10 115

10_114

10_116

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 115's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 115 at Knotilus!

[edit 10 115 Quick Notes]

[edit 10 115 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₆₂₇₁ X_14,6,15,5 X_20,15,1,16 X_16,7,17,8 X_8,19,9,20 X_18,11,19,12 X_10,4,11,3 X_4,10,5,9 X_12,17,13,18 X_2,14,3,13
Gauss code	1, -10, 7, -8, 2, -1, 4, -5, 8, -7, 6, -9, 10, -2, 3, -4, 9, -6, 5, -3
Dowker-Thistlethwaite code	6 10 14 16 4 18 2 20 12 8
Conway Notation	[8*20.20]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 12, width is 5,

Braid index is 5

[{3, 11}, {2, 9}, {8, 10}, {9, 12}, {11, 4}, {5, 3}, {4, 7}, {6, 8}, {7, 13}, {12, 6}, {1, 5}, {13, 2}, {10, 1}]

[edit Notes on presentations of 10 115]

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["10 115"];

In[4]:= PD[K]

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

Out[4]= X₆₂₇₁ X_14,6,15,5 X_20,15,1,16 X_16,7,17,8 X_8,19,9,20 X_18,11,19,12 X_10,4,11,3 X_4,10,5,9 X_12,17,13,18 X_2,14,3,13

In[5]:= GaussCode[K]

Out[5]= 1, -10, 7, -8, 2, -1, 4, -5, 8, -7, 6, -9, 10, -2, 3, -4, 9, -6, 5, -3

In[6]:= DTCode[K]

Out[6]= 6 10 14 16 4 18 2 20 12 8

(The path below may be different on your system)

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

In[8]:= ConwayNotation[K]

Out[8]= [8*20.20]

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}(5,\{1,-2,1,3,2,2,-4,-3,2,-3,-3,-4\})}

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]= { 5, 12, 5 }

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[{3, 11}, {2, 9}, {8, 10}, {9, 12}, {11, 4}, {5, 3}, {4, 7}, {6, 8}, {7, 13}, {12, 6}, {1, 5}, {13, 2}, {10, 1}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Negative amphicheiral
Unknotting number	2
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-6][-6]
Hyperbolic Volume	16.638
A-Polynomial	See Data:10 115/A-polynomial

[edit Notes for 10 115'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 3}
Rasmussen s-Invariant	0

[edit Notes for 10 115's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^3+9 t^2-26 t+37-26 t^{-1} +9 t^{-2} - t^{-3} }
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^6+3 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 \left\{2,t^2+t+1\right\}}
Determinant and Signature	{ 109, 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+4 q^4-9 q^3+14 q^2-17 q+19-17 q^{-1} +14 q^{-2} -9 q^{-3} +4 q^{-4} - q^{-5} }
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^6+2 a^2 z^4+2 z^4 a^{-2} -z^4-a^4 z^2+a^2 z^2+z^2 a^{-2} -z^2 a^{-4} +z^2-a^2- a^{-2} +3}
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 a z^9+3 z^9 a^{-1} +8 a^2 z^8+8 z^8 a^{-2} +16 z^8+8 a^3 z^7+13 a z^7+13 z^7 a^{-1} +8 z^7 a^{-3} +4 a^4 z^6-9 a^2 z^6-9 z^6 a^{-2} +4 z^6 a^{-4} -26 z^6+a^5 z^5-13 a^3 z^5-34 a z^5-34 z^5 a^{-1} -13 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4+a^2 z^4+z^4 a^{-2} -5 z^4 a^{-4} +12 z^4-a^5 z^3+8 a^3 z^3+22 a z^3+22 z^3 a^{-1} +8 z^3 a^{-3} -z^3 a^{-5} +2 a^4 z^2-a^2 z^2-z^2 a^{-2} +2 z^2 a^{-4} -6 z^2-2 a^3 z-5 a z-5 z a^{-1} -2 z a^{-3} +a^2+ a^{-2} +3}
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{16}+q^{14}+2 q^{12}-4 q^{10}+2 q^8-q^6-2 q^4+5 q^2-1+5 q^{-2} -2 q^{-4} - q^{-6} +2 q^{-8} -4 q^{-10} +2 q^{-12} + q^{-14} - q^{-16} }
The G2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+16 q^{72}-17 q^{70}+8 q^{68}+17 q^{66}-53 q^{64}+98 q^{62}-130 q^{60}+121 q^{58}-62 q^{56}-61 q^{54}+225 q^{52}-360 q^{50}+410 q^{48}-311 q^{46}+62 q^{44}+258 q^{42}-536 q^{40}+646 q^{38}-522 q^{36}+193 q^{34}+206 q^{32}-514 q^{30}+589 q^{28}-396 q^{26}+28 q^{24}+339 q^{22}-530 q^{20}+436 q^{18}-110 q^{16}-314 q^{14}+652 q^{12}-743 q^{10}+555 q^8-133 q^6-361 q^4+759 q^2-907+759 q^{-2} -361 q^{-4} -133 q^{-6} +555 q^{-8} -743 q^{-10} +652 q^{-12} -314 q^{-14} -110 q^{-16} +436 q^{-18} -530 q^{-20} +339 q^{-22} +28 q^{-24} -396 q^{-26} +589 q^{-28} -514 q^{-30} +206 q^{-32} +193 q^{-34} -522 q^{-36} +646 q^{-38} -536 q^{-40} +258 q^{-42} +62 q^{-44} -311 q^{-46} +410 q^{-48} -360 q^{-50} +225 q^{-52} -61 q^{-54} -62 q^{-56} +121 q^{-58} -130 q^{-60} +98 q^{-62} -53 q^{-64} +17 q^{-66} +8 q^{-68} -17 q^{-70} +16 q^{-72} -13 q^{-74} +7 q^{-76} -3 q^{-78} + q^{-80} }

Further Quantum Invariants

Further quantum knot invariants for 10_115.

A1 Invariants.

Weight	Invariant
1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{11}+3 q^9-5 q^7+5 q^5-3 q^3+2 q+2 q^{-1} -3 q^{-3} +5 q^{-5} -5 q^{-7} +3 q^{-9} - q^{-11} }
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^{32}-3 q^{30}+q^{28}+11 q^{26}-18 q^{24}-9 q^{22}+44 q^{20}-24 q^{18}-43 q^{16}+65 q^{14}-64 q^{10}+42 q^8+27 q^6-45 q^4-2 q^2+37-2 q^{-2} -45 q^{-4} +27 q^{-6} +42 q^{-8} -64 q^{-10} +65 q^{-14} -43 q^{-16} -24 q^{-18} +44 q^{-20} -9 q^{-22} -18 q^{-24} +11 q^{-26} + q^{-28} -3 q^{-30} + q^{-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 -q^{63}+3 q^{61}-q^{59}-7 q^{57}+2 q^{55}+21 q^{53}+4 q^{51}-58 q^{49}-25 q^{47}+108 q^{45}+93 q^{43}-151 q^{41}-226 q^{39}+156 q^{37}+389 q^{35}-73 q^{33}-540 q^{31}-100 q^{29}+629 q^{27}+303 q^{25}-602 q^{23}-495 q^{21}+480 q^{19}+618 q^{17}-298 q^{15}-650 q^{13}+107 q^{11}+599 q^9+81 q^7-504 q^5-239 q^3+379 q+379 q^{-1} -239 q^{-3} -504 q^{-5} +81 q^{-7} +599 q^{-9} +107 q^{-11} -650 q^{-13} -298 q^{-15} +618 q^{-17} +480 q^{-19} -495 q^{-21} -602 q^{-23} +303 q^{-25} +629 q^{-27} -100 q^{-29} -540 q^{-31} -73 q^{-33} +389 q^{-35} +156 q^{-37} -226 q^{-39} -151 q^{-41} +93 q^{-43} +108 q^{-45} -25 q^{-47} -58 q^{-49} +4 q^{-51} +21 q^{-53} +2 q^{-55} -7 q^{-57} - q^{-59} +3 q^{-61} - q^{-63} }

A2 Invariants.

Weight

Invariant

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{16}+q^{14}+2 q^{12}-4 q^{10}+2 q^8-q^6-2 q^4+5 q^2-1+5 q^{-2} -2 q^{-4} - q^{-6} +2 q^{-8} -4 q^{-10} +2 q^{-12} + q^{-14} - q^{-16} }

2,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{42}-q^{40}-3 q^{38}+2 q^{36}+8 q^{34}-17 q^{30}-5 q^{28}+24 q^{26}+4 q^{24}-27 q^{22}-4 q^{20}+33 q^{18}+10 q^{16}-38 q^{14}-q^{12}+26 q^{10}-9 q^8-18 q^6+9 q^4+12 q^2-6+12 q^{-2} +9 q^{-4} -18 q^{-6} -9 q^{-8} +26 q^{-10} - q^{-12} -38 q^{-14} +10 q^{-16} +33 q^{-18} -4 q^{-20} -27 q^{-22} +4 q^{-24} +24 q^{-26} -5 q^{-28} -17 q^{-30} +8 q^{-34} +2 q^{-36} -3 q^{-38} - q^{-40} + q^{-42} }

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-3q^{32}+q^{30}+8q^{28}-15q^{26}+3q^{24}+26q^{22}-35q^{20}+4q^{18}+40q^{16}-49q^{14}+2q^{12}+39q^{10}-35q^{8}-5q^{6}+26q^{4}-q^{2}-8-q^{-2}+26q^{-4}-5q^{-6}-35q^{-8}+39q^{-10}+2q^{-12}-49q^{-14}+40q^{-16}+4q^{-18}-35q^{-20}+26q^{-22}+3q^{-24}-15q^{-26}+8q^{-28}+q^{-30}-3q^{-32}+q^{-34}$
1,0,0	$-q^{21}+q^{19}+2q^{15}-4q^{13}+3q^{11}-4q^{9}+q^{7}-2q^{5}+4q^{3}+2q+2q^{-1}+4q^{-3}-2q^{-5}+q^{-7}-4q^{-9}+3q^{-11}-4q^{-13}+2q^{-15}+q^{-19}-q^{-21}$

B2 Invariants.

Weight

Invariant

0,1

-q^{34}+3q^{32}-7q^{30}+14q^{28}-25q^{26}+37q^{24}-48q^{22}+57q^{20}-60q^{18}+54q^{16}-41q^{14}+20q^{12}+7q^{10}-37q^{8}+67q^{6}-90q^{4}+109q^{2}-114+109q^{-2}-90q^{-4}+67q^{-6}-37q^{-8}+7q^{-10}+20q^{-12}-41q^{-14}+54q^{-16}-60q^{-18}+57q^{-20}-48q^{-22}+37q^{-24}-25q^{-26}+14q^{-28}-7q^{-30}+3q^{-32}-q^{-34}

1,0

q^{56}-3q^{52}-3q^{50}+4q^{48}+11q^{46}+q^{44}-20q^{42}-16q^{40}+20q^{38}+37q^{36}-3q^{34}-51q^{32}-25q^{30}+47q^{28}+50q^{26}-25q^{24}-64q^{22}-5q^{20}+59q^{18}+26q^{16}-44q^{14}-37q^{12}+28q^{10}+40q^{8}-13q^{6}-40q^{4}+6q^{2}+43+6q^{-2}-40q^{-4}-13q^{-6}+40q^{-8}+28q^{-10}-37q^{-12}-44q^{-14}+26q^{-16}+59q^{-18}-5q^{-20}-64q^{-22}-25q^{-24}+50q^{-26}+47q^{-28}-25q^{-30}-51q^{-32}-3q^{-34}+37q^{-36}+20q^{-38}-16q^{-40}-20q^{-42}+q^{-44}+11q^{-46}+4q^{-48}-3q^{-50}-3q^{-52}+q^{-56}

G2 Invariants.

Weight

Invariant

1,0

q^{80}-3q^{78}+7q^{76}-13q^{74}+16q^{72}-17q^{70}+8q^{68}+17q^{66}-53q^{64}+98q^{62}-130q^{60}+121q^{58}-62q^{56}-61q^{54}+225q^{52}-360q^{50}+410q^{48}-311q^{46}+62q^{44}+258q^{42}-536q^{40}+646q^{38}-522q^{36}+193q^{34}+206q^{32}-514q^{30}+589q^{28}-396q^{26}+28q^{24}+339q^{22}-530q^{20}+436q^{18}-110q^{16}-314q^{14}+652q^{12}-743q^{10}+555q^{8}-133q^{6}-361q^{4}+759q^{2}-907+759q^{-2}-361q^{-4}-133q^{-6}+555q^{-8}-743q^{-10}+652q^{-12}-314q^{-14}-110q^{-16}+436q^{-18}-530q^{-20}+339q^{-22}+28q^{-24}-396q^{-26}+589q^{-28}-514q^{-30}+206q^{-32}+193q^{-34}-522q^{-36}+646q^{-38}-536q^{-40}+258q^{-42}+62q^{-44}-311q^{-46}+410q^{-48}-360q^{-50}+225q^{-52}-61q^{-54}-62q^{-56}+121q^{-58}-130q^{-60}+98q^{-62}-53q^{-64}+17q^{-66}+8q^{-68}-17q^{-70}+16q^{-72}-13q^{-74}+7q^{-76}-3q^{-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.

In[3]:= K = Knot["10 115"];

In[4]:= Alexander[K][t]

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

Out[4]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^3+9 t^2-26 t+37-26 t^{-1} +9 t^{-2} - t^{-3} }

In[5]:= Conway[K][z]

Out[5]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^6+3 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 \left\{2,t^2+t+1\right\}}

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 109, 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+4 q^4-9 q^3+14 q^2-17 q+19-17 q^{-1} +14 q^{-2} -9 q^{-3} +4 q^{-4} - q^{-5} }

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

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

Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^6+2 a^2 z^4+2 z^4 a^{-2} -z^4-a^4 z^2+a^2 z^2+z^2 a^{-2} -z^2 a^{-4} +z^2-a^2- a^{-2} +3}

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

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

Out[10]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 a z^9+3 z^9 a^{-1} +8 a^2 z^8+8 z^8 a^{-2} +16 z^8+8 a^3 z^7+13 a z^7+13 z^7 a^{-1} +8 z^7 a^{-3} +4 a^4 z^6-9 a^2 z^6-9 z^6 a^{-2} +4 z^6 a^{-4} -26 z^6+a^5 z^5-13 a^3 z^5-34 a z^5-34 z^5 a^{-1} -13 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4+a^2 z^4+z^4 a^{-2} -5 z^4 a^{-4} +12 z^4-a^5 z^3+8 a^3 z^3+22 a z^3+22 z^3 a^{-1} +8 z^3 a^{-3} -z^3 a^{-5} +2 a^4 z^2-a^2 z^2-z^2 a^{-2} +2 z^2 a^{-4} -6 z^2-2 a^3 z-5 a z-5 z a^{-1} -2 z a^{-3} +a^2+ a^{-2} +3}

"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.

In[3]:= K = Knot["10 115"];

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]= { $-t^{3}+9t^{2}-26t+37-26t^{-1}+9t^{-2}-t^{-3}$ , $-q^{5}+4q^{4}-9q^{3}+14q^{2}-17q+19-17q^{-1}+14q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$ }

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, 0)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

4

0

8

-{\frac {34}{3}}

-{\frac {62}{3}}

0

0

0

0

{\frac {32}{3}}

0

-{\frac {136}{3}}

-{\frac {248}{3}}

-{\frac {1649}{30}}

{\frac {566}{5}}

-{\frac {9538}{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{305}{18}}

-{\frac {2609}{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

\chi

(fixed

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

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

s=

0 is the signature of 10 115. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

3

7

6

1

-5

5

8

3

5

3

9

6

-3

1

10

8

2

-1

8

10

2

-3

6

9

-3

-5

3

8

5

-7

1

6

-5

-9

3

-11

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
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=-5}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-4}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (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}^{6}\oplus{\mathbb Z}_2^{3}}	${\mathbb {Z} }^{3}$
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=-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}^{8}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 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}^{9}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 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}^{10}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{10}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}^{8}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 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}^{6}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 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}^{3}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 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^{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=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}-4 q^{14}+4 q^{13}+11 q^{12}-33 q^{11}+13 q^{10}+64 q^9-101 q^8-6 q^7+172 q^6-166 q^5-70 q^4+278 q^3-181 q^2-142 q+321-142 q^{-1} -181 q^{-2} +278 q^{-3} -70 q^{-4} -166 q^{-5} +172 q^{-6} -6 q^{-7} -101 q^{-8} +64 q^{-9} +13 q^{-10} -33 q^{-11} +11 q^{-12} +4 q^{-13} -4 q^{-14} + q^{-15} }
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}+4 q^{29}-4 q^{28}-6 q^{27}+8 q^{26}+23 q^{25}-21 q^{24}-68 q^{23}+41 q^{22}+156 q^{21}-36 q^{20}-312 q^{19}-34 q^{18}+538 q^{17}+197 q^{16}-774 q^{15}-501 q^{14}+978 q^{13}+926 q^{12}-1100 q^{11}-1406 q^{10}+1085 q^9+1901 q^8-962 q^7-2322 q^6+733 q^5+2658 q^4-470 q^3-2840 q^2+148 q+2923+148 q^{-1} -2840 q^{-2} -470 q^{-3} +2658 q^{-4} +733 q^{-5} -2322 q^{-6} -962 q^{-7} +1901 q^{-8} +1085 q^{-9} -1406 q^{-10} -1100 q^{-11} +926 q^{-12} +978 q^{-13} -501 q^{-14} -774 q^{-15} +197 q^{-16} +538 q^{-17} -34 q^{-18} -312 q^{-19} -36 q^{-20} +156 q^{-21} +41 q^{-22} -68 q^{-23} -21 q^{-24} +23 q^{-25} +8 q^{-26} -6 q^{-27} -4 q^{-28} +4 q^{-29} - q^{-30} }
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}-4 q^{49}+4 q^{48}+6 q^{47}-13 q^{46}+2 q^{45}-15 q^{44}+40 q^{43}+49 q^{42}-94 q^{41}-61 q^{40}-89 q^{39}+246 q^{38}+385 q^{37}-253 q^{36}-518 q^{35}-744 q^{34}+642 q^{33}+1807 q^{32}+368 q^{31}-1400 q^{30}-3385 q^{29}-217 q^{28}+4490 q^{27}+3818 q^{26}-590 q^{25}-8228 q^{24}-5045 q^{23}+5642 q^{22}+10246 q^{21}+5143 q^{20}-11875 q^{19}-13751 q^{18}+1667 q^{17}+15820 q^{16}+15318 q^{15}-10566 q^{14}-22033 q^{13}-6857 q^{12}+16840 q^{11}+25391 q^{10}-4814 q^9-26115 q^8-15813 q^7+13604 q^6+31686 q^5+2140 q^4-25795 q^3-22277 q^2+8365 q+33665+8365 q^{-1} -22277 q^{-2} -25795 q^{-3} +2140 q^{-4} +31686 q^{-5} +13604 q^{-6} -15813 q^{-7} -26115 q^{-8} -4814 q^{-9} +25391 q^{-10} +16840 q^{-11} -6857 q^{-12} -22033 q^{-13} -10566 q^{-14} +15318 q^{-15} +15820 q^{-16} +1667 q^{-17} -13751 q^{-18} -11875 q^{-19} +5143 q^{-20} +10246 q^{-21} +5642 q^{-22} -5045 q^{-23} -8228 q^{-24} -590 q^{-25} +3818 q^{-26} +4490 q^{-27} -217 q^{-28} -3385 q^{-29} -1400 q^{-30} +368 q^{-31} +1807 q^{-32} +642 q^{-33} -744 q^{-34} -518 q^{-35} -253 q^{-36} +385 q^{-37} +246 q^{-38} -89 q^{-39} -61 q^{-40} -94 q^{-41} +49 q^{-42} +40 q^{-43} -15 q^{-44} +2 q^{-45} -13 q^{-46} +6 q^{-47} +4 q^{-48} -4 q^{-49} + q^{-50} }

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

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Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

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