K11a35
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a35's page at Knotilus! Visit K11a35's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8493 X14,5,15,6 X2837 X16,9,17,10 X18,12,19,11 X22,13,1,14 X6,15,7,16 X20,18,21,17 X12,20,13,19 X10,21,11,22 |
| Gauss code | 1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -6, 10, -9, 11, -7 |
| Dowker-Thistlethwaite code | 4 8 14 2 16 18 22 6 20 12 10 |
| A Braid Representative |
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| A Morse Link Presentation | 
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[edit] Three dimensional invariants
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[edit] Four dimensional invariants
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[edit] Polynomial invariants
| Alexander polynomial | t4−5t3 + 14t2−25t + 31−25t−1 + 14t−2−5t−3 + t−4 |
| Conway polynomial | z8 + 3z6 + 4z4 + 2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 121, 0 } |
| Jones polynomial | q6−4q5 + 8q4−13q3 + 17q2−19q + 20−16q−1 + 12q−2−7q−3 + 3q−4−q−5 |
| HOMFLY-PT polynomial (db, data sources) | z8−a2z6−2z6a−2 + 6z6−4a2z4−8z4a−2 + z4a−4 + 15z4−6a2z2−11z2a−2 + 2z2a−4 + 17z2−3a2−5a−2 + a−4 + 8 |
| Kauffman polynomial (db, data sources) | z10a−2 + z10 + 4az9 + 8z9a−1 + 4z9a−3 + 6a2z8 + 14z8a−2 + 6z8a−4 + 14z8 + 5a3z7 + 2az7−5z7a−1 + 2z7a−3 + 4z7a−5 + 3a4z6−9a2z6−41z6a−2−13z6a−4 + z6a−6−39z6 + a5z5−7a3z5−14az5−22z5a−1−26z5a−3−10z5a−5−5a4z4 + 10a2z4 + 39z4a−2 + 6z4a−4−2z4a−6 + 46z4−2a5z3 + 3a3z3 + 17az3 + 31z3a−1 + 26z3a−3 + 7z3a−5 + 2a4z2−9a2z2−19z2a−2−2z2a−4 + z2a−6−27z2 + a5z−a3z−7az−11za−1−8za−3−2za−5 + 3a2 + 5a−2 + a−4 + 8 |
| The A2 invariant | −q14 + q12−3q10 + q8 + q6−2q4 + 6q2−1 + 4q−2−2q−6 + 2q−8−4q−10 + q−12−q−16 + q−18 |
| The G2 invariant | q80−2q78 + 5q76−8q74 + 9q72−8q70 + q68 + 13q66−29q64 + 47q62−59q60 + 52q58−29q56−18q54 + 82q52−145q50 + 189q48−190q46 + 123q44−q42−162q40 + 313q38−397q36 + 369q34−220q32−26q30 + 284q28−458q26 + 482q24−334q22 + 71q20 + 198q18−370q16 + 362q14−175q12−93q10 + 339q8−419q6 + 293q4 + 9q2−356 + 618q−2−666q−4 + 474q−6−91q−8−335q−10 + 668q−12−772q−14 + 624q−16−278q−18−136q−20 + 452q−22−573q−24 + 464q−26−186q−28−133q−30 + 350q−32−382q−34 + 211q−36 + 72q−38−347q−40 + 480q−42−419q−44 + 178q−46 + 137q−48−410q−50 + 541q−52−482q−54 + 277q−56−11q−58−226q−60 + 352q−62−351q−64 + 255q−66−106q−68−28q−70 + 113q−72−139q−74 + 116q−76−69q−78 + 26q−80 + 7q−82−22q−84 + 21q−86−16q−88 + 8q−90−3q−92 + q−94 |
Further Quantum Invariants
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`
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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
| K = Knot["K11a35"];
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In[4]:=
| Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| t4−5t3 + 14t2−25t + 31−25t−1 + 14t−2−5t−3 + t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z8 + 3z6 + 4z4 + 2z2 + 1 |
In[6]:=
| Alexander[K, 2][t]
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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.
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Out[6]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
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Out[7]=
| { 121, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q6−4q5 + 8q4−13q3 + 17q2−19q + 20−16q−1 + 12q−2−7q−3 + 3q−4−q−5 |
In[9]:=
| HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
| z8−a2z6−2z6a−2 + 6z6−4a2z4−8z4a−2 + z4a−4 + 15z4−6a2z2−11z2a−2 + 2z2a−4 + 17z2−3a2−5a−2 + a−4 + 8 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z10a−2 + z10 + 4az9 + 8z9a−1 + 4z9a−3 + 6a2z8 + 14z8a−2 + 6z8a−4 + 14z8 + 5a3z7 + 2az7−5z7a−1 + 2z7a−3 + 4z7a−5 + 3a4z6−9a2z6−41z6a−2−13z6a−4 + z6a−6−39z6 + a5z5−7a3z5−14az5−22z5a−1−26z5a−3−10z5a−5−5a4z4 + 10a2z4 + 39z4a−2 + 6z4a−4−2z4a−6 + 46z4−2a5z3 + 3a3z3 + 17az3 + 31z3a−1 + 26z3a−3 + 7z3a−5 + 2a4z2−9a2z2−19z2a−2−2z2a−4 + z2a−6−27z2 + a5z−a3z−7az−11za−1−8za−3−2za−5 + 3a2 + 5a−2 + a−4 + 8 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a316,}
Same Jones Polynomial (up to mirroring, 
):
{K11a36, K11a316,}
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`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["K11a35"];
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In[4]:=
| {A = Alexander[K][t], J = Jones[K][q]}
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[4]=
| { t4−5t3 + 14t2−25t + 31−25t−1 + 14t−2−5t−3 + t−4, q6−4q5 + 8q4−13q3 + 17q2−19q + 20−16q−1 + 12q−2−7q−3 + 3q−4−q−5 } |
In[5]:=
| DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
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KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
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KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
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Out[5]=
| {K11a316,} |
In[6]:=
| DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
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KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
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Out[6]=
| {K11a36, K11a316,} |
[edit] Khovanov Homology
| The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j−2r = s + 1 or j−2r = s−1, where s = 0 is the signature of K11a35. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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| Integral Khovanov Homology
(db, data source) |
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[edit] Computer Talk
Much of the above data can be recomputed by Mathematica using the packageKnotTheory`. See A Sample KnotTheory` Session.
[edit] Modifying This Page
| Read me first: Modifying Knot Pages.
See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top. |
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