K11a112
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a112's page at Knotilus! Visit K11a112's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,3,11,4 X14,6,15,5 X16,8,17,7 X18,9,19,10 X2,11,3,12 X20,13,21,14 X6,16,7,15 X22,18,1,17 X12,19,13,20 X8,21,9,22 |
| Gauss code | 1, -6, 2, -1, 3, -8, 4, -11, 5, -2, 6, -10, 7, -3, 8, -4, 9, -5, 10, -7, 11, -9 |
| Dowker-Thistlethwaite code | 4 10 14 16 18 2 20 6 22 12 8 |
| 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 | t4−6t3 + 15t2−25t + 31−25t−1 + 15t−2−6t−3 + t−4 |
| Conway polynomial | z8 + 2z6−z4−3z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 125, 0 } |
| Jones polynomial | −q5 + 4q4−8q3 + 14q2−18q + 20−20q−1 + 17q−2−12q−3 + 7q−4−3q−5 + q−6 |
| HOMFLY-PT polynomial (db, data sources) | z8−2a2z6−z6a−2 + 5z6 + a4z4−8a2z4−3z4a−2 + 9z4 + 3a4z2−10a2z2−2z2a−2 + 6z2 + 2a4−3a2 + a−2 + 1 |
| Kauffman polynomial (db, data sources) | 2a2z10 + 2z10 + 5a3z9 + 11az9 + 6z9a−1 + 5a4z8 + 7a2z8 + 8z8a−2 + 10z8 + 3a5z7−10a3z7−24az7−4z7a−1 + 7z7a−3 + a6z6−13a4z6−30a2z6−10z6a−2 + 4z6a−4−30z6−8a5z5 + 6a3z5 + 19az5−6z5a−1−10z5a−3 + z5a−5−3a6z4 + 12a4z4 + 39a2z4−6z4a−4 + 30z4 + 5a5z3−2a3z3−3az3 + 8z3a−1 + 3z3a−3−z3a−5 + 2a6z2−8a4z2−21a2z2 + 3z2a−2 + 2z2a−4−10z2−a5z−az−3za−1−za−3 + 2a4 + 3a2−a−2 + 1 |
| The A2 invariant | q18 + 2q12−3q10 + 3q8−q6−2q4 + 2q2−5 + 4q−2−2q−4 + 2q−6 + 3q−8−2q−10 + 2q−12−q−14 |
| The G2 invariant | q94−2q92 + 5q90−9q88 + 11q86−12q84 + 5q82 + 11q80−32q78 + 59q76−80q74 + 82q72−56q70−11q68 + 115q66−222q64 + 297q62−283q60 + 156q58 + 70q56−331q54 + 533q52−570q50 + 403q48−73q46−309q44 + 581q42−627q40 + 434q38−68q36−299q34 + 502q32−461q30 + 179q28 + 200q26−506q24 + 596q22−418q20 + 36q18 + 410q16−746q14 + 838q12−646q10 + 216q8 + 289q6−705q4 + 874q2−737 + 363q−2 + 107q−4−487q−6 + 626q−8−488q−10 + 146q−12 + 231q−14−463q−16 + 455q−18−206q−20−151q−22 + 463q−24−584q−26 + 485q−28−208q−30−138q−32 + 410q−34−527q−36 + 476q−38−288q−40 + 60q−42 + 137q−44−258q−46 + 281q−48−233q−50 + 141q−52−39q−54−40q−56 + 83q−58−94q−60 + 78q−62−47q−64 + 20q−66 + 3q−68−14q−70 + 15q−72−13q−74 + 7q−76−3q−78 + q−80 |
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["K11a112"];
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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]=
| t4−6t3 + 15t2−25t + 31−25t−1 + 15t−2−6t−3 + t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z8 + 2z6−z4−3z2 + 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]=
| { 125, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q5 + 4q4−8q3 + 14q2−18q + 20−20q−1 + 17q−2−12q−3 + 7q−4−3q−5 + q−6 |
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]=
| z8−2a2z6−z6a−2 + 5z6 + a4z4−8a2z4−3z4a−2 + 9z4 + 3a4z2−10a2z2−2z2a−2 + 6z2 + 2a4−3a2 + a−2 + 1 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| 2a2z10 + 2z10 + 5a3z9 + 11az9 + 6z9a−1 + 5a4z8 + 7a2z8 + 8z8a−2 + 10z8 + 3a5z7−10a3z7−24az7−4z7a−1 + 7z7a−3 + a6z6−13a4z6−30a2z6−10z6a−2 + 4z6a−4−30z6−8a5z5 + 6a3z5 + 19az5−6z5a−1−10z5a−3 + z5a−5−3a6z4 + 12a4z4 + 39a2z4−6z4a−4 + 30z4 + 5a5z3−2a3z3−3az3 + 8z3a−1 + 3z3a−3−z3a−5 + 2a6z2−8a4z2−21a2z2 + 3z2a−2 + 2z2a−4−10z2−a5z−az−3za−1−za−3 + 2a4 + 3a2−a−2 + 1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, 
):
{K11a5,}
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["K11a112"];
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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]=
| { t4−6t3 + 15t2−25t + 31−25t−1 + 15t−2−6t−3 + t−4, −q5 + 4q4−8q3 + 14q2−18q + 20−20q−1 + 17q−2−12q−3 + 7q−4−3q−5 + q−6 } |
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]=
| {} |
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]=
| {K11a5,} |
[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 K11a112. 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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