6 2

6_1

6_3

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 6 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 6 2 at Knotilus!
	Dror likes to call 6_2 "The Miller Institute Knot", as it is the logo of the Miller Institute for Basic Research.

The Miller Institute Mug [1]

Simple square depiction

3D depiction

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 X7,12,8,1 X11,6,12,7
Gauss code	-1, 4, -3, 1, -2, 6, -5, 3, -4, 2, -6, 5
Dowker-Thistlethwaite code	4 8 10 12 2 6
Conway Notation	[312]

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

[edit Notes on presentations of 6 2]

Knot 6_2.

A graph, knot 6_2.

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["6 2"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 10 12 2 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[312]

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]=

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

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]=

{ 3, 6, 3 }

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

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 $\{3,4\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-1] Hyperbolic Volume 4.40083 A-Polynomial See Data:6 2/A-polynomial

[edit Notes for 6 2's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant -2

[edit Notes for 6 2's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^2+3 t-3+3 t^{-1} - t^{-2}$
Conway polynomial	$-z^4-z^2+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 11, -2 }
Jones polynomial	$q-1+2 q^{-1} -2 q^{-2} +2 q^{-3} -2 q^{-4} + q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^2 a^4+a^4-z^4 a^2-3 z^2 a^2-2 a^2+z^2+2$
Kauffman polynomial (db, data sources)	$z^2 a^6+2 z^3 a^5-z a^5+2 z^4 a^4-2 z^2 a^4+a^4+z^5 a^3-z a^3+3 z^4 a^2-6 z^2 a^2+2 a^2+z^5 a-2 z^3 a+z^4-3 z^2+2$
The A2 invariant	$q^{16}-q^8-q^4+q^2+1+ q^{-2} + q^{-4}$
The G2 invariant	$q^{86}-q^{84}+q^{82}-q^{80}-q^{78}-q^{74}+3 q^{72}-2 q^{70}+q^{68}+2 q^{62}-q^{60}+q^{58}+q^{56}+q^{52}+3 q^{46}-3 q^{44}+q^{42}-q^{40}-q^{38}+2 q^{36}-4 q^{34}+q^{32}-2 q^{30}-3 q^{24}+q^{22}-q^{20}-q^{14}+q^{12}+q^{10}+2 q^6-q^4+2 q^2+1- q^{-2} +3 q^{-4} - q^{-6} +2 q^{-8} - q^{-12} +2 q^{-14} + q^{-18}$

Further Quantum Invariants

Further quantum knot invariants for 6_2.

A1 Invariants.

Weight	Invariant
1	$q^{11}-q^9+q+ q^{-3}$
2	$q^{30}-q^{28}-q^{26}+2 q^{24}-q^{22}-q^{20}+q^{18}+q^{10}-q^6+q^4+q^2-1+ q^{-2} + q^{-4} - q^{-6} + q^{-10}$
3	$q^{57}-q^{55}-q^{53}+q^{51}+q^{49}-q^{47}-3 q^{45}+2 q^{43}+3 q^{41}-q^{39}-2 q^{37}+q^{35}+2 q^{33}-q^{29}-q^{27}-2 q^{19}-q^{17}+2 q^{15}+2 q^{13}-q^{11}+2 q^7+2 q^5-q^3-2 q+ q^{-1} +2 q^{-3} -2 q^{-7} +2 q^{-11} + q^{-13} - q^{-15} - q^{-17} + q^{-21}$
4	$q^{92}-q^{90}-q^{88}+q^{86}+q^{82}-3 q^{80}-q^{78}+3 q^{76}+2 q^{74}+3 q^{72}-5 q^{70}-4 q^{68}+3 q^{66}+4 q^{64}+3 q^{62}-5 q^{60}-5 q^{58}+2 q^{54}+3 q^{52}-q^{50}-2 q^{48}+q^{44}+2 q^{42}+2 q^{40}+q^{38}-2 q^{36}-2 q^{34}+2 q^{30}+q^{28}-3 q^{26}-3 q^{24}-q^{22}+4 q^{20}+2 q^{18}-3 q^{16}-2 q^{14}+5 q^{10}+3 q^8-2 q^6-2 q^4-2 q^2+4+4 q^{-2} -2 q^{-6} -4 q^{-8} + q^{-10} +3 q^{-12} +2 q^{-14} -4 q^{-18} - q^{-20} + q^{-22} +2 q^{-24} +2 q^{-26} - q^{-28} - q^{-30} - q^{-32} + q^{-36}$
5	$q^{135}-q^{133}-q^{131}+q^{129}-q^{123}-q^{121}+3 q^{117}+4 q^{115}-5 q^{111}-5 q^{109}+4 q^{105}+8 q^{103}+3 q^{101}-8 q^{99}-10 q^{97}-5 q^{95}+5 q^{93}+11 q^{91}+7 q^{89}-2 q^{87}-9 q^{85}-6 q^{83}+q^{81}+6 q^{79}+6 q^{77}+q^{75}-4 q^{73}-5 q^{71}-2 q^{69}+q^{67}+2 q^{65}+3 q^{63}+q^{61}-q^{59}-2 q^{57}-3 q^{55}-q^{53}+3 q^{51}+4 q^{49}+q^{47}-3 q^{45}-5 q^{43}-q^{41}+5 q^{39}+5 q^{37}+q^{35}-4 q^{33}-6 q^{31}-q^{29}+5 q^{27}+6 q^{25}-6 q^{21}-7 q^{19}-2 q^{17}+7 q^{15}+8 q^{13}+3 q^{11}-4 q^9-7 q^7-3 q^5+3 q^3+8 q+6 q^{-1} - q^{-3} -6 q^{-5} -6 q^{-7} -2 q^{-9} +4 q^{-11} +7 q^{-13} +4 q^{-15} - q^{-17} -5 q^{-19} -5 q^{-21} - q^{-23} +3 q^{-25} +5 q^{-27} +3 q^{-29} - q^{-31} -4 q^{-33} -3 q^{-35} - q^{-37} + q^{-39} +3 q^{-41} +2 q^{-43} - q^{-47} - q^{-49} - q^{-51} + q^{-55}$
6	$q^{186}-q^{184}-q^{182}+q^{180}-2 q^{174}+q^{172}+5 q^{166}+2 q^{164}-2 q^{162}-6 q^{160}-3 q^{158}-3 q^{156}+3 q^{154}+12 q^{152}+7 q^{150}-3 q^{148}-14 q^{146}-10 q^{144}-8 q^{142}+4 q^{140}+21 q^{138}+17 q^{136}+5 q^{134}-14 q^{132}-18 q^{130}-17 q^{128}-3 q^{126}+18 q^{124}+20 q^{122}+12 q^{120}-5 q^{118}-13 q^{116}-17 q^{114}-9 q^{112}+7 q^{110}+13 q^{108}+12 q^{106}+2 q^{104}-4 q^{102}-10 q^{100}-9 q^{98}-2 q^{96}+3 q^{94}+6 q^{92}+4 q^{90}+3 q^{88}-q^{86}-3 q^{84}-4 q^{82}-3 q^{80}+4 q^{76}+6 q^{74}+4 q^{72}-6 q^{68}-6 q^{66}-3 q^{64}+5 q^{62}+8 q^{60}+5 q^{58}-q^{56}-11 q^{54}-8 q^{52}-q^{50}+8 q^{48}+9 q^{46}+4 q^{44}-4 q^{42}-14 q^{40}-9 q^{38}+q^{36}+12 q^{34}+12 q^{32}+7 q^{30}-4 q^{28}-16 q^{26}-12 q^{24}-q^{22}+12 q^{20}+14 q^{18}+11 q^{16}-15 q^{12}-15 q^{10}-7 q^8+7 q^6+12 q^4+15 q^2+7-8 q^{-2} -13 q^{-4} -12 q^{-6} -2 q^{-8} +4 q^{-10} +14 q^{-12} +13 q^{-14} +2 q^{-16} -5 q^{-18} -11 q^{-20} -9 q^{-22} -6 q^{-24} +5 q^{-26} +10 q^{-28} +8 q^{-30} +4 q^{-32} -2 q^{-34} -6 q^{-36} -10 q^{-38} -3 q^{-40} +2 q^{-42} +5 q^{-44} +6 q^{-46} +4 q^{-48} + q^{-50} -5 q^{-52} -4 q^{-54} -3 q^{-56} - q^{-58} + q^{-60} +3 q^{-62} +3 q^{-64} - q^{-70} - q^{-72} - q^{-74} + q^{-78}$

A2 Invariants.

Weight	Invariant
1,0	$q^{16}-q^8-q^4+q^2+1+ q^{-2} + q^{-4}$
1,1	$q^{44}-2 q^{42}+2 q^{40}-2 q^{38}+5 q^{36}-4 q^{34}+2 q^{32}-4 q^{30}-2 q^{24}+6 q^{22}-4 q^{20}+8 q^{18}-6 q^{16}+6 q^{14}-6 q^{12}+4 q^{10}-2 q^8+q^4-2 q^2+4-4 q^{-2} +4 q^{-4} -2 q^{-6} +4 q^{-8} + q^{-12}$
2,0	$q^{40}-q^{30}-2 q^{28}+q^{20}+2 q^{18}+q^{16}+q^{14}+q^{12}-q^8-q^6-q^2+ q^{-2} + q^{-4} + q^{-8} + q^{-10} + q^{-12}$
3,0	$q^{72}-3 q^{60}-2 q^{58}-q^{56}+2 q^{54}+2 q^{52}+2 q^{46}+4 q^{44}+2 q^{42}-q^{38}-q^{34}-3 q^{32}-4 q^{30}-3 q^{28}-2 q^{26}-q^{24}-q^{22}+2 q^{20}+4 q^{18}+4 q^{16}+3 q^{14}+2 q^{12}+3 q^{10}+q^8-q^6-3 q^4-q^2+1+ q^{-2} - q^{-4} - q^{-6} +2 q^{-10} + q^{-12} - q^{-16} + q^{-20} + q^{-22} + q^{-24}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{36}-q^{34}-q^{32}+q^{30}-q^{28}+2 q^{24}+q^{22}+q^{20}+q^{18}-q^{14}-2 q^{12}-q^{10}-q^8-2 q^6+q^4+2 q^2+1+2 q^{-2} +2 q^{-4} + q^{-8}$
1,0,0	$q^{21}+q^{17}-q^{11}-q^9-q^7-q^5+q^3+q+2 q^{-1} + q^{-3} + q^{-5}$
1,0,1	$q^{58}-2 q^{56}+q^{54}+q^{52}-2 q^{50}+5 q^{48}-2 q^{46}-q^{44}+2 q^{42}-5 q^{40}+3 q^{38}-2 q^{36}-2 q^{34}+3 q^{32}-5 q^{30}+2 q^{28}+q^{26}+4 q^{22}+4 q^{20}+q^{18}+3 q^{16}+2 q^{14}-4 q^{12}+3 q^{10}-7 q^8-q^6-q^4-5 q^2+4-2 q^{-2} +5 q^{-4} +4 q^{-6} +2 q^{-8} +4 q^{-10} + q^{-14}$

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$q^{36}-q^{34}+q^{32}-q^{30}+q^{28}+q^{22}-q^{20}+q^{18}-2 q^{16}+q^{14}-2 q^{12}+q^{10}-q^8+q^4+1+2 q^{-4} + q^{-8}$
1,0	$q^{58}-q^{54}-q^{52}+q^{48}-q^{44}+q^{40}+q^{38}+q^{32}+q^{30}-q^{26}-q^{18}-q^{16}-q^{10}-q^8+q^6+q^4+q^2+ q^{-4} +2 q^{-6} + q^{-14}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{50}-q^{48}-q^{44}+q^{42}-q^{40}+q^{34}+2 q^{32}+q^{30}+2 q^{28}+2 q^{24}-q^{22}-3 q^{18}-q^{16}-3 q^{14}-q^{12}-2 q^{10}-q^8+q^6+q^4+2 q^2+2+3 q^{-2} + q^{-4} +2 q^{-6} + q^{-10}$

G2 Invariants.

Weight	Invariant
1,0	$q^{86}-q^{84}+q^{82}-q^{80}-q^{78}-q^{74}+3 q^{72}-2 q^{70}+q^{68}+2 q^{62}-q^{60}+q^{58}+q^{56}+q^{52}+3 q^{46}-3 q^{44}+q^{42}-q^{40}-q^{38}+2 q^{36}-4 q^{34}+q^{32}-2 q^{30}-3 q^{24}+q^{22}-q^{20}-q^{14}+q^{12}+q^{10}+2 q^6-q^4+2 q^2+1- q^{-2} +3 q^{-4} - q^{-6} +2 q^{-8} - q^{-12} +2 q^{-14} + q^{-18}$

. </div></div>

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["6 2"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^2+3 t-3+3 t^{-1} - t^{-2}$

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]=

{ 11, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q-1+2 q^{-1} -2 q^{-2} +2 q^{-3} -2 q^{-4} + q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^2 a^4+a^4-z^4 a^2-3 z^2 a^2-2 a^2+z^2+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^2 a^6+2 z^3 a^5-z a^5+2 z^4 a^4-2 z^2 a^4+a^4+z^5 a^3-z a^3+3 z^4 a^2-6 z^2 a^2+2 a^2+z^5 a-2 z^3 a+z^4-3 z^2+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

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["6 2"];

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]=

{ $-t^2+3 t-3+3 t^{-1} - t^{-2}$, $q-1+2 q^{-1} -2 q^{-2} +2 q^{-3} -2 q^{-4} + q^{-5}$ }

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]=

{}

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 $-4$ $8$ $8$ $\frac{34}{3}$ $\frac{38}{3}$ $-32$ $-\frac{208}{3}$ $-\frac{64}{3}$ $-24$ $-\frac{32}{3}$ $32$ $-\frac{136}{3}$ $-\frac{152}{3}$ $\frac{2129}{30}$ $\frac{662}{15}$ $-\frac{1862}{45}$ $\frac{463}{18}$ $-\frac{751}{30}$

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 $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$-2 is the signature of 6 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 χ 3             1 1 1               0 -1         2 1   1 -3       1 1     0 -5     1 1       0 -7   1 1         0 -9   1           -1 -11 1             1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

<p>

$n$	$J_n$
2	$q^4-q^3-q^2+3 q-1-3 q^{-1} +5 q^{-2} - q^{-3} -5 q^{-4} +6 q^{-5} -6 q^{-7} +6 q^{-8} -5 q^{-10} +4 q^{-11} -2 q^{-13} + q^{-14}$
3	$q^9-q^8-q^7+3 q^5-3 q^3-2 q^2+5 q+2-4 q^{-1} -5 q^{-2} +6 q^{-3} +5 q^{-4} -4 q^{-5} -7 q^{-6} +5 q^{-7} +8 q^{-8} -4 q^{-9} -10 q^{-10} +4 q^{-11} +10 q^{-12} -4 q^{-13} -10 q^{-14} +3 q^{-15} +10 q^{-16} -3 q^{-17} -8 q^{-18} +2 q^{-19} +7 q^{-20} -2 q^{-21} -4 q^{-22} + q^{-23} +2 q^{-24} -2 q^{-26} + q^{-27}$
4	$q^{16}-q^{15}-q^{14}+4 q^{11}-q^{10}-2 q^9-2 q^8-3 q^7+8 q^6+q^5-q^4-4 q^3-8 q^2+10 q+3+3 q^{-1} -4 q^{-2} -14 q^{-3} +10 q^{-4} +3 q^{-5} +8 q^{-6} -2 q^{-7} -19 q^{-8} +8 q^{-9} +2 q^{-10} +13 q^{-11} -24 q^{-13} +6 q^{-14} +2 q^{-15} +17 q^{-16} + q^{-17} -26 q^{-18} +4 q^{-19} +2 q^{-20} +20 q^{-21} +2 q^{-22} -26 q^{-23} +3 q^{-24} + q^{-25} +18 q^{-26} +3 q^{-27} -22 q^{-28} +2 q^{-29} - q^{-30} +13 q^{-31} +3 q^{-32} -14 q^{-33} +3 q^{-34} -2 q^{-35} +6 q^{-36} +2 q^{-37} -6 q^{-38} +2 q^{-39} - q^{-40} +2 q^{-41} -2 q^{-43} + q^{-44}$
5	$q^{25}-q^{24}-q^{23}+q^{20}+3 q^{19}-3 q^{17}-2 q^{16}-2 q^{15}+6 q^{13}+4 q^{12}-q^{11}-4 q^{10}-6 q^9-4 q^8+6 q^7+8 q^6+4 q^5-q^4-9 q^3-10 q^2+2 q+8+9 q^{-1} +6 q^{-2} -7 q^{-3} -15 q^{-4} -4 q^{-5} +4 q^{-6} +12 q^{-7} +13 q^{-8} -2 q^{-9} -16 q^{-10} -13 q^{-11} - q^{-12} +13 q^{-13} +19 q^{-14} +4 q^{-15} -17 q^{-16} -19 q^{-17} -6 q^{-18} +15 q^{-19} +24 q^{-20} +8 q^{-21} -17 q^{-22} -25 q^{-23} -10 q^{-24} +17 q^{-25} +28 q^{-26} +11 q^{-27} -18 q^{-28} -29 q^{-29} -12 q^{-30} +18 q^{-31} +29 q^{-32} +13 q^{-33} -16 q^{-34} -30 q^{-35} -13 q^{-36} +15 q^{-37} +26 q^{-38} +14 q^{-39} -11 q^{-40} -25 q^{-41} -13 q^{-42} +10 q^{-43} +19 q^{-44} +11 q^{-45} -4 q^{-46} -16 q^{-47} -9 q^{-48} +4 q^{-49} +9 q^{-50} +6 q^{-51} -2 q^{-52} -5 q^{-53} -4 q^{-54} +5 q^{-56} + q^{-57} -2 q^{-58} - q^{-61} +2 q^{-62} -2 q^{-64} + q^{-65}$
6	$q^{36}-q^{35}-q^{34}+q^{31}+4 q^{29}-q^{28}-3 q^{27}-2 q^{26}-2 q^{25}-q^{23}+10 q^{22}+2 q^{21}-q^{20}-3 q^{19}-5 q^{18}-5 q^{17}-8 q^{16}+14 q^{15}+6 q^{14}+5 q^{13}+q^{12}-3 q^{11}-10 q^{10}-19 q^9+11 q^8+4 q^7+11 q^6+8 q^5+8 q^4-9 q^3-29 q^2+5 q-6+10 q^{-1} +13 q^{-2} +23 q^{-3} - q^{-4} -32 q^{-5} -20 q^{-7} +2 q^{-8} +13 q^{-9} +38 q^{-10} +10 q^{-11} -29 q^{-12} -2 q^{-13} -33 q^{-14} -9 q^{-15} +9 q^{-16} +50 q^{-17} +21 q^{-18} -24 q^{-19} -2 q^{-20} -44 q^{-21} -19 q^{-22} +4 q^{-23} +60 q^{-24} +29 q^{-25} -19 q^{-26} -3 q^{-27} -53 q^{-28} -26 q^{-29} + q^{-30} +70 q^{-31} +35 q^{-32} -16 q^{-33} -6 q^{-34} -61 q^{-35} -29 q^{-36} + q^{-37} +76 q^{-38} +39 q^{-39} -14 q^{-40} -8 q^{-41} -65 q^{-42} -32 q^{-43} +77 q^{-45} +41 q^{-46} -10 q^{-47} -7 q^{-48} -63 q^{-49} -35 q^{-50} -5 q^{-51} +70 q^{-52} +40 q^{-53} -4 q^{-54} - q^{-55} -53 q^{-56} -34 q^{-57} -11 q^{-58} +54 q^{-59} +32 q^{-60} +7 q^{-62} -36 q^{-63} -26 q^{-64} -13 q^{-65} +33 q^{-66} +18 q^{-67} - q^{-68} +11 q^{-69} -17 q^{-70} -14 q^{-71} -9 q^{-72} +16 q^{-73} +6 q^{-74} -3 q^{-75} +7 q^{-76} -6 q^{-77} -4 q^{-78} -4 q^{-79} +7 q^{-80} -3 q^{-82} +4 q^{-83} -2 q^{-84} - q^{-86} +2 q^{-87} -2 q^{-89} + q^{-90}$
7	$q^{49}-q^{48}-q^{47}+q^{44}+q^{42}+3 q^{41}-q^{40}-3 q^{39}-2 q^{38}-3 q^{37}+q^{36}+q^{34}+9 q^{33}+3 q^{32}-q^{31}-3 q^{30}-8 q^{29}-3 q^{28}-5 q^{27}-4 q^{26}+12 q^{25}+9 q^{24}+7 q^{23}+5 q^{22}-9 q^{21}-5 q^{20}-12 q^{19}-16 q^{18}+6 q^{17}+7 q^{16}+13 q^{15}+18 q^{14}+q^{13}+3 q^{12}-12 q^{11}-28 q^{10}-6 q^9-7 q^8+6 q^7+25 q^6+14 q^5+21 q^4-29 q^2-14 q-25-15 q^{-1} +19 q^{-2} +19 q^{-3} +40 q^{-4} +22 q^{-5} -18 q^{-6} -11 q^{-7} -41 q^{-8} -38 q^{-9} +2 q^{-10} +12 q^{-11} +51 q^{-12} +46 q^{-13} + q^{-14} - q^{-15} -48 q^{-16} -59 q^{-17} -17 q^{-18} -4 q^{-19} +55 q^{-20} +67 q^{-21} +20 q^{-22} +14 q^{-23} -51 q^{-24} -75 q^{-25} -36 q^{-26} -19 q^{-27} +55 q^{-28} +82 q^{-29} +38 q^{-30} +27 q^{-31} -52 q^{-32} -90 q^{-33} -49 q^{-34} -30 q^{-35} +57 q^{-36} +95 q^{-37} +52 q^{-38} +34 q^{-39} -56 q^{-40} -102 q^{-41} -58 q^{-42} -34 q^{-43} +60 q^{-44} +106 q^{-45} +61 q^{-46} +35 q^{-47} -61 q^{-48} -112 q^{-49} -63 q^{-50} -36 q^{-51} +63 q^{-52} +114 q^{-53} +66 q^{-54} +36 q^{-55} -63 q^{-56} -115 q^{-57} -68 q^{-58} -40 q^{-59} +61 q^{-60} +116 q^{-61} +71 q^{-62} +41 q^{-63} -57 q^{-64} -111 q^{-65} -71 q^{-66} -47 q^{-67} +49 q^{-68} +107 q^{-69} +72 q^{-70} +49 q^{-71} -43 q^{-72} -94 q^{-73} -66 q^{-74} -52 q^{-75} +28 q^{-76} +83 q^{-77} +63 q^{-78} +50 q^{-79} -23 q^{-80} -67 q^{-81} -47 q^{-82} -47 q^{-83} +9 q^{-84} +51 q^{-85} +39 q^{-86} +41 q^{-87} -8 q^{-88} -35 q^{-89} -24 q^{-90} -31 q^{-91} +2 q^{-92} +22 q^{-93} +14 q^{-94} +23 q^{-95} + q^{-96} -16 q^{-97} -8 q^{-98} -13 q^{-99} +3 q^{-100} +7 q^{-101} - q^{-102} +10 q^{-103} +2 q^{-104} -6 q^{-105} -2 q^{-106} -4 q^{-107} +3 q^{-108} +2 q^{-109} -4 q^{-110} +3 q^{-111} +2 q^{-112} -2 q^{-113} - q^{-115} +2 q^{-116} -2 q^{-118} + q^{-119}$

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.

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