (q+1) \left(q^3-1\right) & 0 & 0 & 0 & q \left(q^3-1\right)^2 \left(q^3+q^2+q+1\right) & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^5 & 0 & 0 & 0 & -q^{15/2}-q^{13/2}+q^{7/2}+q^{5/2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^4 & 0 \\ 0 & 0 & 0 & 0 & q^{10} & 0 & 0 & 0 & q^{19/2}-q^{21/2} & 0 & 0 & 0 & (q-1)^2 q^8 (q+1) & 0 & 0 & 0 & -(q-1)^3 q^{11/2} (q+1) \left(q^2+q+1\right) & 0 & 0 & 0 & (q-1)^4 q^2 (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right) & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^8 & 0 & 0 & 0 & q^7-q^9 & 0 & 0 & 0 & q^{10}-q^8-q^7+q^5 & 0 & 0 & 0 & -(q-1)^3 q^2 (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right) & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^6 & 0 & 0 & 0 & q^{9/2}-q^{15/2} & 0 & 0 & 0 & q^9-q^6-q^5+q^2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^4 & 0 & 0 & 0 & q^2-q^6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^2 \end{array} \right),\left( \begin{array}{ccccccccccccccccccccccccc} \frac{1}{q^2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{q^4-1}{q^6} & 0 & 0 & 0 & \frac{1}{q^4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{q^7-q^4-q^3+1}{q^9} & 0 & 0 & 0 & \frac{q^3-1}{q^{13/2}} & 0 & 0 & 0 & \frac{1}{q^6} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{(q-1)^3 (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right)}{q^{11}} & 0 & 0 & 0 & \frac{q^5-q^3-q^2+1}{q^8} & 0 & 0 & 0 & \frac{q^2-1}{q^7} & 0 & 0 & 0 & \frac{1}{q^8} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{(q-1)^4 (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right)}{q^{12}} & 0 & 0 & 0 & \frac{(q-1)^3 (q+1) \left(q^2+q+1\right)}{q^{17/2}} & 0 & 0 & 0 & \frac{(q-1)^2 (q+1)}{q^7} & 0 & 0 & 0 & \frac{q-1}{q^{15/2}} & 0 & 0 & 0 & \frac{1}{q^{10}} & 0 & 0 & 0 & 0 \\ 0 & \frac{1}{q^4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{q^5+q^4-q-1}{q^{17/2}} & 0 & 0 & 0 & \frac{1}{q^5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{\left(q^3-1\right)^2 \left(q^3+q^2+q+1\right)}{q^{12}} & 0 & 0 & 0 & \frac{(q+1) \left(q^3-1\right)}{q^8} & 0 & 0 & 0 & \frac{1}{q^6} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{\left(q^2-1\right)^3 \left(q^2+1\right)^2 \left(q^2+q+1\right)}{q^{29/2}} & 0 & 0 & 0 & \frac{(q+1) \left(q^3-1\right)^2}{q^{10}} & 0 & 0 & 0 & \frac{(q-1) (q+1)^2}{q^{15/2}} & 0 & 0 & 0 & \frac{1}{q^7} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{(q-1)^3 (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right)}{q^{11}} & 0 & 0 & 0 & \frac{q^5-q^3-q^2+1}{q^8} & 0 & 0 & 0 & \frac{q^2-1}{q^7} & 0 & 0 & 0 & \frac{1}{q^8} & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{q^6} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{\left(q^2+q+1\right) \left(q^4-1\right)}{q^{11}} & 0 & 0 & 0 & \frac{1}{q^6} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{(q+1) \left(q^5+q^3-q^2-1\right)^2}{q^{15}} & 0 & 0 & 0 & \frac{(q-1) \left(q^2+q+1\right)^2}{q^{19/2}} & 0 & 0 & 0 & \frac{1}{q^6} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\left(q^3-1\right)^2 \left(q^3+q^2+q+1\right)}{q^{12}} & 0 & 0 & 0 & \frac{(q+1) \left(q^3-1\right)}{q^8} & 0 & 0 & 0 & \frac{1}{q^6} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{q^7-q^4-q^3+1}{q^9} & 0 & 0 & 0 & \frac{q^3-1}{q^{13/2}} & 0 & 0 & 0 & \frac{1}{q^6} & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{q^8} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{(q-1) \left(q^3+q^2+q+1\right)^2}{q^{27/2}} & 0 & 0 & 0 & \frac{1}{q^7} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\left(q^2+q+1\right) \left(q^4-1\right)}{q^{11}} & 0 & 0 & 0 & \frac{1}{q^6} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{q^5+q^4-q-1}{q^{17/2}} & 0 & 0 & 0 & \frac{1}{q^5} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{q^4-1}{q^6} & 0 & 0 & 0 & \frac{1}{q^4} & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{q^{10}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{q^8} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{q^6} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{q^4} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{q^2} \end{array} \right),\left( \begin{array}{ccccc} 0 & 0 & 0 & 0 & \frac{1}{q} \\ 0 & 0 & 0 & \frac{1}{\sqrt{q}} & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & \sqrt{q} & 0 & 0 & 0 \\ q & 0 & 0 & 0 & 0 \end{array} \right),\left( \begin{array}{ccccc} 0 & 0 & 0 & 0 & \frac{1}{q} \\ 0 & 0 & 0 & \frac{1}{\sqrt{q}} & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & \sqrt{q} & 0 & 0 & 0 \\ q & 0 & 0 & 0 & 0 \end{array} \right),\left( \begin{array}{ccccc} 0 & 0 & 0 & 0 & \frac{1}{q} \\ 0 & 0 & 0 & \frac{1}{\sqrt{q}} & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & \sqrt{q} & 0 & 0 & 0 \\ q & 0 & 0 & 0 & 0 \end{array} \right),\left( \begin{array}{ccccc} 0 & 0 & 0 & 0 & \frac{1}{q} \\ 0 & 0 & 0 & \frac{1}{\sqrt{q}} & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & \sqrt{q} & 0 & 0 & 0 \\ q & 0 & 0 & 0 & 0 \end{array} \right)\right)\right]}{q^2+q+1+\frac{1}{q}+\frac{1}{q^2}}\right]}
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 \textrm{Apart}\left[\frac{\textrm{Hold}\left[\textrm{REngine}\left(\textrm{MorseLink}(\textrm{MorseLink::Error: bad input}),\left( \begin{array}{cccccccccccccccccccccccccccccccccccc} q^{5/2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & q^5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{15/2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{10} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{25/2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{15} & 0 & 0 & 0 & 0 & 0 \\ 0 & q^5 & 0 & 0 & 0 & 0 & q^{5/2}-q^{15/2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{13/2} & 0 & 0 & 0 & 0 & -q^{7/2} (q+1) \left(q^5-1\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^8 & 0 & 0 & 0 & 0 & -q^{9/2} \left(q^2+q+1\right) \left(q^5-1\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{19/2} & 0 & 0 & 0 & 0 & -q^{11/2} \left(q^3+q^2+q+1\right) \left(q^5-1\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{11} & 0 & 0 & 0 & 0 & -(q-1) q^{13/2} \left(q^4+q^3+q^2+q+1\right)^2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{25/2} & 0 & 0 & 0 & 0 \\ 0 & 0 & q^{15/2} & 0 & 0 & 0 & 0 & q^{11/2}-q^{19/2} & 0 & 0 & 0 & 0 & q^{23/2}-q^{15/2}-q^{13/2}+q^{5/2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^8 & 0 & 0 & 0 & 0 & -q^{21/2}-q^{19/2}+q^{13/2}+q^{11/2} & 0 & 0 & 0 & 0 & (q-1)^2 q^2 \left(q^2+q+1\right) \left(q^3+q^2+q+1\right) \left(q^4+q^3+q^2+q+1\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{17/2} & 0 & 0 & 0 & 0 & -q^{11/2} \left(q^2+q+1\right) \left(q^4-1\right) & 0 & 0 & 0 & 0 & (q-1)^2 q^{3/2} (q+1) \left(q^2+1\right)^2 \left(q^2+q+1\right) \left(q^4+q^3+q^2+q+1\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^9 & 0 & 0 & 0 & 0 & -(q-1) q^{11/2} \left(q^3+q^2+q+1\right)^2 & 0 & 0 & 0 & 0 & q (q+1) \left(q^2+1\right)^2 \left(q^5-1\right)^2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{19/2} & 0 & 0 & 0 & 0 & -q^{11/2} \left(q^3+q^2+q+1\right) \left(q^5-1\right) & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{10} & 0 & 0 & 0 \\ 0 & 0 & 0 & q^{10} & 0 & 0 & 0 & 0 & q^{17/2}-q^{23/2} & 0 & 0 & 0 & 0 & q^{13}-q^{10}-q^9+q^6 & 0 & 0 & 0 & 0 & -q^{5/2} \left(q^3-1\right) \left(q^4-1\right) \left(q^5-1\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{19/2} & 0 & 0 & 0 & 0 & -q^{15/2} (q+1) \left(q^3-1\right) & 0 & 0 & 0 & 0 & q^{9/2} \left(q^3-1\right)^2 \left(q^3+q^2+q+1\right) & 0 & 0 & 0 & 0 & -(q-1)^3 \left(q^2+q+1\right) \left(q^{13/4}+q^{9/4}+q^{5/4}+\sqrt[4]{q}\right)^2 \left(q^4+q^3+q^2+q+1\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^9 & 0 & 0 & 0 & 0 & -(q-1) q^{13/2} \left(q^2+q+1\right)^2 & 0 & 0 & 0 & 0 & q^3 (q+1) \left(q^5+q^3-q^2-1\right)^2 & 0 & 0 & 0 & 0 & -\frac{(q-1)^3 (q+1) \left(q^2+1\right)^2 \left(q^2+q+1\right) \left(q^4+q^3+q^2+q+1\right)^2}{q^{3/2}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{17/2} & 0 & 0 & 0 & 0 & -q^{11/2} \left(q^2+q+1\right) \left(q^4-1\right) & 0 & 0 & 0 & 0 & (q-1)^2 q^{3/2} (q+1) \left(q^2+1\right)^2 \left(q^2+q+1\right) \left(q^4+q^3+q^2+q+1\right) & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^8 & 0 & 0 & 0 & 0 & -q^{9/2} \left(q^2+q+1\right) \left(q^5-1\right) & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{15/2} & 0 & 0 \\ 0 & 0 & 0 & 0 & q^{25/2} & 0 & 0 & 0 & 0 & q^{23/2}-q^{27/2} & 0 & 0 & 0 & 0 & q^{29/2}-q^{25/2}-q^{23/2}+q^{19/2} & 0 & 0 & 0 & 0 & -(q-1)^3 q^{13/2} (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right) & 0 & 0 & 0 & 0 & (q-1)^4 q^{5/2} (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right) \left(q^4+q^3+q^2+q+1\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{11} & 0 & 0 & 0 & 0 & -(q-1) q^{19/2} (q+1)^2 & 0 & 0 & 0 & 0 & q^7 (q+1) \left(q^3-1\right)^2 & 0 & 0 & 0 & 0 & -q^{7/2} \left(q^2-1\right)^3 \left(q^2+1\right)^2 \left(q^2+q+1\right) & 0 & 0 & 0 & 0 & \frac{(q-1)^4 (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right) \left(q^4+q^3+q^2+q+1\right)^2}{q} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{19/2} & 0 & 0 & 0 & 0 & -q^{15/2} (q+1) \left(q^3-1\right) & 0 & 0 & 0 & 0 & q^{9/2} \left(q^3-1\right)^2 \left(q^3+q^2+q+1\right) & 0 & 0 & 0 & 0 & -(q-1)^3 \left(q^2+q+1\right) \left(q^{13/4}+q^{9/4}+q^{5/4}+\sqrt[4]{q}\right)^2 \left(q^4+q^3+q^2+q+1\right) & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^8 & 0 & 0 & 0 & 0 & -q^{21/2}-q^{19/2}+q^{13/2}+q^{11/2} & 0 & 0 & 0 & 0 & (q-1)^2 q^2 \left(q^2+q+1\right) \left(q^3+q^2+q+1\right) \left(q^4+q^3+q^2+q+1\right) & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{13/2} & 0 & 0 & 0 & 0 & -q^{7/2} (q+1) \left(q^5-1\right) & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^5 & 0 \\ 0 & 0 & 0 & 0 & 0 & q^{15} & 0 & 0 & 0 & 0 & q^{29/2}-q^{31/2} & 0 & 0 & 0 & 0 & (q-1)^2 q^{13} (q+1) & 0 & 0 & 0 & 0 & -(q-1)^3 q^{21/2} (q+1) \left(q^2+q+1\right) & 0 & 0 & 0 & 0 & (q-1)^4 q^7 (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right) & 0 & 0 & 0 & 0 & -(q-1)^5 q^{5/2} (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right) \left(q^4+q^3+q^2+q+1\right) & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{25/2} & 0 & 0 & 0 & 0 & q^{23/2}-q^{27/2} & 0 & 0 & 0 & 0 & q^{29/2}-q^{25/2}-q^{23/2}+q^{19/2} & 0 & 0 & 0 & 0 & -(q-1)^3 q^{13/2} (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right) & 0 & 0 & 0 & 0 & (q-1)^4 q^{5/2} (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right) \left(q^4+q^3+q^2+q+1\right) & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{10} & 0 & 0 & 0 & 0 & q^{17/2}-q^{23/2} & 0 & 0 & 0 & 0 & q^{13}-q^{10}-q^9+q^6 & 0 & 0 & 0 & 0 & -q^{5/2} \left(q^3-1\right) \left(q^4-1\right) \left(q^5-1\right) & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{15/2} & 0 & 0 & 0 & 0 & q^{11/2}-q^{19/2} & 0 & 0 & 0 & 0 & q^{23/2}-q^{15/2}-q^{13/2}+q^{5/2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^5 & 0 & 0 & 0 & 0 & q^{5/2}-q^{15/2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{5/2} \end{array} \right),\left( \begin{array}{cccccccccccccccccccccccccccccccccccc} \frac{1}{q^{5/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{q^5-1}{q^{15/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{q^9-q^5-q^4+1}{q^{23/2}} & 0 & 0 & 0 & 0 & \frac{q^4-1}{q^{17/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^{15/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{\left(1-q^3\right) \left(1-q^4\right) \left(q^5-1\right)}{q^{29/2}} & 0 & 0 & 0 & 0 & \frac{q^7-q^4-q^3+1}{q^{11}} & 0 & 0 & 0 & 0 & \frac{q^3-1}{q^{19/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^{10}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{(q-1)^4 (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right) \left(q^4+q^3+q^2+q+1\right)}{q^{33/2}} & 0 & 0 & 0 & 0 & \frac{(q-1)^3 (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right)}{q^{25/2}} & 0 & 0 & 0 & 0 & \frac{q^5-q^3-q^2+1}{q^{21/2}} & 0 & 0 & 0 & 0 & \frac{q^2-1}{q^{21/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^{25/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{(q-1)^5 (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right) \left(q^4+q^3+q^2+q+1\right)}{q^{35/2}} & 0 & 0 & 0 & 0 & \frac{(q-1)^4 (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right)}{q^{13}} & 0 & 0 & 0 & 0 & \frac{(q-1)^3 (q+1) \left(q^2+q+1\right)}{q^{21/2}} & 0 & 0 & 0 & 0 & \frac{(q-1)^2 (q+1)}{q^{10}} & 0 & 0 & 0 & 0 & \frac{q-1}{q^{23/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^{15}} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{1}{q^5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{(q+1) \left(q^5-1\right)}{q^{21/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^{13/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{(q-1)^2 \left(q^2+q+1\right) \left(q^3+q^2+q+1\right) \left(q^4+q^3+q^2+q+1\right)}{q^{15}} & 0 & 0 & 0 & 0 & \frac{q^5+q^4-q-1}{q^{21/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^8} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{(q-1)^3 \left(q^2+q+1\right) \left(q^3+q^2+q+1\right)^2 \left(q^4+q^3+q^2+q+1\right)}{q^{37/2}} & 0 & 0 & 0 & 0 & \frac{\left(q^3-1\right)^2 \left(q^3+q^2+q+1\right)}{q^{27/2}} & 0 & 0 & 0 & 0 & \frac{(q+1) \left(q^3-1\right)}{q^{21/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^{19/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{(q-1)^4 (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right) \left(q^4+q^3+q^2+q+1\right)^2}{q^{21}} & 0 & 0 & 0 & 0 & \frac{\left(q^2-1\right)^3 \left(q^2+1\right)^2 \left(q^2+q+1\right)}{q^{31/2}} & 0 & 0 & 0 & 0 & \frac{(q+1) \left(q^3-1\right)^2}{q^{12}} & 0 & 0 & 0 & 0 & \frac{(q-1) (q+1)^2}{q^{21/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^{11}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{(q-1)^4 (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right) \left(q^4+q^3+q^2+q+1\right)}{q^{33/2}} & 0 & 0 & 0 & 0 & \frac{(q-1)^3 (q+1)^2 \left(q^2+1\right) \left(q^2+q+1\right)}{q^{25/2}} & 0 & 0 & 0 & 0 & \frac{q^5-q^3-q^2+1}{q^{21/2}} & 0 & 0 & 0 & 0 & \frac{q^2-1}{q^{21/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^{25/2}} & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{q^{15/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{\left(q^2+q+1\right) \left(q^5-1\right)}{q^{27/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^8} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{(q-1)^2 (q+1) \left(q^2+1\right)^2 \left(q^2+q+1\right) \left(q^4+q^3+q^2+q+1\right)}{q^{37/2}} & 0 & 0 & 0 & 0 & \frac{\left(q^2+q+1\right) \left(q^4-1\right)}{q^{25/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^{17/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{(q-1)^3 (q+1) \left(q^2+1\right)^2 \left(q^2+q+1\right) \left(q^4+q^3+q^2+q+1\right)^2}{q^{45/2}} & 0 & 0 & 0 & 0 & \frac{(q+1) \left(q^5+q^3-q^2-1\right)^2}{q^{16}} & 0 & 0 & 0 & 0 & \frac{(q-1) \left(q^2+q+1\right)^2}{q^{23/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^9} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{(q-1)^3 \left(q^2+q+1\right) \left(q^3+q^2+q+1\right)^2 \left(q^4+q^3+q^2+q+1\right)}{q^{37/2}} & 0 & 0 & 0 & 0 & \frac{\left(q^3-1\right)^2 \left(q^3+q^2+q+1\right)}{q^{27/2}} & 0 & 0 & 0 & 0 & \frac{(q+1) \left(q^3-1\right)}{q^{21/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^{19/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\left(1-q^3\right) \left(1-q^4\right) \left(q^5-1\right)}{q^{29/2}} & 0 & 0 & 0 & 0 & \frac{q^7-q^4-q^3+1}{q^{11}} & 0 & 0 & 0 & 0 & \frac{q^3-1}{q^{19/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^{10}} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{q^{10}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{\left(q^3+q^2+q+1\right) \left(q^5-1\right)}{q^{33/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^{19/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{(q+1) \left(q^2+1\right)^2 \left(q^5-1\right)^2}{q^{22}} & 0 & 0 & 0 & 0 & \frac{(q-1) \left(q^3+q^2+q+1\right)^2}{q^{29/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^9} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{(q-1)^2 (q+1) \left(q^2+1\right)^2 \left(q^2+q+1\right) \left(q^4+q^3+q^2+q+1\right)}{q^{37/2}} & 0 & 0 & 0 & 0 & \frac{\left(q^2+q+1\right) \left(q^4-1\right)}{q^{25/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^{17/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{(q-1)^2 \left(q^2+q+1\right) \left(q^3+q^2+q+1\right) \left(q^4+q^3+q^2+q+1\right)}{q^{15}} & 0 & 0 & 0 & 0 & \frac{q^5+q^4-q-1}{q^{21/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^8} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{q^9-q^5-q^4+1}{q^{23/2}} & 0 & 0 & 0 & 0 & \frac{q^4-1}{q^{17/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^{15/2}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{q^{25/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{(q-1) \left(q^4+q^3+q^2+q+1\right)^2}{q^{39/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^{11}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\left(q^3+q^2+q+1\right) \left(q^5-1\right)}{q^{33/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^{19/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\left(q^2+q+1\right) \left(q^5-1\right)}{q^{27/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^8} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{(q+1) \left(q^5-1\right)}{q^{21/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^{13/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{q^5-1}{q^{15/2}} & 0 & 0 & 0 & 0 & \frac{1}{q^5} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{q^{15}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{q^{25/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{q^{10}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{q^{15/2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{q^5} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{q^{5/2}} \end{array} \right),\left( \begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & \frac{1}{q^{5/4}} \\ 0 & 0 & 0 & 0 & \frac{1}{q^{3/4}} & 0 \\ 0 & 0 & 0 & \frac{1}{\sqrt[4]{q}} & 0 & 0 \\ 0 & 0 & \sqrt[4]{q} & 0 & 0 & 0 \\ 0 & q^{3/4} & 0 & 0 & 0 & 0 \\ q^{5/4} & 0 & 0 & 0 & 0 & 0 \end{array} \right),\left( \begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & \frac{1}{q^{5/4}} \\ 0 & 0 & 0 & 0 & \frac{1}{q^{3/4}} & 0 \\ 0 & 0 & 0 & \frac{1}{\sqrt[4]{q}} & 0 & 0 \\ 0 & 0 & \sqrt[4]{q} & 0 & 0 & 0 \\ 0 & q^{3/4} & 0 & 0 & 0 & 0 \\ q^{5/4} & 0 & 0 & 0 & 0 & 0 \end{array} \right),\left( \begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & \frac{1}{q^{5/4}} \\ 0 & 0 & 0 & 0 & \frac{1}{q^{3/4}} & 0 \\ 0 & 0 & 0 & \frac{1}{\sqrt[4]{q}} & 0 & 0 \\ 0 & 0 & \sqrt[4]{q} & 0 & 0 & 0 \\ 0 & q^{3/4} & 0 & 0 & 0 & 0 \\ q^{5/4} & 0 & 0 & 0 & 0 & 0 \end{array} \right),\left( \begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & \frac{1}{q^{5/4}} \\ 0 & 0 & 0 & 0 & \frac{1}{q^{3/4}} & 0 \\ 0 & 0 & 0 & \frac{1}{\sqrt[4]{q}} & 0 & 0 \\ 0 & 0 & \sqrt[4]{q} & 0 & 0 & 0 \\ 0 & q^{3/4} & 0 & 0 & 0 & 0 \\ q^{5/4} & 0 & 0 & 0 & 0 & 0 \end{array} \right)\right)\right]}{q^{5/2}+q^{3/2}+\sqrt{q}+\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}+\frac{1}{q^{5/2}}}\right]}
6	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. TeX parse error: MathJax internal buffer size exceeded; is there a recursive macro call?"): {\displaystyle {\textrm {Apart}}\left[{\frac {{\textrm {Hold}}\left[{\textrm {REngine}}\left({\textrm {MorseLink}}({\textrm 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7	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. TeX parse error: MathJax internal buffer size exceeded; is there a recursive macro call?"): {\displaystyle {\textrm {Apart}}\left[{\frac {{\textrm {Hold}}\left[{\textrm {REngine}}\left({\textrm {MorseLink}}({\textrm 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{(q-1)^{2}\left(q^{2}+1\right)\left(q^{4}+q^{3}+q^{2}+q+1\right)\left(q^{5}+q^{4}+q^{3}+q^{2}+q+1\right)\left(q^{6}+q^{5}+q^{4}+q^{3}+q^{2}+q+1\right)}{q^{30}}}&0&0&0&0&0&0&{\frac {(q-1)(q+1)^{2}\left(q^{2}+1\right)\left(q^{4}+q^{2}+1\right)}{q^{43/2}}}&0&0&0&0&0&0&{\frac {1}{q^{15}}}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&{\frac {\left(q^{2}+1\right)\left(q^{3}-1\right)^{2}\left(q^{3}+1\right)\left(q^{6}+q^{5}+q^{4}+q^{3}+q^{2}+q+1\right)}{q^{51/2}}}&0&0&0&0&0&0&{\frac {\left(q^{2}+q+1\right)^{2}\left(q^{4}-q^{3}+q-1\right)}{q^{37/2}}}&0&0&0&0&0&0&{\frac {1}{q^{27/2}}}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&{\frac {\left(q^{3}-1\right)^{2}\left(q^{3}+1\right)\left(q^{6}+q^{5}+q^{4}+q^{3}+q^{2}+q+1\right)}{q^{21}}}&0&0&0&0&0&0&{\frac {q^{7}+q^{6}-q-1}{q^{31/2}}}&0&0&0&0&0&0&{\frac {1}{q^{12}}}&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&{\frac {q^{13}-q^{7}-q^{6}+1}{q^{33/2}}}&0&0&0&0&0&0&{\frac {q^{6}-1}{q^{25/2}}}&0&0&0&0&0&0&{\frac {1}{q^{21/2}}}&0&0\\0&0&0&0&0&0&{\frac {1}{q^{49/2}}}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&{\frac {\left(q^{7}-1\right)^{2}}{(q-1)q^{69/2}}}&0&0&0&0&0&0&{\frac {1}{q^{22}}}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&{\frac {\left(q^{5}+q^{4}+q^{3}+q^{2}+q+1\right)\left(q^{7}-1\right)}{q^{61/2}}}&0&0&0&0&0&0&{\frac {1}{q^{39/2}}}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&{\frac {\left(q^{4}+q^{3}+q^{2}+q+1\right)\left(q^{7}-1\right)}{q^{53/2}}}&0&0&0&0&0&0&{\frac 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{1}{\sqrt[{4}]{q}}}&0&0&0\\0&0&0&{\sqrt[{4}]{q}}&0&0&0&0\\0&0&q^{3/4}&0&0&0&0&0\\0&q^{5/4}&0&0&0&0&0&0\\q^{7/4}&0&0&0&0&0&0&0\end{array}}\right),\left({\begin{array}{cccccccc}0&0&0&0&0&0&0&{\frac {1}{q^{7/4}}}\\0&0&0&0&0&0&{\frac {1}{q^{5/4}}}&0\\0&0&0&0&0&{\frac {1}{q^{3/4}}}&0&0\\0&0&0&0&{\frac {1}{\sqrt[{4}]{q}}}&0&0&0\\0&0&0&{\sqrt[{4}]{q}}&0&0&0&0\\0&0&q^{3/4}&0&0&0&0&0\\0&q^{5/4}&0&0&0&0&0&0\\q^{7/4}&0&0&0&0&0&0&0\end{array}}\right)\right)\right]}{q^{7/2}+q^{5/2}+q^{3/2}+{\sqrt {q}}+{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}+{\frac {1}{q^{5/2}}}+{\frac {1}{q^{7/2}}}}}\right]}

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).

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4_1

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5 1

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

Modifying This Page

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