K11a177
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a177's page at Knotilus! Visit K11a177's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X12,4,13,3 X14,5,15,6 X16,8,17,7 X18,10,19,9 X2,12,3,11 X22,13,1,14 X20,16,21,15 X8,18,9,17 X10,20,11,19 X6,21,7,22 |
| Gauss code | 1, -6, 2, -1, 3, -11, 4, -9, 5, -10, 6, -2, 7, -3, 8, -4, 9, -5, 10, -8, 11, -7 |
| Dowker-Thistlethwaite code | 4 12 14 16 18 2 22 20 8 10 6 |
| 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 + 13t2−19t + 21−19t−1 + 13t−2−5t−3 + t−4 |
| Conway polynomial | z8 + 3z6 + 3z4 + 4z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 97, 4 } |
| Jones polynomial | q10−4q9 + 7q8−11q7 + 14q6−15q5 + 15q4−12q3 + 9q2−5q + 3−q−1 |
| HOMFLY-PT polynomial (db, data sources) | z8a−4−z6a−2 + 6z6a−4−2z6a−6−4z4a−2 + 14z4a−4−8z4a−6 + z4a−8−4z2a−2 + 15z2a−4−9z2a−6 + 2z2a−8−a−2 + 5a−4−3a−6 |
| Kauffman polynomial (db, data sources) | z10a−4 + z10a−6 + 3z9a−3 + 7z9a−5 + 4z9a−7 + 3z8a−2 + 6z8a−4 + 10z8a−6 + 7z8a−8 + z7a−1−9z7a−3−18z7a−5 + 8z7a−9−13z6a−2−34z6a−4−36z6a−6−8z6a−8 + 7z6a−10−4z5a−1 + 3z5a−3 + 7z5a−5−12z5a−7−8z5a−9 + 4z5a−11 + 17z4a−2 + 47z4a−4 + 39z4a−6 + z4a−8−7z4a−10 + z4a−12 + 4z3a−1 + 6z3a−3 + 8z3a−5 + 8z3a−7−z3a−9−3z3a−11−8z2a−2−25z2a−4−18z2a−6 + z2a−10−za−1−3za−3−5za−5−za−7 + 2za−9 + a−2 + 5a−4 + 3a−6 |
| The A2 invariant | −q2 + 1−q−2 + q−4 + 2q−6−q−8 + 4q−10−2q−12 + 2q−14 + q−16−q−18 + 2q−20−3q−22−q−26−q−28 + q−30 |
| The G2 invariant | q12−2q10 + 5q8−9q6 + 10q4−11q2 + 3 + 14q−2−35q−4 + 56q−6−66q−8 + 50q−10−12q−12−51q−14 + 116q−16−156q−18 + 154q−20−93q−22−8q−24 + 123q−26−205q−28 + 225q−30−168q−32 + 57q−34 + 71q−36−167q−38 + 196q−40−142q−42 + 46q−44 + 69q−46−136q−48 + 131q−50−62q−52−50q−54 + 153q−56−199q−58 + 170q−60−60q−62−87q−64 + 224q−66−291q−68 + 264q−70−149q−72−16q−74 + 171q−76−265q−78 + 267q−80−178q−82 + 44q−84 + 84q−86−158q−88 + 150q−90−84q−92−12q−94 + 86q−96−112q−98 + 76q−100−q−102−88q−104 + 148q−106−157q−108 + 113q−110−33q−112−62q−114 + 131q−116−166q−118 + 158q−120−107q−122 + 37q−124 + 37q−126−91q−128 + 114q−130−107q−132 + 78q−134−35q−136−6q−138 + 34q−140−50q−142 + 47q−144−32q−146 + 18q−148−2q−150−6q−152 + 9q−154−10q−156 + 6q−158−3q−160 + q−162 |
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["K11a177"];
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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 + 13t2−19t + 21−19t−1 + 13t−2−5t−3 + t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z8 + 3z6 + 3z4 + 4z2 + 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]=
| { 97, 4 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q10−4q9 + 7q8−11q7 + 14q6−15q5 + 15q4−12q3 + 9q2−5q + 3−q−1 |
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]=
| z8a−4−z6a−2 + 6z6a−4−2z6a−6−4z4a−2 + 14z4a−4−8z4a−6 + z4a−8−4z2a−2 + 15z2a−4−9z2a−6 + 2z2a−8−a−2 + 5a−4−3a−6 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z10a−4 + z10a−6 + 3z9a−3 + 7z9a−5 + 4z9a−7 + 3z8a−2 + 6z8a−4 + 10z8a−6 + 7z8a−8 + z7a−1−9z7a−3−18z7a−5 + 8z7a−9−13z6a−2−34z6a−4−36z6a−6−8z6a−8 + 7z6a−10−4z5a−1 + 3z5a−3 + 7z5a−5−12z5a−7−8z5a−9 + 4z5a−11 + 17z4a−2 + 47z4a−4 + 39z4a−6 + z4a−8−7z4a−10 + z4a−12 + 4z3a−1 + 6z3a−3 + 8z3a−5 + 8z3a−7−z3a−9−3z3a−11−8z2a−2−25z2a−4−18z2a−6 + z2a−10−za−1−3za−3−5za−5−za−7 + 2za−9 + a−2 + 5a−4 + 3a−6 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, 
):
{}
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["K11a177"];
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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 + 13t2−19t + 21−19t−1 + 13t−2−5t−3 + t−4, q10−4q9 + 7q8−11q7 + 14q6−15q5 + 15q4−12q3 + 9q2−5q + 3−q−1 } |
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]=
| {} |
[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 = 4 is the signature of K11a177. 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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