8 8

8_7

8_9

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 8 8's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 8 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3849 X11,15,12,14 X5,13,6,12 X13,7,14,6 X9,1,10,16 X15,11,16,10 X7283
Gauss code	-1, 8, -2, 1, -4, 5, -8, 2, -6, 7, -3, 4, -5, 3, -7, 6
Dowker-Thistlethwaite code	4 8 12 2 16 14 6 10
Conway Notation	[2312]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 9, width is 4, Braid index is 4		[{10, 2}, {1, 8}, {4, 9}, {8, 10}, {3, 5}, {2, 4}, {6, 3}, {5, 7}, {9, 6}, {7, 1}]

[edit Notes on presentations of 8 8]

Knot 8_8.

A graph, knot 8_8.

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["8 8"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3849 X11,15,12,14 X5,13,6,12 X13,7,14,6 X9,1,10,16 X15,11,16,10 X7283

In[5]:=

GaussCode[K]

Out[5]=

-1, 8, -2, 1, -4, 5, -8, 2, -6, 7, -3, 4, -5, 3, -7, 6

In[6]:=

DTCode[K]

Out[6]=

4 8 12 2 16 14 6 10

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[2312]

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}(4,\{1,1,1,2,-1,-3,2,-3,-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, 9, 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[{10, 2}, {1, 8}, {4, 9}, {8, 10}, {3, 5}, {2, 4}, {6, 3}, {5, 7}, {9, 6}, {7, 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 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 \{4,5\}} Nakanishi index 1 Maximal Thurston-Bennequin number [-4][-6] Hyperbolic Volume 7.80134 A-Polynomial See Data:8 8/A-polynomial

[edit Notes for 8 8'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 0} 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 0} Concordance genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} Rasmussen s-Invariant 0

[edit Notes for 8 8'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 2 t^2-6 t+9-6 t^{-1} +2 t^{-2} }
Conway polynomial	$2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 25, 0 }
Jones polynomial	$-q^{5}+2q^{4}-3q^{3}+4q^{2}-4q+5-3q^{-1}+2q^{-2}-q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-2}+z^{4}-a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}+2z^{2}-a^{2}+a^{-2}-a^{-4}+2$
Kauffman polynomial (db, data sources)	$z^{7}a^{-1}+z^{7}a^{-3}+4z^{6}a^{-2}+2z^{6}a^{-4}+2z^{6}+2az^{5}+z^{5}a^{-1}+z^{5}a^{-5}+2a^{2}z^{4}-9z^{4}a^{-2}-6z^{4}a^{-4}-z^{4}+a^{3}z^{3}-3z^{3}a^{-1}-5z^{3}a^{-3}-3z^{3}a^{-5}-2a^{2}z^{2}+5z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}-a^{3}z-az+za^{-1}+3za^{-3}+2za^{-5}+a^{2}-a^{-2}-a^{-4}+2$
The A2 invariant	$-q^{10}-q^{4}+2q^{2}+1+2q^{-2}+q^{-4}+q^{-8}-q^{-10}-q^{-16}$
The G2 invariant	$q^{52}-q^{50}+2q^{48}-2q^{46}-3q^{40}+4q^{38}-5q^{36}+4q^{34}-5q^{32}+q^{30}+3q^{28}-6q^{26}+8q^{24}-9q^{22}+7q^{20}-4q^{18}-3q^{16}+7q^{14}-8q^{12}+9q^{10}-q^{8}-3q^{6}+6q^{4}-3q^{2}+1+6q^{-2}-10q^{-4}+12q^{-6}-5q^{-8}+10q^{-12}-14q^{-14}+17q^{-16}-11q^{-18}+3q^{-20}+4q^{-22}-9q^{-24}+14q^{-26}-10q^{-28}+6q^{-30}+q^{-32}-5q^{-34}+7q^{-36}-5q^{-38}+4q^{-42}-8q^{-44}+7q^{-46}-3q^{-48}-4q^{-50}+10q^{-52}-13q^{-54}+10q^{-56}-5q^{-58}-4q^{-60}+7q^{-62}-10q^{-64}+9q^{-66}-5q^{-68}+2q^{-72}-4q^{-74}+3q^{-76}-q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 8_8.

A1 Invariants.

Weight	Invariant
1	$-q^{7}+q^{5}-q^{3}+2q+q^{-1}+q^{-5}-q^{-7}+q^{-9}-q^{-11}$
2	$q^{20}-q^{18}-q^{16}+2q^{14}-3q^{12}-q^{10}+5q^{8}-3q^{6}-2q^{4}+5q^{2}-q^{-2}+q^{-4}+3q^{-6}-2q^{-10}+3q^{-12}-5q^{-16}+2q^{-18}+2q^{-20}-4q^{-22}+q^{-24}+3q^{-26}-2q^{-28}-q^{-30}+q^{-32}$
3	$-q^{39}+q^{37}+q^{35}-q^{31}+q^{29}+q^{27}-4q^{25}-q^{23}+5q^{21}+2q^{19}-9q^{17}-4q^{15}+10q^{13}+7q^{11}-11q^{9}-7q^{7}+7q^{5}+11q^{3}-3q-7q^{-1}+q^{-3}+6q^{-5}+5q^{-7}-3q^{-9}-6q^{-11}+q^{-13}+9q^{-15}-q^{-17}-9q^{-19}-2q^{-21}+10q^{-23}+3q^{-25}-10q^{-27}-6q^{-29}+9q^{-31}+7q^{-33}-6q^{-35}-10q^{-37}+3q^{-39}+10q^{-41}-8q^{-45}-3q^{-47}+6q^{-49}+5q^{-51}-3q^{-53}-4q^{-55}+2q^{-59}+q^{-61}-q^{-63}$
4	$q^{64}-q^{62}-q^{60}-q^{56}+3q^{54}-q^{52}+q^{50}+2q^{48}-4q^{46}+q^{44}-4q^{42}+5q^{40}+10q^{38}-6q^{36}-8q^{34}-14q^{32}+9q^{30}+26q^{28}+q^{26}-17q^{24}-34q^{22}+3q^{20}+40q^{18}+18q^{16}-11q^{14}-43q^{12}-12q^{10}+30q^{8}+27q^{6}+10q^{4}-27q^{2}-21+6q^{-2}+18q^{-4}+18q^{-6}-3q^{-8}-16q^{-10}-15q^{-12}+2q^{-14}+19q^{-16}+14q^{-18}-10q^{-20}-25q^{-22}-3q^{-24}+20q^{-26}+24q^{-28}-8q^{-30}-33q^{-32}-8q^{-34}+19q^{-36}+31q^{-38}-3q^{-40}-35q^{-42}-17q^{-44}+9q^{-46}+37q^{-48}+12q^{-50}-26q^{-52}-24q^{-54}-9q^{-56}+29q^{-58}+24q^{-60}-4q^{-62}-19q^{-64}-26q^{-66}+8q^{-68}+21q^{-70}+13q^{-72}-q^{-74}-22q^{-76}-8q^{-78}+4q^{-80}+12q^{-82}+11q^{-84}-7q^{-86}-7q^{-88}-5q^{-90}+q^{-92}+6q^{-94}+q^{-96}-2q^{-100}-q^{-102}+q^{-104}$
5	$-q^{95}+q^{93}+q^{91}+q^{87}-q^{85}-3q^{83}-q^{81}+q^{79}+4q^{75}+3q^{73}-2q^{71}-6q^{69}-6q^{67}+9q^{63}+15q^{61}+7q^{59}-16q^{57}-27q^{55}-13q^{53}+18q^{51}+43q^{49}+34q^{47}-19q^{45}-68q^{43}-55q^{41}+10q^{39}+83q^{37}+87q^{35}+7q^{33}-94q^{31}-118q^{29}-36q^{27}+85q^{25}+139q^{23}+63q^{21}-64q^{19}-131q^{17}-93q^{15}+31q^{13}+120q^{11}+99q^{9}+3q^{7}-78q^{5}-93q^{3}-33q+47q^{-1}+77q^{-3}+48q^{-5}-9q^{-7}-50q^{-9}-53q^{-11}-20q^{-13}+30q^{-15}+53q^{-17}+35q^{-19}-10q^{-21}-54q^{-23}-52q^{-25}+3q^{-27}+59q^{-29}+57q^{-31}+2q^{-33}-62q^{-35}-71q^{-37}-3q^{-39}+74q^{-41}+77q^{-43}+9q^{-45}-78q^{-47}-92q^{-49}-16q^{-51}+80q^{-53}+103q^{-55}+31q^{-57}-73q^{-59}-114q^{-61}-50q^{-63}+57q^{-65}+111q^{-67}+73q^{-69}-30q^{-71}-106q^{-73}-89q^{-75}+82q^{-79}+96q^{-81}+36q^{-83}-52q^{-85}-89q^{-87}-58q^{-89}+15q^{-91}+68q^{-93}+67q^{-95}+21q^{-97}-38q^{-99}-63q^{-101}-41q^{-103}+7q^{-105}+42q^{-107}+45q^{-109}+19q^{-111}-19q^{-113}-37q^{-115}-28q^{-117}-q^{-119}+20q^{-121}+25q^{-123}+15q^{-125}-6q^{-127}-17q^{-129}-14q^{-131}-3q^{-133}+5q^{-135}+9q^{-137}+7q^{-139}-q^{-141}-4q^{-143}-3q^{-145}-q^{-147}+2q^{-151}+q^{-153}-q^{-155}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{10}-q^{4}+2q^{2}+1+2q^{-2}+q^{-4}+q^{-8}-q^{-10}-q^{-16}$
1,1	$q^{28}-2q^{26}+4q^{24}-6q^{22}+9q^{20}-12q^{18}+14q^{16}-20q^{14}+19q^{12}-24q^{10}+22q^{8}-22q^{6}+18q^{4}-4q^{2}+20q^{-2}-27q^{-4}+44q^{-6}-48q^{-8}+56q^{-10}-52q^{-12}+48q^{-14}-40q^{-16}+24q^{-18}-15q^{-20}-4q^{-22}+14q^{-24}-24q^{-26}+31q^{-28}-32q^{-30}+30q^{-32}-24q^{-34}+17q^{-36}-12q^{-38}+6q^{-40}-2q^{-42}+q^{-44}$
2,0	$q^{26}-q^{22}+q^{18}-q^{16}-3q^{14}+2q^{10}-3q^{8}-3q^{6}+2q^{4}+2q^{2}+2+2q^{-2}+5q^{-4}+2q^{-6}+2q^{-8}+3q^{-10}+q^{-12}-2q^{-14}-4q^{-20}-3q^{-22}+q^{-26}-q^{-28}-q^{-30}+2q^{-32}+q^{-34}-q^{-36}-q^{-38}+q^{-42}$
3,0	$-q^{48}+q^{44}+q^{42}-2q^{38}+2q^{36}+3q^{34}+q^{32}-4q^{30}-5q^{28}+4q^{26}+6q^{24}-q^{22}-10q^{20}-8q^{18}+8q^{16}+9q^{14}-3q^{12}-13q^{10}-8q^{8}+8q^{6}+6q^{4}-4+4q^{-2}+10q^{-4}+8q^{-6}+3q^{-8}+3q^{-10}+7q^{-12}+2q^{-14}-3q^{-16}-2q^{-18}+3q^{-20}+2q^{-22}-8q^{-24}-9q^{-26}+6q^{-30}+q^{-32}-10q^{-34}-7q^{-36}+4q^{-38}+8q^{-40}+2q^{-42}-8q^{-44}-5q^{-46}+3q^{-48}+7q^{-50}+4q^{-52}-5q^{-54}-4q^{-56}+5q^{-60}+4q^{-62}-2q^{-64}-3q^{-66}-3q^{-68}+q^{-70}+2q^{-72}+q^{-74}-q^{-78}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{22}-q^{20}+2q^{16}-3q^{14}-3q^{12}+2q^{10}-3q^{8}-3q^{6}+5q^{4}+q^{2}+2+4q^{-2}+3q^{-4}+q^{-6}+3q^{-10}+2q^{-12}-3q^{-14}+q^{-16}+q^{-18}-5q^{-20}-q^{-22}+q^{-24}-3q^{-26}+q^{-28}+q^{-30}-q^{-32}+q^{-34}$
1,0,0	$-q^{13}-q^{9}-q^{5}+2q^{3}+q+2q^{-1}+2q^{-3}+q^{-5}+q^{-7}+q^{-11}-q^{-13}-q^{-17}-q^{-21}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{28}+q^{22}+q^{20}-2q^{18}-4q^{16}-2q^{14}-q^{12}-4q^{10}-4q^{8}+3q^{6}+2q^{4}+q^{2}+4+7q^{-2}+4q^{-4}+3q^{-6}+5q^{-8}+4q^{-10}+2q^{-14}+4q^{-16}-2q^{-18}-3q^{-20}+q^{-22}-2q^{-24}-6q^{-26}-4q^{-28}-q^{-32}-2q^{-34}+q^{-36}+2q^{-38}+q^{-44}$
1,0,0,0	$-q^{16}-q^{12}-q^{10}-q^{6}+2q^{4}+q^{2}+2+2q^{-2}+2q^{-4}+q^{-6}+q^{-8}+q^{-10}+q^{-14}-q^{-16}-q^{-20}-q^{-22}-q^{-26}$

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

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}-q^{28}+q^{26}-q^{24}+2 q^{22}-3 q^{20}-4 q^{16}+2 q^{14}-4 q^{12}-2 q^8+2 q^6+3 q^4+q^2+5- q^{-2} +7 q^{-4} -2 q^{-6} +6 q^{-8} -4 q^{-10} +6 q^{-12} -2 q^{-14} +5 q^{-16} -2 q^{-18} +2 q^{-20} - q^{-22} - q^{-24} - q^{-26} -4 q^{-28} + q^{-30} -4 q^{-32} + q^{-34} -3 q^{-36} +3 q^{-38} -2 q^{-40} +2 q^{-42} - q^{-44} + q^{-46} }

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}-q^{50}+2 q^{48}-2 q^{46}-3 q^{40}+4 q^{38}-5 q^{36}+4 q^{34}-5 q^{32}+q^{30}+3 q^{28}-6 q^{26}+8 q^{24}-9 q^{22}+7 q^{20}-4 q^{18}-3 q^{16}+7 q^{14}-8 q^{12}+9 q^{10}-q^8-3 q^6+6 q^4-3 q^2+1+6 q^{-2} -10 q^{-4} +12 q^{-6} -5 q^{-8} +10 q^{-12} -14 q^{-14} +17 q^{-16} -11 q^{-18} +3 q^{-20} +4 q^{-22} -9 q^{-24} +14 q^{-26} -10 q^{-28} +6 q^{-30} + q^{-32} -5 q^{-34} +7 q^{-36} -5 q^{-38} +4 q^{-42} -8 q^{-44} +7 q^{-46} -3 q^{-48} -4 q^{-50} +10 q^{-52} -13 q^{-54} +10 q^{-56} -5 q^{-58} -4 q^{-60} +7 q^{-62} -10 q^{-64} +9 q^{-66} -5 q^{-68} +2 q^{-72} -4 q^{-74} +3 q^{-76} - q^{-78} + q^{-80} }

.

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["8 8"];

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 2 t^2-6 t+9-6 t^{-1} +2 t^{-2} }

In[5]:=

Conway[K][z]

Out[5]=

$2z^{4}+2z^{2}+1$

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]=

{ 25, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{7}a^{-1}+z^{7}a^{-3}+4z^{6}a^{-2}+2z^{6}a^{-4}+2z^{6}+2az^{5}+z^{5}a^{-1}+z^{5}a^{-5}+2a^{2}z^{4}-9z^{4}a^{-2}-6z^{4}a^{-4}-z^{4}+a^{3}z^{3}-3z^{3}a^{-1}-5z^{3}a^{-3}-3z^{3}a^{-5}-2a^{2}z^{2}+5z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}-a^{3}z-az+za^{-1}+3za^{-3}+2za^{-5}+a^{2}-a^{-2}-a^{-4}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_129, K11n39, K11n45, K11n50, K11n132,}

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}} ): {10_129,}

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["8 8"];

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 2 t^2-6 t+9-6 t^{-1} +2 t^{-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^5+2 q^4-3 q^3+4 q^2-4 q+5-3 q^{-1} +2 q^{-2} - q^{-3} } }

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]=

{10_129, K11n39, K11n45, K11n50, K11n132,}

In[6]:=

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

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

Out[6]=

{10_129,}

Vassiliev invariants

V2 and V3:

(2, 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 8} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 32} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{124}{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{4}{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 64} 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{272}{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{32}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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{256}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 32} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{992}{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{32}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{6271}{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{1516}{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{484}{45}} 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{113}{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 -\frac{449}{15}}

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 8 8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 χ 11                 1 -1 9               1   1 7             2 1   -1 5           2 1     1 3         2 2       0 1       3 2         1 -1     1 3           2 -3   1 2             -1 -5   1               1 -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}\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}^{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=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^{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}^{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=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}^{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=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^{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=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}\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=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}_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^{15}-2 q^{14}-q^{13}+6 q^{12}-4 q^{11}-6 q^{10}+12 q^9-4 q^8-13 q^7+17 q^6-q^5-18 q^4+19 q^3+2 q^2-20 q+17+3 q^{-1} -15 q^{-2} +10 q^{-3} +2 q^{-4} -7 q^{-5} +4 q^{-6} -2 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^{30}+2 q^{29}+q^{28}-2 q^{27}-5 q^{26}+3 q^{25}+9 q^{24}-q^{23}-14 q^{22}-2 q^{21}+17 q^{20}+9 q^{19}-21 q^{18}-15 q^{17}+21 q^{16}+22 q^{15}-19 q^{14}-30 q^{13}+17 q^{12}+35 q^{11}-12 q^{10}-42 q^9+10 q^8+43 q^7-2 q^6-50 q^5+3 q^4+46 q^3+6 q^2-49 q-2+38 q^{-1} +10 q^{-2} -35 q^{-3} -6 q^{-4} +24 q^{-5} +6 q^{-6} -17 q^{-7} -3 q^{-8} +10 q^{-9} + q^{-10} -6 q^{-11} +4 q^{-13} -2 q^{-14} - q^{-15} +2 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^{50}-2 q^{49}-q^{48}+2 q^{47}+q^{46}+6 q^{45}-7 q^{44}-7 q^{43}+q^{41}+24 q^{40}-6 q^{39}-15 q^{38}-12 q^{37}-13 q^{36}+45 q^{35}+8 q^{34}-7 q^{33}-25 q^{32}-47 q^{31}+52 q^{30}+23 q^{29}+21 q^{28}-20 q^{27}-85 q^{26}+37 q^{25}+21 q^{24}+59 q^{23}+5 q^{22}-113 q^{21}+11 q^{20}+3 q^{19}+91 q^{18}+39 q^{17}-125 q^{16}-16 q^{15}-22 q^{14}+116 q^{13}+71 q^{12}-129 q^{11}-39 q^{10}-44 q^9+131 q^8+95 q^7-124 q^6-56 q^5-61 q^4+130 q^3+108 q^2-103 q-56-73 q^{-1} +103 q^{-2} +102 q^{-3} -66 q^{-4} -39 q^{-5} -70 q^{-6} +61 q^{-7} +71 q^{-8} -34 q^{-9} -10 q^{-10} -48 q^{-11} +24 q^{-12} +34 q^{-13} -17 q^{-14} +8 q^{-15} -23 q^{-16} +7 q^{-17} +11 q^{-18} -11 q^{-19} +10 q^{-20} -7 q^{-21} +2 q^{-22} +2 q^{-23} -6 q^{-24} +5 q^{-25} - q^{-26} + q^{-27} -2 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^{75}+2 q^{74}+q^{73}-2 q^{72}-q^{71}-2 q^{70}-2 q^{69}+5 q^{68}+9 q^{67}-5 q^{65}-10 q^{64}-13 q^{63}+2 q^{62}+20 q^{61}+21 q^{60}+5 q^{59}-15 q^{58}-34 q^{57}-25 q^{56}+11 q^{55}+39 q^{54}+43 q^{53}+11 q^{52}-37 q^{51}-60 q^{50}-37 q^{49}+17 q^{48}+68 q^{47}+70 q^{46}+9 q^{45}-59 q^{44}-90 q^{43}-56 q^{42}+37 q^{41}+107 q^{40}+97 q^{39}+q^{38}-104 q^{37}-138 q^{36}-52 q^{35}+90 q^{34}+173 q^{33}+104 q^{32}-66 q^{31}-192 q^{30}-159 q^{29}+26 q^{28}+214 q^{27}+208 q^{26}+6 q^{25}-215 q^{24}-255 q^{23}-50 q^{22}+228 q^{21}+295 q^{20}+74 q^{19}-218 q^{18}-332 q^{17}-118 q^{16}+237 q^{15}+359 q^{14}+129 q^{13}-216 q^{12}-388 q^{11}-173 q^{10}+235 q^9+403 q^8+174 q^7-198 q^6-411 q^5-223 q^4+202 q^3+406 q^2+215 q-141-382 q^{-1} -253 q^{-2} +122 q^{-3} +346 q^{-4} +230 q^{-5} -60 q^{-6} -286 q^{-7} -232 q^{-8} +33 q^{-9} +222 q^{-10} +192 q^{-11} +7 q^{-12} -159 q^{-13} -156 q^{-14} -21 q^{-15} +101 q^{-16} +110 q^{-17} +31 q^{-18} -58 q^{-19} -76 q^{-20} -25 q^{-21} +28 q^{-22} +45 q^{-23} +18 q^{-24} -9 q^{-25} -23 q^{-26} -16 q^{-27} +3 q^{-28} +14 q^{-29} +4 q^{-30} +2 q^{-31} -8 q^{-33} -3 q^{-34} +5 q^{-35} -2 q^{-36} +2 q^{-37} +4 q^{-38} -3 q^{-39} -2 q^{-40} + q^{-41} - q^{-42} +2 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^{105}-2 q^{104}-q^{103}+2 q^{102}+q^{101}+2 q^{100}-2 q^{99}+4 q^{98}-7 q^{97}-9 q^{96}+3 q^{95}+4 q^{94}+11 q^{93}+2 q^{92}+18 q^{91}-14 q^{90}-27 q^{89}-13 q^{88}-7 q^{87}+16 q^{86}+9 q^{85}+65 q^{84}+6 q^{83}-30 q^{82}-37 q^{81}-47 q^{80}-21 q^{79}-24 q^{78}+111 q^{77}+60 q^{76}+30 q^{75}-10 q^{74}-58 q^{73}-90 q^{72}-139 q^{71}+78 q^{70}+64 q^{69}+116 q^{68}+100 q^{67}+49 q^{66}-82 q^{65}-267 q^{64}-54 q^{63}-70 q^{62}+100 q^{61}+200 q^{60}+269 q^{59}+82 q^{58}-267 q^{57}-171 q^{56}-313 q^{55}-89 q^{54}+154 q^{53}+472 q^{52}+354 q^{51}-85 q^{50}-146 q^{49}-531 q^{48}-389 q^{47}-69 q^{46}+540 q^{45}+606 q^{44}+210 q^{43}+35 q^{42}-628 q^{41}-682 q^{40}-390 q^{39}+472 q^{38}+759 q^{37}+508 q^{36}+292 q^{35}-615 q^{34}-899 q^{33}-705 q^{32}+340 q^{31}+827 q^{30}+750 q^{29}+534 q^{28}-557 q^{27}-1050 q^{26}-957 q^{25}+220 q^{24}+859 q^{23}+934 q^{22}+715 q^{21}-508 q^{20}-1162 q^{19}-1146 q^{18}+133 q^{17}+887 q^{16}+1075 q^{15}+844 q^{14}-464 q^{13}-1239 q^{12}-1287 q^{11}+41 q^{10}+880 q^9+1171 q^8+956 q^7-367 q^6-1237 q^5-1374 q^4-105 q^3+768 q^2+1170 q+1047-174 q^{-1} -1085 q^{-2} -1342 q^{-3} -283 q^{-4} +519 q^{-5} +1000 q^{-6} +1029 q^{-7} +62 q^{-8} -766 q^{-9} -1118 q^{-10} -382 q^{-11} +213 q^{-12} +667 q^{-13} +839 q^{-14} +210 q^{-15} -397 q^{-16} -743 q^{-17} -328 q^{-18} +309 q^{-20} +534 q^{-21} +212 q^{-22} -137 q^{-23} -382 q^{-24} -181 q^{-25} -65 q^{-26} +79 q^{-27} +264 q^{-28} +128 q^{-29} -25 q^{-30} -157 q^{-31} -59 q^{-32} -46 q^{-33} -11 q^{-34} +111 q^{-35} +54 q^{-36} + q^{-37} -56 q^{-38} -7 q^{-39} -19 q^{-40} -25 q^{-41} +43 q^{-42} +18 q^{-43} +4 q^{-44} -19 q^{-45} +5 q^{-46} -7 q^{-47} -17 q^{-48} +16 q^{-49} +4 q^{-50} +4 q^{-51} -6 q^{-52} +4 q^{-53} -2 q^{-54} -8 q^{-55} +5 q^{-56} +2 q^{-58} - q^{-59} + q^{-60} -2 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^{140}+2 q^{139}+q^{138}-2 q^{137}-q^{136}-2 q^{135}+2 q^{134}-2 q^{132}+7 q^{131}+6 q^{130}-2 q^{129}-4 q^{128}-13 q^{127}-4 q^{126}-9 q^{124}+18 q^{123}+23 q^{122}+16 q^{121}+9 q^{120}-27 q^{119}-24 q^{118}-21 q^{117}-43 q^{116}+4 q^{115}+33 q^{114}+49 q^{113}+77 q^{112}+7 q^{111}-12 q^{110}-32 q^{109}-109 q^{108}-69 q^{107}-40 q^{106}+15 q^{105}+130 q^{104}+97 q^{103}+97 q^{102}+76 q^{101}-89 q^{100}-116 q^{99}-177 q^{98}-180 q^{97}+12 q^{96}+60 q^{95}+195 q^{94}+296 q^{93}+139 q^{92}+67 q^{91}-148 q^{90}-373 q^{89}-293 q^{88}-273 q^{87}-3 q^{86}+366 q^{85}+411 q^{84}+514 q^{83}+262 q^{82}-228 q^{81}-460 q^{80}-750 q^{79}-577 q^{78}-18 q^{77}+361 q^{76}+896 q^{75}+931 q^{74}+387 q^{73}-134 q^{72}-950 q^{71}-1232 q^{70}-794 q^{69}-228 q^{68}+840 q^{67}+1448 q^{66}+1226 q^{65}+685 q^{64}-599 q^{63}-1559 q^{62}-1605 q^{61}-1183 q^{60}+250 q^{59}+1527 q^{58}+1903 q^{57}+1700 q^{56}+190 q^{55}-1405 q^{54}-2132 q^{53}-2160 q^{52}-646 q^{51}+1183 q^{50}+2250 q^{49}+2583 q^{48}+1126 q^{47}-933 q^{46}-2318 q^{45}-2934 q^{44}-1548 q^{43}+654 q^{42}+2317 q^{41}+3230 q^{40}+1952 q^{39}-396 q^{38}-2323 q^{37}-3464 q^{36}-2271 q^{35}+160 q^{34}+2287 q^{33}+3675 q^{32}+2578 q^{31}+14 q^{30}-2306 q^{29}-3832 q^{28}-2785 q^{27}-181 q^{26}+2282 q^{25}+4005 q^{24}+3018 q^{23}+281 q^{22}-2337 q^{21}-4118 q^{20}-3156 q^{19}-411 q^{18}+2302 q^{17}+4257 q^{16}+3370 q^{15}+515 q^{14}-2338 q^{13}-4324 q^{12}-3478 q^{11}-681 q^{10}+2199 q^9+4373 q^8+3687 q^7+877 q^6-2110 q^5-4318 q^4-3738 q^3-1121 q^2+1787 q+4157+3852 q^{-1} +1387 q^{-2} -1500 q^{-3} -3863 q^{-4} -3738 q^{-5} -1611 q^{-6} +1019 q^{-7} +3423 q^{-8} +3585 q^{-9} +1790 q^{-10} -617 q^{-11} -2879 q^{-12} -3209 q^{-13} -1837 q^{-14} +175 q^{-15} +2267 q^{-16} +2766 q^{-17} +1751 q^{-18} +142 q^{-19} -1658 q^{-20} -2213 q^{-21} -1558 q^{-22} -363 q^{-23} +1133 q^{-24} +1662 q^{-25} +1269 q^{-26} +446 q^{-27} -695 q^{-28} -1157 q^{-29} -964 q^{-30} -445 q^{-31} +396 q^{-32} +763 q^{-33} +664 q^{-34} +369 q^{-35} -210 q^{-36} -452 q^{-37} -424 q^{-38} -286 q^{-39} +103 q^{-40} +261 q^{-41} +256 q^{-42} +195 q^{-43} -61 q^{-44} -142 q^{-45} -133 q^{-46} -121 q^{-47} +29 q^{-48} +67 q^{-49} +74 q^{-50} +90 q^{-51} -30 q^{-52} -47 q^{-53} -32 q^{-54} -40 q^{-55} +18 q^{-56} +8 q^{-57} +17 q^{-58} +44 q^{-59} -14 q^{-60} -21 q^{-61} -8 q^{-62} -13 q^{-63} +11 q^{-64} -3 q^{-65} +2 q^{-66} +22 q^{-67} -5 q^{-68} -8 q^{-69} -2 q^{-70} -6 q^{-71} +5 q^{-72} -3 q^{-73} +8 q^{-75} - q^{-76} -2 q^{-77} -2 q^{-79} + q^{-80} - q^{-81} +2 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.

8_7 8_9