10 137
From Knot Atlas
Jump to navigationJump to search
|
|
|
 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 137's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X1425 X5,10,6,11 X3948 X9,3,10,2 X16,12,17,11 X14,7,15,8 X6,15,7,16 X20,18,1,17 X18,13,19,14 X12,19,13,20 |
| Gauss code | -1, 4, -3, 1, -2, -7, 6, 3, -4, 2, 5, -10, 9, -6, 7, -5, 8, -9, 10, -8 |
| Dowker-Thistlethwaite code | 4 8 10 -14 2 -16 -18 -6 -20 -12 |
| Conway Notation | [22,211,2-] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 10, width is 5, Braid index is 5 |
|
 [{12, 2}, {1, 10}, {11, 6}, {10, 12}, {9, 3}, {2, 8}, {7, 9}, {8, 11}, {5, 1}, {6, 4}, {3, 5}, {4, 7}] |
[edit Notes on presentations of 10 137]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
|
K = Knot["10 137"];
|
In[4]:=
|
PD[K]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
X1425 X5,10,6,11 X3948 X9,3,10,2 X16,12,17,11 X14,7,15,8 X6,15,7,16 X20,18,1,17 X18,13,19,14 X12,19,13,20 |
In[5]:=
|
GaussCode[K]
|
Out[5]=
|
-1, 4, -3, 1, -2, -7, 6, 3, -4, 2, 5, -10, 9, -6, 7, -5, 8, -9, 10, -8 |
In[6]:=
|
DTCode[K]
|
Out[6]=
|
4 8 10 -14 2 -16 -18 -6 -20 -12 |
(The path below may be different on your system)
In[7]:=
|
AppendTo[$Path, "C:/bin/LinKnot/"];
|
In[8]:=
|
ConwayNotation[K]
|
Out[8]=
|
[22,211,2-] |
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,-1,2,-3,-2,-2,4,-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, 10, 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[{12, 2}, {1, 10}, {11, 6}, {10, 12}, {9, 3}, {2, 8}, {7, 9}, {8, 11}, {5, 1}, {6, 4}, {3, 5}, {4, 7}] |
In[14]:=
|
Draw[ap]
|
|
|
Out[14]=
|
-Graphics- |
Three dimensional invariants
|
Four dimensional invariants
|
Polynomial invariants
| Alexander polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^2-6 t+11-6 t^{-1} + 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 z^4-2 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 | { 25, 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^2-2 q+4-4 q^{-1} +4 q^{-2} -4 q^{-3} +3 q^{-4} -2 q^{-5} + q^{-6} } |
| 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 a^6-2 z^2 a^4-2 a^4+z^4 a^2+2 z^2 a^2+2 a^2-2 z^2-1+ a^{-2} } |
| 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 a^4 z^8+a^2 z^8+2 a^5 z^7+4 a^3 z^7+2 a z^7+a^6 z^6-a^4 z^6-a^2 z^6+z^6-8 a^5 z^5-15 a^3 z^5-7 a z^5-4 a^6 z^4-7 a^4 z^4-5 a^2 z^4-2 z^4+8 a^5 z^3+15 a^3 z^3+9 a z^3+2 z^3 a^{-1} +4 a^6 z^2+8 a^4 z^2+7 a^2 z^2+z^2 a^{-2} +4 z^2-3 a^5 z-5 a^3 z-3 a z-z a^{-1} -a^6-2 a^4-2 a^2- a^{-2} -1} |
| 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^{20}+q^{18}-q^{16}-q^{12}-q^{10}+q^8+q^4+ q^{-2} - q^{-4} + q^{-6} + q^{-8} } |
| 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^{94}-q^{92}+3 q^{90}-4 q^{88}+3 q^{86}-q^{84}-4 q^{82}+10 q^{80}-10 q^{78}+10 q^{76}-4 q^{74}-4 q^{72}+11 q^{70}-12 q^{68}+8 q^{66}-q^{64}-6 q^{62}+8 q^{60}-6 q^{58}-2 q^{56}+9 q^{54}-13 q^{52}+10 q^{50}-5 q^{48}-6 q^{46}+12 q^{44}-15 q^{42}+13 q^{40}-9 q^{38}+3 q^{36}+5 q^{34}-9 q^{32}+11 q^{30}-9 q^{28}+5 q^{26}+3 q^{24}-6 q^{22}+6 q^{20}-2 q^{18}-3 q^{16}+10 q^{14}-11 q^{12}+7 q^{10}+q^8-10 q^6+14 q^4-13 q^2+7+ q^{-2} -7 q^{-4} +7 q^{-6} -5 q^{-8} +3 q^{-10} + q^{-12} -2 q^{-14} + q^{-16} -2 q^{-20} +3 q^{-22} +2 q^{-28} - q^{-30} + q^{-32} + q^{-38} } |
Further Quantum Invariants
Further quantum knot invariants for 10_137.
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^{13}-q^{11}+q^9-q^7+2 q^{-1} - q^{-3} + q^{-5} } |
| 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^{38}-q^{36}-2 q^{34}+3 q^{32}+q^{30}-4 q^{28}+q^{26}+3 q^{24}-2 q^{22}-2 q^{20}+3 q^{18}+q^{16}-3 q^{14}+2 q^{12}+2 q^{10}-3 q^8-q^6+3 q^4-3+3 q^{-2} +2 q^{-4} -3 q^{-6} +2 q^{-10} } |
| 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^{75}-q^{73}-2 q^{71}+4 q^{67}+3 q^{65}-5 q^{63}-6 q^{61}+2 q^{59}+8 q^{57}+3 q^{55}-7 q^{53}-7 q^{51}+2 q^{49}+10 q^{47}+5 q^{45}-10 q^{43}-10 q^{41}+5 q^{39}+14 q^{37}-2 q^{35}-15 q^{33}-q^{31}+14 q^{29}+4 q^{27}-12 q^{25}-4 q^{23}+11 q^{21}+5 q^{19}-10 q^{17}-5 q^{15}+6 q^{13}+7 q^{11}-3 q^9-8 q^7-4 q^5+10 q^3+11 q-5 q^{-1} -15 q^{-3} +18 q^{-7} +3 q^{-9} -14 q^{-11} -7 q^{-13} +9 q^{-15} +8 q^{-17} -4 q^{-19} -5 q^{-21} + q^{-23} +2 q^{-25} + q^{-27} - q^{-29} } |
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^{20}+q^{18}-q^{16}-q^{12}-q^{10}+q^8+q^4+ q^{-2} - q^{-4} + q^{-6} + q^{-8} } |
| 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^{52}+q^{50}-3 q^{46}-2 q^{44}+q^{42}+2 q^{40}-q^{36}+2 q^{34}+2 q^{32}-3 q^{28}-q^{26}+q^{22}+q^{18}+3 q^{16}-2 q^{10}-q^8+q^4-q^2-1+2 q^{-2} +3 q^{-4} -3 q^{-8} + q^{-10} +2 q^{-12} + q^{-20} } |
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^{40}-q^{38}+q^{36}+q^{34}-2 q^{32}+q^{30}-q^{28}-q^{26}+2 q^{24}+2 q^{18}-q^{14}-q^{10}-q^6+2 q^2-1+ q^{-2} +2 q^{-4} -2 q^{-6} + q^{-8} +2 q^{-10} + q^{-16} } |
| 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^{27}+q^{25}+q^{23}-q^{21}-2 q^{17}-q^{15}-q^{13}+q^{11}+q^9+q^7+q^5- q^{-1} + q^{-3} - q^{-5} + q^{-7} + q^{-9} + q^{-11} } |
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^{40}-q^{38}+3 q^{36}-3 q^{34}+4 q^{32}-3 q^{30}+3 q^{28}-3 q^{26}-4 q^{20}+4 q^{18}-6 q^{16}+7 q^{14}-6 q^{12}+7 q^{10}-4 q^8+3 q^6+1-3 q^{-2} +4 q^{-4} -4 q^{-6} +3 q^{-8} -2 q^{-10} +2 q^{-12} + q^{-16} } |
| 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^{66}-q^{62}-q^{60}+2 q^{58}+2 q^{56}-2 q^{54}-3 q^{52}+q^{50}+3 q^{48}-4 q^{44}-2 q^{42}+4 q^{40}+3 q^{38}-q^{36}-3 q^{34}+q^{32}+3 q^{30}+q^{28}-2 q^{26}-q^{24}+2 q^{22}+q^{20}-2 q^{18}-3 q^{16}+q^{14}+3 q^{12}-q^{10}-4 q^8+3 q^4+2 q^2-2-2 q^{-2} +2 q^{-4} +4 q^{-6} - q^{-8} -3 q^{-10} +2 q^{-14} +2 q^{-16} - q^{-20} + q^{-26} } |
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^{94}-q^{92}+3 q^{90}-4 q^{88}+3 q^{86}-q^{84}-4 q^{82}+10 q^{80}-10 q^{78}+10 q^{76}-4 q^{74}-4 q^{72}+11 q^{70}-12 q^{68}+8 q^{66}-q^{64}-6 q^{62}+8 q^{60}-6 q^{58}-2 q^{56}+9 q^{54}-13 q^{52}+10 q^{50}-5 q^{48}-6 q^{46}+12 q^{44}-15 q^{42}+13 q^{40}-9 q^{38}+3 q^{36}+5 q^{34}-9 q^{32}+11 q^{30}-9 q^{28}+5 q^{26}+3 q^{24}-6 q^{22}+6 q^{20}-2 q^{18}-3 q^{16}+10 q^{14}-11 q^{12}+7 q^{10}+q^8-10 q^6+14 q^4-13 q^2+7+ q^{-2} -7 q^{-4} +7 q^{-6} -5 q^{-8} +3 q^{-10} + q^{-12} -2 q^{-14} + q^{-16} -2 q^{-20} +3 q^{-22} +2 q^{-28} - q^{-30} + q^{-32} + q^{-38} } |
.
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["10 137"];
|
In[4]:=
|
Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^2-6 t+11-6 t^{-1} + 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 z^4-2 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]=
|
{ 25, 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^2-2 q+4-4 q^{-1} +4 q^{-2} -4 q^{-3} +3 q^{-4} -2 q^{-5} + q^{-6} } |
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 a^6-2 z^2 a^4-2 a^4+z^4 a^2+2 z^2 a^2+2 a^2-2 z^2-1+ a^{-2} } |
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 a^4 z^8+a^2 z^8+2 a^5 z^7+4 a^3 z^7+2 a z^7+a^6 z^6-a^4 z^6-a^2 z^6+z^6-8 a^5 z^5-15 a^3 z^5-7 a z^5-4 a^6 z^4-7 a^4 z^4-5 a^2 z^4-2 z^4+8 a^5 z^3+15 a^3 z^3+9 a z^3+2 z^3 a^{-1} +4 a^6 z^2+8 a^4 z^2+7 a^2 z^2+z^2 a^{-2} +4 z^2-3 a^5 z-5 a^3 z-3 a z-z a^{-1} -a^6-2 a^4-2 a^2- a^{-2} -1} |
"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}} ): {10_155, K11n37,}
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["10 137"];
|
In[4]:=
|
{A = Alexander[K][t], J = Jones[K][q]}
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[4]=
|
{ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^2-6 t+11-6 t^{-1} + 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^2-2 q+4-4 q^{-1} +4 q^{-2} -4 q^{-3} +3 q^{-4} -2 q^{-5} + q^{-6} } } |
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]=
|
{10_155, K11n37,} |
Vassiliev invariants
| V2 and V3: | (-2, 2) |
| V2,1 through V6,9: |
|
V2,1 through V6,9 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 10 137. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
|
| Integral Khovanov Homology
(db, data source) |
|
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 2 q^4-2 q^3-3 q^2+7 q-1-9 q^{-1} +10 q^{-2} +2 q^{-3} -13 q^{-4} +8 q^{-5} +7 q^{-6} -13 q^{-7} +3 q^{-8} +11 q^{-9} -11 q^{-10} -2 q^{-11} +11 q^{-12} -6 q^{-13} -4 q^{-14} +6 q^{-15} - q^{-16} -2 q^{-17} + q^{-18} } |
| 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^{13}+2 q^{12}+q^{11}-q^{10}-7 q^9+3 q^8+13 q^7-23 q^5-4 q^4+30 q^3+15 q^2-41 q-19+40 q^{-1} +31 q^{-2} -42 q^{-3} -33 q^{-4} +36 q^{-5} +36 q^{-6} -32 q^{-7} -34 q^{-8} +25 q^{-9} +31 q^{-10} -17 q^{-11} -28 q^{-12} +10 q^{-13} +23 q^{-14} - q^{-15} -18 q^{-16} -5 q^{-17} +9 q^{-18} +12 q^{-19} -2 q^{-20} -14 q^{-21} -6 q^{-22} +12 q^{-23} +13 q^{-24} -9 q^{-25} -14 q^{-26} +3 q^{-27} +13 q^{-28} + q^{-29} -9 q^{-30} -3 q^{-31} +5 q^{-32} +2 q^{-33} - q^{-34} -2 q^{-35} + q^{-36} } |
| 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^{22}+2 q^{21}+q^{20}-3 q^{19}-q^{18}-4 q^{17}+11 q^{16}+9 q^{15}-10 q^{14}-17 q^{13}-22 q^{12}+36 q^{11}+48 q^{10}-7 q^9-58 q^8-81 q^7+53 q^6+124 q^5+37 q^4-94 q^3-174 q^2+29 q+192+105 q^{-1} -89 q^{-2} -241 q^{-3} -22 q^{-4} +210 q^{-5} +150 q^{-6} -58 q^{-7} -253 q^{-8} -59 q^{-9} +193 q^{-10} +154 q^{-11} -27 q^{-12} -228 q^{-13} -79 q^{-14} +162 q^{-15} +140 q^{-16} +5 q^{-17} -190 q^{-18} -96 q^{-19} +119 q^{-20} +118 q^{-21} +45 q^{-22} -136 q^{-23} -111 q^{-24} +62 q^{-25} +82 q^{-26} +77 q^{-27} -65 q^{-28} -99 q^{-29} +11 q^{-30} +24 q^{-31} +74 q^{-32} -53 q^{-34} -4 q^{-35} -28 q^{-36} +32 q^{-37} +21 q^{-38} -5 q^{-39} +18 q^{-40} -38 q^{-41} -7 q^{-42} +2 q^{-43} +6 q^{-44} +35 q^{-45} -14 q^{-46} -11 q^{-47} -13 q^{-48} -5 q^{-49} +23 q^{-50} + q^{-51} -7 q^{-53} -7 q^{-54} +6 q^{-55} + q^{-56} +2 q^{-57} - q^{-58} -2 q^{-59} + q^{-60} } |
| 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^{30}-q^{29}+4 q^{28}+4 q^{27}-q^{26}-6 q^{25}-13 q^{24}-10 q^{23}+20 q^{22}+36 q^{21}+23 q^{20}-23 q^{19}-73 q^{18}-75 q^{17}+18 q^{16}+133 q^{15}+150 q^{14}+20 q^{13}-184 q^{12}-269 q^{11}-100 q^{10}+223 q^9+402 q^8+220 q^7-218 q^6-534 q^5-365 q^4+176 q^3+619 q^2+524 q-95-679 q^{-1} -640 q^{-2} +3 q^{-3} +669 q^{-4} +733 q^{-5} +94 q^{-6} -652 q^{-7} -769 q^{-8} -158 q^{-9} +595 q^{-10} +777 q^{-11} +212 q^{-12} -550 q^{-13} -761 q^{-14} -234 q^{-15} +496 q^{-16} +729 q^{-17} +259 q^{-18} -445 q^{-19} -699 q^{-20} -276 q^{-21} +386 q^{-22} +660 q^{-23} +308 q^{-24} -316 q^{-25} -622 q^{-26} -344 q^{-27} +233 q^{-28} +567 q^{-29} +385 q^{-30} -131 q^{-31} -503 q^{-32} -415 q^{-33} +23 q^{-34} +412 q^{-35} +428 q^{-36} +88 q^{-37} -302 q^{-38} -415 q^{-39} -177 q^{-40} +176 q^{-41} +361 q^{-42} +243 q^{-43} -54 q^{-44} -279 q^{-45} -259 q^{-46} -47 q^{-47} +170 q^{-48} +234 q^{-49} +113 q^{-50} -70 q^{-51} -171 q^{-52} -129 q^{-53} -8 q^{-54} +89 q^{-55} +108 q^{-56} +50 q^{-57} -24 q^{-58} -59 q^{-59} -49 q^{-60} -20 q^{-61} +10 q^{-62} +27 q^{-63} +29 q^{-64} +22 q^{-65} + q^{-66} -16 q^{-67} -28 q^{-68} -25 q^{-69} - q^{-70} +23 q^{-71} +26 q^{-72} +12 q^{-73} -5 q^{-74} -21 q^{-75} -18 q^{-76} - q^{-77} +12 q^{-78} +10 q^{-79} +5 q^{-80} -9 q^{-82} -5 q^{-83} +2 q^{-84} +2 q^{-85} + q^{-86} +2 q^{-87} - q^{-88} -2 q^{-89} + q^{-90} } |
| 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^{47}-2 q^{46}-q^{45}+2 q^{44}+q^{43}+q^{42}+6 q^{40}-11 q^{39}-14 q^{38}+2 q^{37}+10 q^{36}+19 q^{35}+21 q^{34}+28 q^{33}-45 q^{32}-84 q^{31}-54 q^{30}+2 q^{29}+88 q^{28}+161 q^{27}+187 q^{26}-51 q^{25}-274 q^{24}-341 q^{23}-222 q^{22}+104 q^{21}+495 q^{20}+730 q^{19}+270 q^{18}-401 q^{17}-916 q^{16}-952 q^{15}-318 q^{14}+746 q^{13}+1625 q^{12}+1181 q^{11}-19 q^{10}-1366 q^9-2008 q^8-1358 q^7+433 q^6+2310 q^5+2334 q^4+940 q^3-1207 q^2-2730 q-2508-396 q^{-1} +2325 q^{-2} +3042 q^{-3} +1903 q^{-4} -580 q^{-5} -2772 q^{-6} -3140 q^{-7} -1158 q^{-8} +1899 q^{-9} +3101 q^{-10} +2364 q^{-11} -28 q^{-12} -2441 q^{-13} -3187 q^{-14} -1493 q^{-15} +1503 q^{-16} +2850 q^{-17} +2379 q^{-18} +223 q^{-19} -2118 q^{-20} -2984 q^{-21} -1537 q^{-22} +1251 q^{-23} +2574 q^{-24} +2265 q^{-25} +360 q^{-26} -1835 q^{-27} -2759 q^{-28} -1574 q^{-29} +955 q^{-30} +2268 q^{-31} +2192 q^{-32} +615 q^{-33} -1423 q^{-34} -2495 q^{-35} -1720 q^{-36} +450 q^{-37} +1799 q^{-38} +2109 q^{-39} +1029 q^{-40} -769 q^{-41} -2050 q^{-42} -1864 q^{-43} -244 q^{-44} +1065 q^{-45} +1826 q^{-46} +1422 q^{-47} +73 q^{-48} -1293 q^{-49} -1748 q^{-50} -885 q^{-51} +131 q^{-52} +1167 q^{-53} +1470 q^{-54} +810 q^{-55} -300 q^{-56} -1165 q^{-57} -1093 q^{-58} -661 q^{-59} +238 q^{-60} +966 q^{-61} +1025 q^{-62} +519 q^{-63} -286 q^{-64} -684 q^{-65} -867 q^{-66} -478 q^{-67} +169 q^{-68} +598 q^{-69} +709 q^{-70} +334 q^{-71} -17 q^{-72} -456 q^{-73} -556 q^{-74} -320 q^{-75} -2 q^{-76} +342 q^{-77} +337 q^{-78} +305 q^{-79} +28 q^{-80} -192 q^{-81} -254 q^{-82} -217 q^{-83} -4 q^{-84} +32 q^{-85} +170 q^{-86} +132 q^{-87} +54 q^{-88} -16 q^{-89} -80 q^{-90} -23 q^{-91} -95 q^{-92} -11 q^{-93} +8 q^{-94} +30 q^{-95} +33 q^{-96} +26 q^{-97} +63 q^{-98} -34 q^{-99} -19 q^{-100} -38 q^{-101} -25 q^{-102} -18 q^{-103} +6 q^{-104} +57 q^{-105} +8 q^{-106} +14 q^{-107} -8 q^{-108} -13 q^{-109} -25 q^{-110} -13 q^{-111} +17 q^{-112} +2 q^{-113} +11 q^{-114} +4 q^{-115} +3 q^{-116} -9 q^{-117} -7 q^{-118} +4 q^{-119} -2 q^{-120} +2 q^{-121} + q^{-122} +2 q^{-123} - q^{-124} -2 q^{-125} + q^{-126} } |
| 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^{63}-2 q^{62}-q^{61}+2 q^{60}+2 q^{59}+2 q^{58}-4 q^{57}-2 q^{56}+q^{55}-7 q^{54}-4 q^{53}+9 q^{52}+18 q^{51}+24 q^{50}-7 q^{49}-26 q^{48}-32 q^{47}-57 q^{46}-31 q^{45}+32 q^{44}+113 q^{43}+171 q^{42}+96 q^{41}-46 q^{40}-196 q^{39}-361 q^{38}-327 q^{37}-77 q^{36}+314 q^{35}+736 q^{34}+766 q^{33}+386 q^{32}-325 q^{31}-1177 q^{30}-1526 q^{29}-1104 q^{28}+54 q^{27}+1636 q^{26}+2583 q^{25}+2298 q^{24}+690 q^{23}-1856 q^{22}-3765 q^{21}-3981 q^{20}-2063 q^{19}+1569 q^{18}+4835 q^{17}+5982 q^{16}+4054 q^{15}-635 q^{14}-5495 q^{13}-7943 q^{12}-6433 q^{11}-970 q^{10}+5444 q^9+9541 q^8+8929 q^7+3064 q^6-4769 q^5-10520 q^4-11014 q^3-5274 q^2+3459 q+10717+12593 q^{-1} +7327 q^{-2} -1972 q^{-3} -10356 q^{-4} -13394 q^{-5} -8824 q^{-6} +461 q^{-7} +9547 q^{-8} +13611 q^{-9} +9820 q^{-10} +719 q^{-11} -8687 q^{-12} -13368 q^{-13} -10222 q^{-14} -1525 q^{-15} +7866 q^{-16} +12914 q^{-17} +10277 q^{-18} +1973 q^{-19} -7261 q^{-20} -12424 q^{-21} -10106 q^{-22} -2157 q^{-23} +6826 q^{-24} +11968 q^{-25} +9874 q^{-26} +2241 q^{-27} -6488 q^{-28} -11579 q^{-29} -9683 q^{-30} -2335 q^{-31} +6162 q^{-32} +11209 q^{-33} +9546 q^{-34} +2537 q^{-35} -5713 q^{-36} -10806 q^{-37} -9500 q^{-38} -2896 q^{-39} +5114 q^{-40} +10301 q^{-41} +9466 q^{-42} +3436 q^{-43} -4273 q^{-44} -9642 q^{-45} -9446 q^{-46} -4111 q^{-47} +3213 q^{-48} +8744 q^{-49} +9326 q^{-50} +4896 q^{-51} -1899 q^{-52} -7614 q^{-53} -9036 q^{-54} -5653 q^{-55} +418 q^{-56} +6158 q^{-57} +8457 q^{-58} +6312 q^{-59} +1147 q^{-60} -4450 q^{-61} -7524 q^{-62} -6644 q^{-63} -2632 q^{-64} +2511 q^{-65} +6174 q^{-66} +6557 q^{-67} +3871 q^{-68} -556 q^{-69} -4471 q^{-70} -5923 q^{-71} -4609 q^{-72} -1234 q^{-73} +2520 q^{-74} +4757 q^{-75} +4746 q^{-76} +2606 q^{-77} -626 q^{-78} -3184 q^{-79} -4198 q^{-80} -3323 q^{-81} -979 q^{-82} +1440 q^{-83} +3101 q^{-84} +3336 q^{-85} +2023 q^{-86} +110 q^{-87} -1703 q^{-88} -2692 q^{-89} -2345 q^{-90} -1223 q^{-91} +310 q^{-92} +1662 q^{-93} +2053 q^{-94} +1709 q^{-95} +695 q^{-96} -559 q^{-97} -1304 q^{-98} -1587 q^{-99} -1215 q^{-100} -304 q^{-101} +472 q^{-102} +1075 q^{-103} +1200 q^{-104} +731 q^{-105} +196 q^{-106} -441 q^{-107} -835 q^{-108} -750 q^{-109} -535 q^{-110} -65 q^{-111} +390 q^{-112} +508 q^{-113} +530 q^{-114} +296 q^{-115} -32 q^{-116} -172 q^{-117} -350 q^{-118} -318 q^{-119} -126 q^{-120} -29 q^{-121} +132 q^{-122} +166 q^{-123} +109 q^{-124} +135 q^{-125} +28 q^{-126} -53 q^{-127} -40 q^{-128} -93 q^{-129} -50 q^{-130} -37 q^{-131} -56 q^{-132} +34 q^{-133} +49 q^{-134} +45 q^{-135} +65 q^{-136} +14 q^{-137} +6 q^{-138} -13 q^{-139} -69 q^{-140} -37 q^{-141} -22 q^{-142} -2 q^{-143} +37 q^{-144} +20 q^{-145} +24 q^{-146} +24 q^{-147} -10 q^{-148} -16 q^{-149} -21 q^{-150} -17 q^{-151} +8 q^{-152} +4 q^{-154} +13 q^{-155} +4 q^{-156} +2 q^{-157} -6 q^{-158} -7 q^{-159} +2 q^{-160} -2 q^{-162} +2 q^{-163} + q^{-164} +2 q^{-165} - q^{-166} -2 q^{-167} + q^{-168} } |
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. |
|
