10 138
From Knot Atlas
Jump to navigationJump to search
|
|
|
 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 138's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X4251 X10,6,11,5 X8394 X2,9,3,10 X16,12,17,11 X7,15,8,14 X15,7,16,6 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 | [211,211,2-] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 10, width is 5, Braid index is 5 |
|
 [{2, 11}, {1, 7}, {10, 6}, {11, 9}, {8, 3}, {7, 10}, {5, 2}, {6, 4}, {3, 5}, {4, 8}, {9, 1}] |
[edit Notes on presentations of 10 138]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
|
K = Knot["10 138"];
|
In[4]:=
|
PD[K]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
X4251 X10,6,11,5 X8394 X2,9,3,10 X16,12,17,11 X7,15,8,14 X15,7,16,6 X20,18,1,17 X18,13,19,14 X12,19,13,20 |
In[5]:=
|
GaussCode[K]
|
Out[5]=
|
1, -4, 3, -1, 2, 7, -6, -3, 4, -2, 5, -10, 9, 6, -7, -5, 8, -9, 10, -8 |
In[6]:=
|
DTCode[K]
|
Out[6]=
|
4 8 10 -14 2 16 18 -6 20 12 |
(The path below may be different on your system)
In[7]:=
|
AppendTo[$Path, "C:/bin/LinKnot/"];
|
In[8]:=
|
ConwayNotation[K]
|
Out[8]=
|
[211,211,2-] |
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[{2, 11}, {1, 7}, {10, 6}, {11, 9}, {8, 3}, {7, 10}, {5, 2}, {6, 4}, {3, 5}, {4, 8}, {9, 1}] |
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^3-5 t^2+8 t-7+8 t^{-1} -5 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+z^4-3 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 | { 35, 2 } |
| 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 2 q^5-4 q^4+5 q^3-6 q^2+6 q-5+4 q^{-1} -2 q^{-2} + q^{-3} } |
| 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 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -2 z^4+a^2 z^2+5 z^2 a^{-2} -3 z^2 a^{-4} -6 z^2+2 a^2+3 a^{-2} -2 a^{-4} + a^{-6} -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 z^8 a^{-2} +z^8+2 a z^7+5 z^7 a^{-1} +3 z^7 a^{-3} +a^2 z^6+3 z^6 a^{-2} +3 z^6 a^{-4} +z^6-7 a z^5-14 z^5 a^{-1} -6 z^5 a^{-3} +z^5 a^{-5} -4 a^2 z^4-13 z^4 a^{-2} -5 z^4 a^{-4} -12 z^4+6 a z^3+8 z^3 a^{-1} +5 z^3 a^{-3} +3 z^3 a^{-5} +5 a^2 z^2+10 z^2 a^{-2} +6 z^2 a^{-4} +3 z^2 a^{-6} +12 z^2-a z-z a^{-1} -2 z a^{-3} -2 z a^{-5} -2 a^2-3 a^{-2} -2 a^{-4} - a^{-6} -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^{10}+q^8+q^4-q^2- q^{-4} +2 q^{-6} - q^{-8} + q^{-10} - q^{-12} - q^{-14} + q^{-16} + q^{-20} } |
| 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^{46}-q^{44}+4 q^{42}-5 q^{40}+5 q^{38}-2 q^{36}-4 q^{34}+14 q^{32}-18 q^{30}+20 q^{28}-11 q^{26}-4 q^{24}+19 q^{22}-27 q^{20}+29 q^{18}-16 q^{16}+17 q^{12}-25 q^{10}+19 q^8-5 q^6-13 q^4+20 q^2-21+7 q^{-2} +8 q^{-4} -25 q^{-6} +33 q^{-8} -29 q^{-10} +14 q^{-12} +5 q^{-14} -25 q^{-16} +35 q^{-18} -34 q^{-20} +24 q^{-22} -4 q^{-24} -12 q^{-26} +28 q^{-28} -27 q^{-30} +17 q^{-32} + q^{-34} -15 q^{-36} +22 q^{-38} -17 q^{-40} +16 q^{-44} -24 q^{-46} +25 q^{-48} -15 q^{-50} -5 q^{-52} +18 q^{-54} -26 q^{-56} +22 q^{-58} -13 q^{-60} + q^{-62} +9 q^{-64} -12 q^{-66} +11 q^{-68} -8 q^{-70} +5 q^{-72} + q^{-74} -3 q^{-76} + q^{-78} -2 q^{-80} +2 q^{-82} - q^{-84} +2 q^{-86} + q^{-88} } |
Further Quantum Invariants
Further quantum knot invariants for 10_138.
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^7-q^5+2 q^3-q+ q^{-1} - q^{-5} + q^{-7} -2 q^{-9} +2 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^{22}-q^{20}-2 q^{18}+4 q^{16}+q^{14}-6 q^{12}+4 q^{10}+5 q^8-7 q^6-q^4+7 q^2-3-4 q^{-2} +5 q^{-4} + q^{-6} -5 q^{-8} + q^{-10} +6 q^{-12} -2 q^{-14} -5 q^{-16} +7 q^{-18} + q^{-20} -8 q^{-22} +4 q^{-24} +2 q^{-26} -4 q^{-28} + 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^{45}-q^{43}-2 q^{41}+5 q^{37}+3 q^{35}-7 q^{33}-8 q^{31}+6 q^{29}+14 q^{27}-19 q^{23}-8 q^{21}+17 q^{19}+19 q^{17}-10 q^{15}-26 q^{13}+q^{11}+28 q^9+10 q^7-27 q^5-19 q^3+22 q+24 q^{-1} -15 q^{-3} -25 q^{-5} +11 q^{-7} +28 q^{-9} -6 q^{-11} -24 q^{-13} - q^{-15} +22 q^{-17} +7 q^{-19} -18 q^{-21} -17 q^{-23} +10 q^{-25} +24 q^{-27} + q^{-29} -28 q^{-31} -13 q^{-33} +28 q^{-35} +21 q^{-37} -21 q^{-39} -24 q^{-41} +14 q^{-43} +21 q^{-45} -3 q^{-47} -17 q^{-49} +8 q^{-53} -2 q^{-57} -2 q^{-59} +2 q^{-61} } |
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^{10}+q^8+q^4-q^2- q^{-4} +2 q^{-6} - q^{-8} + q^{-10} - q^{-12} - q^{-14} + q^{-16} + q^{-20} } |
| 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^{28}+q^{26}-2 q^{22}+3 q^{18}+q^{16}-3 q^{14}-q^{12}+3 q^{10}+2 q^8-3 q^6+3 q^2-2 q^{-2} - q^{-6} -2 q^{-8} + q^{-10} +2 q^{-16} +7 q^{-18} +2 q^{-20} -4 q^{-22} + q^{-26} -3 q^{-28} -4 q^{-30} +2 q^{-34} +2 q^{-36} - q^{-48} + q^{-52} } |
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^{20}-q^{18}+2 q^{16}+2 q^{14}-2 q^{12}+4 q^{10}+q^8-4 q^6+4 q^4-2 q^2-5+3 q^{-2} -3 q^{-6} +2 q^{-8} +2 q^{-10} + q^{-12} - q^{-14} +4 q^{-18} -4 q^{-20} +5 q^{-24} -5 q^{-26} - q^{-28} +4 q^{-30} -3 q^{-32} - q^{-34} +3 q^{-36} } |
| 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}+2 q^9+q^5-2 q^3-2 q^{-1} + q^{-7} +2 q^{-9} + q^{-13} -2 q^{-15} -2 q^{-19} + q^{-21} + q^{-25} + q^{-27} } |
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^{20}-q^{18}+4 q^{16}-4 q^{14}+6 q^{12}-6 q^{10}+7 q^8-6 q^6+4 q^4-2 q^2-3+5 q^{-2} -8 q^{-4} +11 q^{-6} -12 q^{-8} +14 q^{-10} -11 q^{-12} +9 q^{-14} -6 q^{-16} +2 q^{-18} -4 q^{-22} +5 q^{-24} -7 q^{-26} +7 q^{-28} -6 q^{-30} +5 q^{-32} -3 q^{-34} +3 q^{-36} } |
| 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^{34}-q^{30}-q^{28}+3 q^{26}+3 q^{24}-2 q^{22}-4 q^{20}+q^{18}+6 q^{16}+3 q^{14}-6 q^{12}-5 q^{10}+4 q^8+7 q^6-q^4-8 q^2-2+5 q^{-2} +3 q^{-4} -3 q^{-6} -4 q^{-8} +2 q^{-10} +4 q^{-12} - q^{-14} -5 q^{-16} + q^{-18} +6 q^{-20} + q^{-22} -5 q^{-24} -3 q^{-26} +5 q^{-28} +5 q^{-30} -3 q^{-32} -6 q^{-34} +2 q^{-36} +7 q^{-38} + q^{-40} -6 q^{-42} -4 q^{-44} +3 q^{-46} +5 q^{-48} - q^{-50} -4 q^{-52} -2 q^{-54} + q^{-56} +3 q^{-58} } |
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^{46}-q^{44}+4 q^{42}-5 q^{40}+5 q^{38}-2 q^{36}-4 q^{34}+14 q^{32}-18 q^{30}+20 q^{28}-11 q^{26}-4 q^{24}+19 q^{22}-27 q^{20}+29 q^{18}-16 q^{16}+17 q^{12}-25 q^{10}+19 q^8-5 q^6-13 q^4+20 q^2-21+7 q^{-2} +8 q^{-4} -25 q^{-6} +33 q^{-8} -29 q^{-10} +14 q^{-12} +5 q^{-14} -25 q^{-16} +35 q^{-18} -34 q^{-20} +24 q^{-22} -4 q^{-24} -12 q^{-26} +28 q^{-28} -27 q^{-30} +17 q^{-32} + q^{-34} -15 q^{-36} +22 q^{-38} -17 q^{-40} +16 q^{-44} -24 q^{-46} +25 q^{-48} -15 q^{-50} -5 q^{-52} +18 q^{-54} -26 q^{-56} +22 q^{-58} -13 q^{-60} + q^{-62} +9 q^{-64} -12 q^{-66} +11 q^{-68} -8 q^{-70} +5 q^{-72} + q^{-74} -3 q^{-76} + q^{-78} -2 q^{-80} +2 q^{-82} - q^{-84} +2 q^{-86} + q^{-88} } |
.
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["10 138"];
|
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^3-5 t^2+8 t-7+8 t^{-1} -5 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+z^4-3 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]=
|
{ 35, 2 } |
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 2 q^5-4 q^4+5 q^3-6 q^2+6 q-5+4 q^{-1} -2 q^{-2} + q^{-3} } |
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 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -2 z^4+a^2 z^2+5 z^2 a^{-2} -3 z^2 a^{-4} -6 z^2+2 a^2+3 a^{-2} -2 a^{-4} + a^{-6} -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 z^8 a^{-2} +z^8+2 a z^7+5 z^7 a^{-1} +3 z^7 a^{-3} +a^2 z^6+3 z^6 a^{-2} +3 z^6 a^{-4} +z^6-7 a z^5-14 z^5 a^{-1} -6 z^5 a^{-3} +z^5 a^{-5} -4 a^2 z^4-13 z^4 a^{-2} -5 z^4 a^{-4} -12 z^4+6 a z^3+8 z^3 a^{-1} +5 z^3 a^{-3} +3 z^3 a^{-5} +5 a^2 z^2+10 z^2 a^{-2} +6 z^2 a^{-4} +3 z^2 a^{-6} +12 z^2-a z-z a^{-1} -2 z a^{-3} -2 z a^{-5} -2 a^2-3 a^{-2} -2 a^{-4} - a^{-6} -3} |
"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}} ): {K11n117,}
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["10 138"];
|
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^3-5 t^2+8 t-7+8 t^{-1} -5 t^{-2} + t^{-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 2 q^5-4 q^4+5 q^3-6 q^2+6 q-5+4 q^{-1} -2 q^{-2} + q^{-3} } } |
In[5]:=
|
DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
|
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
|
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
|
Out[5]=
|
{} |
In[6]:=
|
DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
Out[6]=
|
{K11n117,} |
Vassiliev invariants
| V2 and V3: | (-3, -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=} 2 is the signature of 10 138. 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 q^{15}-5 q^{13}+7 q^{12}+2 q^{11}-17 q^{10}+16 q^9+8 q^8-29 q^7+19 q^6+16 q^5-34 q^4+13 q^3+22 q^2-30 q+4+23 q^{-1} -20 q^{-2} -4 q^{-3} +17 q^{-4} -8 q^{-5} -5 q^{-6} +7 q^{-7} - q^{-8} -2 q^{-9} + 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 2 q^{29}-4 q^{28}+2 q^{26}+10 q^{25}-12 q^{24}-17 q^{23}+16 q^{22}+34 q^{21}-19 q^{20}-55 q^{19}+19 q^{18}+76 q^{17}-12 q^{16}-96 q^{15}+4 q^{14}+105 q^{13}+11 q^{12}-110 q^{11}-23 q^{10}+104 q^9+36 q^8-95 q^7-46 q^6+81 q^5+54 q^4-61 q^3-63 q^2+45 q+64-22 q^{-1} -65 q^{-2} +4 q^{-3} +56 q^{-4} +15 q^{-5} -47 q^{-6} -23 q^{-7} +29 q^{-8} +31 q^{-9} -18 q^{-10} -25 q^{-11} +4 q^{-12} +20 q^{-13} + q^{-14} -11 q^{-15} -4 q^{-16} +6 q^{-17} +2 q^{-18} - q^{-19} -2 q^{-20} + q^{-21} } |
| 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^{48}-5 q^{46}+11 q^{44}+3 q^{43}-6 q^{42}-29 q^{41}-7 q^{40}+56 q^{39}+34 q^{38}-17 q^{37}-113 q^{36}-58 q^{35}+145 q^{34}+140 q^{33}+4 q^{32}-253 q^{31}-201 q^{30}+215 q^{29}+311 q^{28}+107 q^{27}-367 q^{26}-400 q^{25}+200 q^{24}+440 q^{23}+259 q^{22}-376 q^{21}-547 q^{20}+120 q^{19}+456 q^{18}+371 q^{17}-302 q^{16}-582 q^{15}+38 q^{14}+376 q^{13}+414 q^{12}-193 q^{11}-538 q^{10}-31 q^9+257 q^8+413 q^7-72 q^6-448 q^5-96 q^4+114 q^3+378 q^2+57 q-315-140 q^{-1} -40 q^{-2} +285 q^{-3} +153 q^{-4} -145 q^{-5} -117 q^{-6} -156 q^{-7} +136 q^{-8} +159 q^{-9} +2 q^{-10} -25 q^{-11} -170 q^{-12} +78 q^{-14} +56 q^{-15} +59 q^{-16} -96 q^{-17} -46 q^{-18} -3 q^{-19} +26 q^{-20} +68 q^{-21} -21 q^{-22} -22 q^{-23} -23 q^{-24} -6 q^{-25} +32 q^{-26} +2 q^{-27} -9 q^{-29} -8 q^{-30} +7 q^{-31} + q^{-32} +2 q^{-33} - q^{-34} -2 q^{-35} + q^{-36} } |
| 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 q^{71}-4 q^{70}+2 q^{67}+12 q^{66}+2 q^{65}-28 q^{64}-20 q^{63}+8 q^{62}+38 q^{61}+73 q^{60}+6 q^{59}-123 q^{58}-140 q^{57}-16 q^{56}+187 q^{55}+299 q^{54}+100 q^{53}-316 q^{52}-530 q^{51}-235 q^{50}+404 q^{49}+833 q^{48}+502 q^{47}-451 q^{46}-1189 q^{45}-861 q^{44}+414 q^{43}+1507 q^{42}+1294 q^{41}-244 q^{40}-1769 q^{39}-1743 q^{38}-q^{37}+1904 q^{36}+2127 q^{35}+318 q^{34}-1915 q^{33}-2422 q^{32}-619 q^{31}+1817 q^{30}+2580 q^{29}+893 q^{28}-1660 q^{27}-2628 q^{26}-1082 q^{25}+1461 q^{24}+2577 q^{23}+1219 q^{22}-1262 q^{21}-2476 q^{20}-1292 q^{19}+1064 q^{18}+2326 q^{17}+1350 q^{16}-856 q^{15}-2166 q^{14}-1398 q^{13}+639 q^{12}+1983 q^{11}+1437 q^{10}-384 q^9-1769 q^8-1486 q^7+117 q^6+1520 q^5+1487 q^4+173 q^3-1204 q^2-1465 q-441+865 q^{-1} +1334 q^{-2} +675 q^{-3} -484 q^{-4} -1144 q^{-5} -811 q^{-6} +128 q^{-7} +853 q^{-8} +845 q^{-9} +180 q^{-10} -542 q^{-11} -745 q^{-12} -375 q^{-13} +209 q^{-14} +569 q^{-15} +458 q^{-16} +29 q^{-17} -328 q^{-18} -402 q^{-19} -207 q^{-20} +113 q^{-21} +295 q^{-22} +237 q^{-23} +48 q^{-24} -133 q^{-25} -213 q^{-26} -127 q^{-27} +23 q^{-28} +123 q^{-29} +133 q^{-30} +53 q^{-31} -49 q^{-32} -93 q^{-33} -71 q^{-34} -9 q^{-35} +51 q^{-36} +60 q^{-37} +22 q^{-38} -11 q^{-39} -33 q^{-40} -30 q^{-41} -2 q^{-42} +19 q^{-43} +13 q^{-44} +6 q^{-45} -11 q^{-47} -6 q^{-48} +3 q^{-49} +2 q^{-50} + q^{-51} +2 q^{-52} - q^{-53} -2 q^{-54} + q^{-55} } |
| 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^{99}-5 q^{97}+7 q^{95}+4 q^{94}-q^{92}-10 q^{91}-33 q^{90}-10 q^{89}+59 q^{88}+67 q^{87}+20 q^{86}-31 q^{85}-122 q^{84}-195 q^{83}-62 q^{82}+257 q^{81}+407 q^{80}+252 q^{79}-79 q^{78}-560 q^{77}-895 q^{76}-468 q^{75}+628 q^{74}+1434 q^{73}+1313 q^{72}+304 q^{71}-1360 q^{70}-2689 q^{69}-2062 q^{68}+516 q^{67}+3106 q^{66}+3836 q^{65}+2149 q^{64}-1635 q^{63}-5303 q^{62}-5388 q^{61}-1267 q^{60}+4189 q^{59}+7192 q^{58}+5816 q^{57}-106 q^{56}-7123 q^{55}-9375 q^{54}-4840 q^{53}+3324 q^{52}+9467 q^{51}+9864 q^{50}+3090 q^{49}-6812 q^{48}-11955 q^{47}-8532 q^{46}+850 q^{45}+9496 q^{44}+12299 q^{43}+6222 q^{42}-4911 q^{41}-12255 q^{40}-10591 q^{39}-1587 q^{38}+8003 q^{37}+12603 q^{36}+7896 q^{35}-2919 q^{34}-11135 q^{33}-10849 q^{32}-2970 q^{31}+6298 q^{30}+11702 q^{29}+8245 q^{28}-1516 q^{27}-9694 q^{26}-10261 q^{25}-3638 q^{24}+4811 q^{23}+10513 q^{22}+8180 q^{21}-286 q^{20}-8186 q^{19}-9566 q^{18}-4371 q^{17}+3130 q^{16}+9158 q^{15}+8220 q^{14}+1363 q^{13}-6217 q^{12}-8719 q^{11}-5424 q^{10}+830 q^9+7204 q^8+8087 q^7+3427 q^6-3442 q^5-7140 q^4-6256 q^3-1911 q^2+4286 q+7006+5123 q^{-1} -154 q^{-2} -4390 q^{-3} -5880 q^{-4} -4121 q^{-5} +780 q^{-6} +4481 q^{-7} +5297 q^{-8} +2493 q^{-9} -950 q^{-10} -3787 q^{-11} -4539 q^{-12} -1980 q^{-13} +1169 q^{-14} +3515 q^{-15} +3167 q^{-16} +1655 q^{-17} -857 q^{-18} -2896 q^{-19} -2648 q^{-20} -1220 q^{-21} +916 q^{-22} +1813 q^{-23} +2148 q^{-24} +1081 q^{-25} -601 q^{-26} -1432 q^{-27} -1556 q^{-28} -641 q^{-29} +27 q^{-30} +1024 q^{-31} +1156 q^{-32} +582 q^{-33} -26 q^{-34} -604 q^{-35} -600 q^{-36} -665 q^{-37} -45 q^{-38} +337 q^{-39} +442 q^{-40} +379 q^{-41} +118 q^{-42} -21 q^{-43} -382 q^{-44} -249 q^{-45} -127 q^{-46} +25 q^{-47} +135 q^{-48} +168 q^{-49} +188 q^{-50} -49 q^{-51} -62 q^{-52} -105 q^{-53} -77 q^{-54} -40 q^{-55} +26 q^{-56} +102 q^{-57} +21 q^{-58} +24 q^{-59} -13 q^{-60} -24 q^{-61} -39 q^{-62} -16 q^{-63} +24 q^{-64} +3 q^{-65} +14 q^{-66} +5 q^{-67} +3 q^{-68} -11 q^{-69} -8 q^{-70} +5 q^{-71} -2 q^{-72} +2 q^{-73} + q^{-74} +2 q^{-75} - q^{-76} -2 q^{-77} + q^{-78} } |
| 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 2 q^{131}-4 q^{130}+10 q^{126}+4 q^{125}-4 q^{124}-16 q^{123}-22 q^{122}-4 q^{121}+12 q^{120}+34 q^{119}+78 q^{118}+53 q^{117}-55 q^{116}-161 q^{115}-201 q^{114}-82 q^{113}+110 q^{112}+332 q^{111}+519 q^{110}+367 q^{109}-196 q^{108}-861 q^{107}-1194 q^{106}-828 q^{105}+220 q^{104}+1498 q^{103}+2463 q^{102}+2111 q^{101}+138 q^{100}-2584 q^{99}-4616 q^{98}-4296 q^{97}-1200 q^{96}+3502 q^{95}+7564 q^{94}+8110 q^{93}+3737 q^{92}-3940 q^{91}-11278 q^{90}-13524 q^{89}-8126 q^{88}+3039 q^{87}+14879 q^{86}+20312 q^{85}+14871 q^{84}+11 q^{83}-17596 q^{82}-27866 q^{81}-23558 q^{80}-5459 q^{79}+18330 q^{78}+34791 q^{77}+33434 q^{76}+13450 q^{75}-16471 q^{74}-40128 q^{73}-43282 q^{72}-22929 q^{71}+11993 q^{70}+42776 q^{69}+51651 q^{68}+32875 q^{67}-5362 q^{66}-42614 q^{65}-57642 q^{64}-41859 q^{63}-2210 q^{62}+39922 q^{61}+60703 q^{60}+48891 q^{59}+9628 q^{58}-35597 q^{57}-61109 q^{56}-53422 q^{55}-15874 q^{54}+30643 q^{53}+59505 q^{52}+55554 q^{51}+20410 q^{50}-25991 q^{49}-56715 q^{48}-55711 q^{47}-23237 q^{46}+22058 q^{45}+53549 q^{44}+54696 q^{43}+24672 q^{42}-19075 q^{41}-50470 q^{40}-53065 q^{39}-25281 q^{38}+16697 q^{37}+47676 q^{36}+51393 q^{35}+25669 q^{34}-14603 q^{33}-45126 q^{32}-49896 q^{31}-26220 q^{30}+12340 q^{29}+42479 q^{28}+48603 q^{27}+27295 q^{26}-9483 q^{25}-39537 q^{24}-47428 q^{23}-28846 q^{22}+5877 q^{21}+35879 q^{20}+45981 q^{19}+30851 q^{18}-1327 q^{17}-31309 q^{16}-44046 q^{15}-32974 q^{14}-3902 q^{13}+25631 q^{12}+41104 q^{11}+34720 q^{10}+9706 q^9-18807 q^8-36959 q^7-35645 q^6-15403 q^5+11148 q^4+31224 q^3+35068 q^2+20477 q-2969-24086 q^{-1} -32665 q^{-2} -24040 q^{-3} -4890 q^{-4} +15793 q^{-5} +28117 q^{-6} +25535 q^{-7} +11601 q^{-8} -7124 q^{-9} -21699 q^{-10} -24446 q^{-11} -16263 q^{-12} -1023 q^{-13} +14079 q^{-14} +20880 q^{-15} +18220 q^{-16} +7466 q^{-17} -6217 q^{-18} -15283 q^{-19} -17356 q^{-20} -11524 q^{-21} -640 q^{-22} +8867 q^{-23} +14043 q^{-24} +12633 q^{-25} +5489 q^{-26} -2628 q^{-27} -9241 q^{-28} -11298 q^{-29} -7856 q^{-30} -2056 q^{-31} +4272 q^{-32} +8104 q^{-33} +7715 q^{-34} +4766 q^{-35} -93 q^{-36} -4413 q^{-37} -5942 q^{-38} -5309 q^{-39} -2389 q^{-40} +1154 q^{-41} +3337 q^{-42} +4322 q^{-43} +3255 q^{-44} +957 q^{-45} -1012 q^{-46} -2621 q^{-47} -2798 q^{-48} -1757 q^{-49} -543 q^{-50} +966 q^{-51} +1729 q^{-52} +1595 q^{-53} +1179 q^{-54} +141 q^{-55} -678 q^{-56} -979 q^{-57} -1080 q^{-58} -580 q^{-59} -48 q^{-60} +299 q^{-61} +691 q^{-62} +603 q^{-63} +327 q^{-64} +83 q^{-65} -282 q^{-66} -339 q^{-67} -312 q^{-68} -270 q^{-69} +2 q^{-70} +146 q^{-71} +203 q^{-72} +222 q^{-73} +72 q^{-74} +9 q^{-75} -48 q^{-76} -149 q^{-77} -99 q^{-78} -53 q^{-79} +6 q^{-80} +73 q^{-81} +40 q^{-82} +40 q^{-83} +37 q^{-84} -15 q^{-85} -27 q^{-86} -34 q^{-87} -22 q^{-88} +13 q^{-89} + q^{-90} +5 q^{-91} +16 q^{-92} +5 q^{-93} +2 q^{-94} -8 q^{-95} -8 q^{-96} +3 q^{-97} -2 q^{-99} +2 q^{-100} + q^{-101} +2 q^{-102} - q^{-103} -2 q^{-104} + q^{-105} } |
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. |
|
