8 13

8_12

8_14

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 8 13's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 13 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X11,1,12,16 X5,13,6,12 X15,7,16,6 X7,15,8,14 X13,9,14,8 X9,2,10,3
Gauss code	-1, 8, -2, 1, -4, 5, -6, 7, -8, 2, -3, 4, -7, 6, -5, 3
Dowker-Thistlethwaite code	4 10 12 14 2 16 8 6
Conway Notation	[31112]

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

[edit Notes on presentations of 8 13]

Knot 8_13.

A graph, knot 8_13.

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 13"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3,10,4,11 X11,1,12,16 X5,13,6,12 X15,7,16,6 X7,15,8,14 X13,9,14,8 X9,2,10,3

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 12 14 2 16 8 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[31112]

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,-1,2,-1,2,2,3,-2,3\})}

In[10]:=

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

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

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

Out[10]=

{ 4, 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, 5}, {1, 8}, {6, 9}, {8, 10}, {9, 4}, {5, 2}, {3, 1}, {4, 7}, {2, 6}, {7, 3}]

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 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 8.53123 A-Polynomial See Data:8 13/A-polynomial

[edit Notes for 8 13'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 1} 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 1} 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 2} Rasmussen s-Invariant 0

[edit Notes for 8 13'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^2-7 t+11-7 t^{-1} +2 t^{-2} }
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 2 z^4+z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 29, 0 }
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^5+2 q^4-3 q^3+5 q^2-5 q+5-4 q^{-1} +3 q^{-2} - q^{-3} }
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-2}+z^{4}-a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}+z^{2}+2a^{-2}-a^{-4}$
Kauffman polynomial (db, data sources)	$z^{7}a^{-1}+z^{7}a^{-3}+5z^{6}a^{-2}+2z^{6}a^{-4}+3z^{6}+4az^{5}+4z^{5}a^{-1}+z^{5}a^{-3}+z^{5}a^{-5}+3a^{2}z^{4}-11z^{4}a^{-2}-6z^{4}a^{-4}-2z^{4}+a^{3}z^{3}-4az^{3}-9z^{3}a^{-1}-7z^{3}a^{-3}-3z^{3}a^{-5}-2a^{2}z^{2}+7z^{2}a^{-2}+5z^{2}a^{-4}+az+3za^{-1}+4za^{-3}+2za^{-5}-2a^{-2}-a^{-4}$
The A2 invariant	$-q^{10}+q^{8}+q^{6}-q^{4}+q^{2}-1+q^{-2}+q^{-4}+q^{-6}+2q^{-8}-q^{-10}-q^{-16}$
The G2 invariant	$q^{52}-2q^{50}+3q^{48}-4q^{46}+q^{44}-3q^{40}+9q^{38}-11q^{36}+12q^{34}-8q^{32}+6q^{28}-13q^{26}+18q^{24}-16q^{22}+10q^{20}-8q^{16}+14q^{14}-10q^{12}+4q^{10}+2q^{8}-10q^{6}+9q^{4}-3q^{2}-7+17q^{-2}-22q^{-4}+19q^{-6}-6q^{-8}-9q^{-10}+19q^{-12}-26q^{-14}+26q^{-16}-14q^{-18}+3q^{-20}+11q^{-22}-16q^{-24}+21q^{-26}-10q^{-28}+q^{-30}+6q^{-32}-9q^{-34}+8q^{-36}-6q^{-40}+14q^{-42}-15q^{-44}+9q^{-46}+q^{-48}-13q^{-50}+18q^{-52}-19q^{-54}+11q^{-56}-4q^{-58}-6q^{-60}+11q^{-62}-13q^{-64}+10q^{-66}-4q^{-68}-q^{-70}+2q^{-72}-4q^{-74}+3q^{-76}-q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 8_13.

A1 Invariants.

Weight	Invariant
1	$-q^{7}+2q^{5}-q^{3}+q+2q^{-5}-q^{-7}+q^{-9}-q^{-11}$
2	$q^{20}-2q^{18}-q^{16}+5q^{14}-4q^{12}-3q^{10}+7q^{8}-3q^{6}-3q^{4}+6q^{2}+1-2q^{-2}+3q^{-6}-q^{-8}-5q^{-10}+4q^{-12}+3q^{-14}-6q^{-16}+3q^{-18}+4q^{-20}-5q^{-22}+3q^{-26}-2q^{-28}-q^{-30}+q^{-32}$
3	$-q^{39}+2q^{37}+q^{35}-3q^{33}-2q^{31}+4q^{29}+6q^{27}-9q^{25}-8q^{23}+10q^{21}+11q^{19}-12q^{17}-15q^{15}+12q^{13}+19q^{11}-8q^{9}-18q^{7}+5q^{5}+16q^{3}+q-11q^{-1}-6q^{-3}+7q^{-5}+10q^{-7}-q^{-9}-12q^{-11}-3q^{-13}+14q^{-15}+8q^{-17}-15q^{-19}-11q^{-21}+13q^{-23}+13q^{-25}-10q^{-27}-17q^{-29}+8q^{-31}+17q^{-33}-2q^{-35}-17q^{-37}-q^{-39}+14q^{-41}+5q^{-43}-10q^{-45}-7q^{-47}+5q^{-49}+6q^{-51}-2q^{-53}-4q^{-55}+2q^{-59}+q^{-61}-q^{-63}$
4	$q^{64}-2q^{62}-q^{60}+3q^{58}+2q^{54}-7q^{52}-q^{50}+10q^{48}+q^{46}+3q^{44}-21q^{42}-6q^{40}+26q^{38}+13q^{36}-47q^{32}-19q^{30}+43q^{28}+40q^{26}+11q^{24}-67q^{22}-47q^{20}+36q^{18}+60q^{16}+35q^{14}-55q^{12}-60q^{10}+6q^{8}+46q^{6}+49q^{4}-18q^{2}-43-22q^{-2}+11q^{-4}+39q^{-6}+19q^{-8}-13q^{-10}-39q^{-12}-19q^{-14}+24q^{-16}+43q^{-18}+9q^{-20}-44q^{-22}-35q^{-24}+9q^{-26}+59q^{-28}+26q^{-30}-46q^{-32}-47q^{-34}-7q^{-36}+64q^{-38}+41q^{-40}-34q^{-42}-52q^{-44}-32q^{-46}+51q^{-48}+56q^{-50}-4q^{-52}-42q^{-54}-54q^{-56}+18q^{-58}+50q^{-60}+26q^{-62}-12q^{-64}-51q^{-66}-12q^{-68}+21q^{-70}+29q^{-72}+14q^{-74}-25q^{-76}-18q^{-78}-4q^{-80}+12q^{-82}+16q^{-84}-4q^{-86}-6q^{-88}-6q^{-90}+6q^{-94}+q^{-96}-2q^{-100}-q^{-102}+q^{-104}$
5	$-q^{95}+2q^{93}+q^{91}-3q^{89}+q^{83}+2q^{81}-6q^{77}-4q^{75}+7q^{73}+12q^{71}+4q^{69}-13q^{67}-20q^{65}-14q^{63}+20q^{61}+52q^{59}+24q^{57}-36q^{55}-78q^{53}-56q^{51}+38q^{49}+124q^{47}+100q^{45}-36q^{43}-161q^{41}-154q^{39}+5q^{37}+184q^{35}+211q^{33}+41q^{31}-183q^{29}-251q^{27}-94q^{25}+149q^{23}+266q^{21}+146q^{19}-95q^{17}-245q^{15}-175q^{13}+34q^{11}+195q^{9}+183q^{7}+27q^{5}-131q^{3}-166q-71q^{-1}+63q^{-3}+132q^{-5}+100q^{-7}+q^{-9}-98q^{-11}-115q^{-13}-48q^{-15}+65q^{-17}+128q^{-19}+82q^{-21}-39q^{-23}-136q^{-25}-111q^{-27}+27q^{-29}+148q^{-31}+130q^{-33}-18q^{-35}-157q^{-37}-154q^{-39}+6q^{-41}+171q^{-43}+177q^{-45}+11q^{-47}-176q^{-49}-200q^{-51}-41q^{-53}+166q^{-55}+225q^{-57}+77q^{-59}-142q^{-61}-231q^{-63}-119q^{-65}+90q^{-67}+226q^{-69}+164q^{-71}-37q^{-73}-191q^{-75}-185q^{-77}-36q^{-79}+139q^{-81}+190q^{-83}+86q^{-85}-73q^{-87}-162q^{-89}-121q^{-91}+8q^{-93}+117q^{-95}+127q^{-97}+42q^{-99}-62q^{-101}-107q^{-103}-66q^{-105}+13q^{-107}+70q^{-109}+69q^{-111}+18q^{-113}-35q^{-115}-53q^{-117}-29q^{-119}+7q^{-121}+30q^{-123}+28q^{-125}+6q^{-127}-14q^{-129}-17q^{-131}-7q^{-133}+2q^{-135}+8q^{-137}+8q^{-139}-4q^{-143}-3q^{-145}-q^{-147}+2q^{-151}+q^{-153}-q^{-155}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{10}+q^{8}+q^{6}-q^{4}+q^{2}-1+q^{-2}+q^{-4}+q^{-6}+2q^{-8}-q^{-10}-q^{-16}$
1,1	$q^{28}-4q^{26}+8q^{24}-12q^{22}+20q^{20}-32q^{18}+38q^{16}-44q^{14}+51q^{12}-52q^{10}+44q^{8}-30q^{6}+13q^{4}+12q^{2}-36+66q^{-2}-82q^{-4}+100q^{-6}-102q^{-8}+98q^{-10}-86q^{-12}+66q^{-14}-42q^{-16}+12q^{-18}+10q^{-20}-28q^{-22}+46q^{-24}-54q^{-26}+55q^{-28}-48q^{-30}+40q^{-32}-32q^{-34}+19q^{-36}-12q^{-38}+6q^{-40}-2q^{-42}+q^{-44}$
2,0	$q^{26}-q^{24}-2q^{22}+q^{20}+3q^{18}-5q^{14}+3q^{10}-2q^{8}-2q^{6}+4q^{4}+4q^{2}+1+q^{-2}+2q^{-4}-q^{-6}-2q^{-8}+q^{-10}-q^{-12}-3q^{-14}+3q^{-16}+4q^{-18}-q^{-20}-2q^{-22}+2q^{-24}+2q^{-26}-2q^{-28}-3q^{-30}+q^{-32}+q^{-34}-q^{-36}-q^{-38}+q^{-42}$

A3 Invariants.

Weight	Invariant
0,1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{22}-2 q^{20}-q^{18}+4 q^{16}-3 q^{14}-q^{12}+6 q^{10}-3 q^8-3 q^6+4 q^4-2 q^2-2+2 q^{-2} +2 q^{-4} + q^{-6} +5 q^{-10} +4 q^{-12} -4 q^{-14} +3 q^{-16} +2 q^{-18} -6 q^{-20} + q^{-24} -4 q^{-26} + q^{-28} + q^{-30} - q^{-32} + q^{-34} }
1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{13}+q^{11}+q^7-q^5+q^3-q+ q^{-3} + q^{-5} +2 q^{-7} + q^{-9} +2 q^{-11} - q^{-13} - q^{-17} - q^{-21} }

A4 Invariants.

Weight	Invariant
0,1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{28}-q^{26}-2 q^{24}+q^{22}+2 q^{20}-q^{18}-2 q^{16}+3 q^{14}+3 q^{12}-3 q^{10}-3 q^8+4 q^6-4 q^2+1+2 q^{-2} -2 q^{-4} - q^{-6} +4 q^{-8} +3 q^{-10} + q^{-12} +7 q^{-14} +8 q^{-16} - q^{-20} +4 q^{-22} -2 q^{-24} -7 q^{-26} -4 q^{-28} -2 q^{-32} -3 q^{-34} + q^{-36} +2 q^{-38} + q^{-44} }
1,0,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{16}+q^{14}+q^8-q^6+q^4-q^2+ q^{-4} + q^{-6} +2 q^{-8} +2 q^{-10} + q^{-12} +2 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}+2 q^{20}-3 q^{18}+4 q^{16}-5 q^{14}+5 q^{12}-4 q^{10}+3 q^8-q^6+4 q^2-6+8 q^{-2} -8 q^{-4} +9 q^{-6} -8 q^{-8} +7 q^{-10} -4 q^{-12} +2 q^{-14} + q^{-16} -2 q^{-18} +4 q^{-20} -4 q^{-22} +5 q^{-24} -4 q^{-26} +3 q^{-28} -3 q^{-30} + q^{-32} - q^{-34} }
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{36}-2 q^{32}-2 q^{30}+q^{28}+4 q^{26}+q^{24}-4 q^{22}-3 q^{20}+3 q^{18}+5 q^{16}-5 q^{12}-2 q^{10}+3 q^8+3 q^6-3 q^4-3 q^2+2+4 q^{-2} -3 q^{-6} +4 q^{-10} +2 q^{-12} -2 q^{-14} - q^{-16} +4 q^{-18} +3 q^{-20} -2 q^{-22} -4 q^{-24} +2 q^{-26} +5 q^{-28} + q^{-30} -5 q^{-32} -4 q^{-34} +2 q^{-36} +4 q^{-38} - q^{-40} -4 q^{-42} -2 q^{-44} +2 q^{-46} +2 q^{-48} - q^{-50} - q^{-52} + q^{-56} }

D4 Invariants.

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

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}-2 q^{50}+3 q^{48}-4 q^{46}+q^{44}-3 q^{40}+9 q^{38}-11 q^{36}+12 q^{34}-8 q^{32}+6 q^{28}-13 q^{26}+18 q^{24}-16 q^{22}+10 q^{20}-8 q^{16}+14 q^{14}-10 q^{12}+4 q^{10}+2 q^8-10 q^6+9 q^4-3 q^2-7+17 q^{-2} -22 q^{-4} +19 q^{-6} -6 q^{-8} -9 q^{-10} +19 q^{-12} -26 q^{-14} +26 q^{-16} -14 q^{-18} +3 q^{-20} +11 q^{-22} -16 q^{-24} +21 q^{-26} -10 q^{-28} + q^{-30} +6 q^{-32} -9 q^{-34} +8 q^{-36} -6 q^{-40} +14 q^{-42} -15 q^{-44} +9 q^{-46} + q^{-48} -13 q^{-50} +18 q^{-52} -19 q^{-54} +11 q^{-56} -4 q^{-58} -6 q^{-60} +11 q^{-62} -13 q^{-64} +10 q^{-66} -4 q^{-68} - q^{-70} +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 13"];

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^2-7 t+11-7 t^{-1} +2 t^{-2} }

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 2 z^4+z^2+1}

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 29, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^5+2 q^4-3 q^3+5 q^2-5 q+5-4 q^{-1} +3 q^{-2} - q^{-3} }

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{7}a^{-1}+z^{7}a^{-3}+5z^{6}a^{-2}+2z^{6}a^{-4}+3z^{6}+4az^{5}+4z^{5}a^{-1}+z^{5}a^{-3}+z^{5}a^{-5}+3a^{2}z^{4}-11z^{4}a^{-2}-6z^{4}a^{-4}-2z^{4}+a^{3}z^{3}-4az^{3}-9z^{3}a^{-1}-7z^{3}a^{-3}-3z^{3}a^{-5}-2a^{2}z^{2}+7z^{2}a^{-2}+5z^{2}a^{-4}+az+3za^{-1}+4za^{-3}+2za^{-5}-2a^{-2}-a^{-4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {}

Computer Talk

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

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

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

In[3]:=

K = Knot["8 13"];

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^2-7 t+11-7 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+5 q^2-5 q+5-4 q^{-1} +3 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]=

{}

In[6]:=

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

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

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(1, 1)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8} 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{14}{3}} $-{\frac {38}{3}}$ 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{80}{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 -24} 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 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{56}{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{152}{3}} ${\frac {1951}{30}}$ 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{526}{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 -\frac{6058}{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{31}{18}} 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{1409}{30}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 0 is the signature of 8 13. 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           3 1     2 3         2 2       0 1       3 3         0 -1     2 3           1 -3   1 2             -1 -5   2               2 -7 1                 -1

Integral Khovanov Homology (db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{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=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}^{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}^{2}} $r=2$ 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^{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}^{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=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}-5 q^{11}-6 q^{10}+15 q^9-6 q^8-15 q^7+24 q^6-5 q^5-24 q^4+28 q^3-q^2-27 q+26+2 q^{-1} -22 q^{-2} +17 q^{-3} +2 q^{-4} -12 q^{-5} +7 q^{-6} + q^{-7} -3 q^{-8} + q^{-9} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{30}+2 q^{29}+q^{28}-2 q^{27}-5 q^{26}+4 q^{25}+9 q^{24}-3 q^{23}-17 q^{22}+q^{21}+24 q^{20}+6 q^{19}-32 q^{18}-15 q^{17}+39 q^{16}+25 q^{15}-41 q^{14}-40 q^{13}+46 q^{12}+48 q^{11}-41 q^{10}-64 q^9+42 q^8+71 q^7-35 q^6-81 q^5+33 q^4+82 q^3-24 q^2-84 q+20+77 q^{-1} -12 q^{-2} -69 q^{-3} +9 q^{-4} +54 q^{-5} -2 q^{-6} -42 q^{-7} +2 q^{-8} +27 q^{-9} + q^{-10} -19 q^{-11} + q^{-12} +9 q^{-13} -4 q^{-15} - q^{-16} +3 q^{-17} - q^{-18} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{50}-2 q^{49}-q^{48}+2 q^{47}+q^{46}+6 q^{45}-8 q^{44}-7 q^{43}+2 q^{42}+3 q^{41}+26 q^{40}-12 q^{39}-23 q^{38}-12 q^{37}-4 q^{36}+65 q^{35}+3 q^{34}-31 q^{33}-45 q^{32}-43 q^{31}+104 q^{30}+41 q^{29}-7 q^{28}-77 q^{27}-115 q^{26}+116 q^{25}+79 q^{24}+53 q^{23}-82 q^{22}-198 q^{21}+96 q^{20}+97 q^{19}+128 q^{18}-59 q^{17}-269 q^{16}+56 q^{15}+98 q^{14}+200 q^{13}-26 q^{12}-319 q^{11}+12 q^{10}+89 q^9+253 q^8+8 q^7-338 q^6-31 q^5+69 q^4+279 q^3+40 q^2-318 q-59+36 q^{-1} +258 q^{-2} +65 q^{-3} -251 q^{-4} -62 q^{-5} -4 q^{-6} +192 q^{-7} +70 q^{-8} -161 q^{-9} -37 q^{-10} -28 q^{-11} +109 q^{-12} +50 q^{-13} -83 q^{-14} -8 q^{-15} -25 q^{-16} +47 q^{-17} +22 q^{-18} -36 q^{-19} +5 q^{-20} -12 q^{-21} +15 q^{-22} +7 q^{-23} -12 q^{-24} +3 q^{-25} -3 q^{-26} +4 q^{-27} + q^{-28} -3 q^{-29} + q^{-30} }
5	$-q^{75}+2q^{74}+q^{73}-2q^{72}-q^{71}-2q^{70}-2q^{69}+6q^{68}+9q^{67}-2q^{66}-7q^{65}-11q^{64}-12q^{63}+9q^{62}+29q^{61}+20q^{60}-5q^{59}-34q^{58}-48q^{57}-15q^{56}+47q^{55}+73q^{54}+46q^{53}-33q^{52}-105q^{51}-94q^{50}+6q^{49}+118q^{48}+150q^{47}+52q^{46}-115q^{45}-203q^{44}-123q^{43}+77q^{42}+239q^{41}+211q^{40}-11q^{39}-254q^{38}-298q^{37}-72q^{36}+233q^{35}+365q^{34}+190q^{33}-192q^{32}-434q^{31}-281q^{30}+121q^{29}+454q^{28}+409q^{27}-44q^{26}-493q^{25}-488q^{24}-38q^{23}+478q^{22}+596q^{21}+122q^{20}-499q^{19}-653q^{18}-198q^{17}+475q^{16}+735q^{15}+270q^{14}-481q^{13}-774q^{12}-336q^{11}+450q^{10}+832q^{9}+391q^{8}-435q^{7}-837q^{6}-449q^{5}+383q^{4}+849q^{3}+490q^{2}-336q-805-518q^{-1}+249q^{-2}+754q^{-3}+525q^{-4}-178q^{-5}-649q^{-6}-506q^{-7}+88q^{-8}+545q^{-9}+455q^{-10}-28q^{-11}-408q^{-12}-386q^{-13}-29q^{-14}+302q^{-15}+298q^{-16}+40q^{-17}-184q^{-18}-216q^{-19}-56q^{-20}+123q^{-21}+139q^{-22}+33q^{-23}-59q^{-24}-80q^{-25}-32q^{-26}+37q^{-27}+45q^{-28}+11q^{-29}-17q^{-30}-20q^{-31}-4q^{-32}+5q^{-33}+11q^{-34}+5q^{-35}-10q^{-36}-3q^{-37}+4q^{-38}+3q^{-41}-4q^{-42}-q^{-43}+3q^{-44}-q^{-45}$
6	$q^{105}-2q^{104}-q^{103}+2q^{102}+q^{101}+2q^{100}-2q^{99}+4q^{98}-8q^{97}-9q^{96}+5q^{95}+6q^{94}+12q^{93}+16q^{91}-22q^{90}-35q^{89}-9q^{88}+5q^{87}+35q^{86}+22q^{85}+72q^{84}-21q^{83}-80q^{82}-71q^{81}-50q^{80}+25q^{79}+48q^{78}+207q^{77}+70q^{76}-60q^{75}-148q^{74}-188q^{73}-118q^{72}-42q^{71}+346q^{70}+265q^{69}+146q^{68}-77q^{67}-283q^{66}-387q^{65}-373q^{64}+283q^{63}+390q^{62}+491q^{61}+264q^{60}-98q^{59}-565q^{58}-852q^{57}-94q^{56}+195q^{55}+718q^{54}+750q^{53}+454q^{52}-405q^{51}-1201q^{50}-632q^{49}-372q^{48}+593q^{47}+1106q^{46}+1193q^{45}+118q^{44}-1213q^{43}-1073q^{42}-1118q^{41}+124q^{40}+1165q^{39}+1866q^{38}+808q^{37}-921q^{36}-1286q^{35}-1807q^{34}-489q^{33}+981q^{32}+2349q^{31}+1453q^{30}-504q^{29}-1325q^{28}-2337q^{27}-1057q^{26}+720q^{25}+2671q^{24}+1961q^{23}-114q^{22}-1313q^{21}-2725q^{20}-1506q^{19}+492q^{18}+2897q^{17}+2343q^{16}+215q^{15}-1294q^{14}-3006q^{13}-1861q^{12}+281q^{11}+3008q^{10}+2627q^{9}+542q^{8}-1193q^{7}-3132q^{6}-2158q^{5}-11q^{4}+2891q^{3}+2759q^{2}+906q-888-2973q^{-1}-2331q^{-2}-420q^{-3}+2429q^{-4}+2600q^{-5}+1207q^{-6}-376q^{-7}-2423q^{-8}-2213q^{-9}-802q^{-10}+1667q^{-11}+2056q^{-12}+1246q^{-13}+147q^{-14}-1585q^{-15}-1725q^{-16}-928q^{-17}+876q^{-18}+1271q^{-19}+950q^{-20}+421q^{-21}-775q^{-22}-1040q^{-23}-739q^{-24}+343q^{-25}+570q^{-26}+506q^{-27}+391q^{-28}-266q^{-29}-463q^{-30}-418q^{-31}+119q^{-32}+167q^{-33}+168q^{-34}+228q^{-35}-63q^{-36}-148q^{-37}-172q^{-38}+57q^{-39}+25q^{-40}+19q^{-41}+94q^{-42}-12q^{-43}-35q^{-44}-54q^{-45}+34q^{-46}-2q^{-47}-10q^{-48}+28q^{-49}-5q^{-50}-5q^{-51}-15q^{-52}+17q^{-53}-2q^{-54}-8q^{-55}+8q^{-56}-3q^{-57}-3q^{-59}+4q^{-60}+q^{-61}-3q^{-62}+q^{-63}$
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}+8 q^{131}+6 q^{130}-4 q^{129}-6 q^{128}-14 q^{127}-2 q^{126}+3 q^{125}-6 q^{124}+25 q^{123}+28 q^{122}+10 q^{121}-5 q^{120}-48 q^{119}-35 q^{118}-22 q^{117}-32 q^{116}+45 q^{115}+85 q^{114}+82 q^{113}+69 q^{112}-57 q^{111}-101 q^{110}-121 q^{109}-170 q^{108}-25 q^{107}+102 q^{106}+211 q^{105}+300 q^{104}+122 q^{103}-33 q^{102}-201 q^{101}-442 q^{100}-335 q^{99}-149 q^{98}+141 q^{97}+551 q^{96}+538 q^{95}+423 q^{94}+101 q^{93}-521 q^{92}-731 q^{91}-781 q^{90}-482 q^{89}+315 q^{88}+768 q^{87}+1112 q^{86}+1011 q^{85}+115 q^{84}-594 q^{83}-1312 q^{82}-1559 q^{81}-751 q^{80}+118 q^{79}+1284 q^{78}+2043 q^{77}+1489 q^{76}+608 q^{75}-924 q^{74}-2293 q^{73}-2229 q^{72}-1563 q^{71}+231 q^{70}+2253 q^{69}+2838 q^{68}+2603 q^{67}+735 q^{66}-1864 q^{65}-3166 q^{64}-3609 q^{63}-1943 q^{62}+1125 q^{61}+3246 q^{60}+4506 q^{59}+3170 q^{58}-168 q^{57}-2947 q^{56}-5125 q^{55}-4448 q^{54}-1027 q^{53}+2453 q^{52}+5588 q^{51}+5550 q^{50}+2183 q^{49}-1703 q^{48}-5715 q^{47}-6547 q^{46}-3436 q^{45}+894 q^{44}+5790 q^{43}+7334 q^{42}+4473 q^{41}-40 q^{40}-5607 q^{39}-7992 q^{38}-5517 q^{37}-752 q^{36}+5502 q^{35}+8509 q^{34}+6296 q^{33}+1474 q^{32}-5273 q^{31}-8946 q^{30}-7071 q^{29}-2070 q^{28}+5177 q^{27}+9302 q^{26}+7641 q^{25}+2595 q^{24}-5016 q^{23}-9657 q^{22}-8219 q^{21}-3010 q^{20}+4948 q^{19}+9926 q^{18}+8672 q^{17}+3456 q^{16}-4798 q^{15}-10186 q^{14}-9151 q^{13}-3854 q^{12}+4617 q^{11}+10293 q^{10}+9531 q^9+4354 q^8-4240 q^7-10292 q^6-9891 q^5-4866 q^4+3744 q^3+10026 q^2+10041 q+5442-2978 q^{-1} -9500 q^{-2} -10054 q^{-3} -5946 q^{-4} +2107 q^{-5} +8639 q^{-6} +9702 q^{-7} +6341 q^{-8} -1071 q^{-9} -7507 q^{-10} -9069 q^{-11} -6480 q^{-12} +92 q^{-13} +6140 q^{-14} +8055 q^{-15} +6347 q^{-16} +783 q^{-17} -4706 q^{-18} -6812 q^{-19} -5852 q^{-20} -1392 q^{-21} +3288 q^{-22} +5393 q^{-23} +5128 q^{-24} +1742 q^{-25} -2107 q^{-26} -4026 q^{-27} -4165 q^{-28} -1758 q^{-29} +1145 q^{-30} +2734 q^{-31} +3215 q^{-32} +1621 q^{-33} -534 q^{-34} -1768 q^{-35} -2280 q^{-36} -1257 q^{-37} +140 q^{-38} +973 q^{-39} +1532 q^{-40} +966 q^{-41} +22 q^{-42} -532 q^{-43} -954 q^{-44} -606 q^{-45} -73 q^{-46} +203 q^{-47} +554 q^{-48} +390 q^{-49} +75 q^{-50} -72 q^{-51} -317 q^{-52} -209 q^{-53} -37 q^{-54} +8 q^{-55} +153 q^{-56} +98 q^{-57} +27 q^{-58} +30 q^{-59} -88 q^{-60} -61 q^{-61} +4 q^{-62} -9 q^{-63} +35 q^{-64} +5 q^{-65} +29 q^{-67} -22 q^{-68} -16 q^{-69} +6 q^{-70} - q^{-71} +9 q^{-72} -7 q^{-73} -5 q^{-74} +13 q^{-75} -4 q^{-76} -5 q^{-77} +3 q^{-78} +3 q^{-80} -4 q^{-81} - q^{-82} +3 q^{-83} - q^{-84} }

Computer Talk

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

Modifying This Page

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

8_12 8_14