K11a142

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K11a141.gif

K11a141

K11a143.gif

K11a143

K11a142.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a142 at Knotilus!

[edit K11a142 Quick Notes]


[edit K11a142 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X16,5,17,6 X18,7,19,8 X12,10,13,9 X2,11,3,12 X20,13,21,14 X22,15,1,16 X6,17,7,18 X8,19,9,20 X14,21,15,22
Gauss code 1, -6, 2, -1, 3, -9, 4, -10, 5, -2, 6, -5, 7, -11, 8, -3, 9, -4, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 10 16 18 12 2 20 22 6 8 14
A Braid Representative
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A Morse Link Presentation K11a142 ML.gif

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [math]\displaystyle{ 4 }[/math]
Rasmussen s-Invariant 6

[edit Notes for K11a142's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^4+5 t^3-8 t^2+10 t-11+10 t^{-1} -8 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ -z^8-3 z^6+2 z^4+7 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 59, -6 }
Jones polynomial [math]\displaystyle{ q^{-1} -2 q^{-2} +4 q^{-3} -6 q^{-4} +8 q^{-5} -8 q^{-6} +9 q^{-7} -8 q^{-8} +6 q^{-9} -4 q^{-10} +2 q^{-11} - q^{-12} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^4 a^{10}-4 z^2 a^{10}-3 a^{10}+2 z^6 a^8+10 z^4 a^8+14 z^2 a^8+5 a^8-z^8 a^6-6 z^6 a^6-12 z^4 a^6-10 z^2 a^6-3 a^6+z^6 a^4+5 z^4 a^4+7 z^2 a^4+2 a^4 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^3 a^{15}-z a^{15}+2 z^4 a^{14}-z^2 a^{14}+3 z^5 a^{13}-2 z^3 a^{13}+z a^{13}+4 z^6 a^{12}-5 z^4 a^{12}+2 z^2 a^{12}+5 z^7 a^{11}-11 z^5 a^{11}+7 z^3 a^{11}-2 z a^{11}+5 z^8 a^{10}-15 z^6 a^{10}+15 z^4 a^{10}-11 z^2 a^{10}+3 a^{10}+3 z^9 a^9-6 z^7 a^9-6 z^5 a^9+10 z^3 a^9-3 z a^9+z^{10} a^8+3 z^8 a^8-26 z^6 a^8+38 z^4 a^8-21 z^2 a^8+5 a^8+5 z^9 a^7-22 z^7 a^7+27 z^5 a^7-11 z^3 a^7+2 z a^7+z^{10} a^6-z^8 a^6-13 z^6 a^6+28 z^4 a^6-16 z^2 a^6+3 a^6+2 z^9 a^5-11 z^7 a^5+19 z^5 a^5-11 z^3 a^5+z a^5+z^8 a^4-6 z^6 a^4+12 z^4 a^4-9 z^2 a^4+2 a^4 }[/math]
The A2 invariant [math]\displaystyle{ -q^{36}-q^{34}-q^{30}+q^{28}-q^{26}+q^{24}+q^{22}+3 q^{18}-q^{16}+q^{14}-q^{12}+q^8+q^4 }[/math]
The G2 invariant [math]\displaystyle{ q^{196}-q^{194}+2 q^{192}-2 q^{190}+q^{188}-2 q^{184}+4 q^{182}-5 q^{180}+6 q^{178}-5 q^{176}+2 q^{174}+2 q^{172}-5 q^{170}+9 q^{168}-11 q^{166}+8 q^{164}-7 q^{162}+5 q^{158}-10 q^{156}+14 q^{154}-13 q^{152}+9 q^{150}-4 q^{148}-5 q^{146}+7 q^{144}-10 q^{142}+12 q^{140}-13 q^{138}+14 q^{136}-11 q^{134}+3 q^{132}+8 q^{130}-19 q^{128}+24 q^{126}-25 q^{124}+11 q^{122}+4 q^{120}-20 q^{118}+24 q^{116}-12 q^{114}-8 q^{112}+23 q^{110}-29 q^{108}+14 q^{106}+11 q^{104}-33 q^{102}+46 q^{100}-40 q^{98}+26 q^{96}+8 q^{94}-30 q^{92}+48 q^{90}-46 q^{88}+35 q^{86}-12 q^{84}-9 q^{82}+27 q^{80}-33 q^{78}+30 q^{76}-12 q^{74}-10 q^{72}+24 q^{70}-29 q^{68}+11 q^{66}+13 q^{64}-34 q^{62}+41 q^{60}-33 q^{58}+6 q^{56}+24 q^{54}-44 q^{52}+49 q^{50}-36 q^{48}+11 q^{46}+14 q^{44}-27 q^{42}+30 q^{40}-21 q^{38}+12 q^{36}+q^{34}-6 q^{32}+7 q^{30}-6 q^{28}+4 q^{26}-q^{24}+q^{22} }[/math]
Further Quantum Invariants
K11a142 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`
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["K11a142"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^4+5 t^3-8 t^2+10 t-11+10 t^{-1} -8 t^{-2} +5 t^{-3} - t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^8-3 z^6+2 z^4+7 z^2+1 }[/math]
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]= [math]\displaystyle{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 59, -6 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^{-1} -2 q^{-2} +4 q^{-3} -6 q^{-4} +8 q^{-5} -8 q^{-6} +9 q^{-7} -8 q^{-8} +6 q^{-9} -4 q^{-10} +2 q^{-11} - q^{-12} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -z^4 a^{10}-4 z^2 a^{10}-3 a^{10}+2 z^6 a^8+10 z^4 a^8+14 z^2 a^8+5 a^8-z^8 a^6-6 z^6 a^6-12 z^4 a^6-10 z^2 a^6-3 a^6+z^6 a^4+5 z^4 a^4+7 z^2 a^4+2 a^4 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^3 a^{15}-z a^{15}+2 z^4 a^{14}-z^2 a^{14}+3 z^5 a^{13}-2 z^3 a^{13}+z a^{13}+4 z^6 a^{12}-5 z^4 a^{12}+2 z^2 a^{12}+5 z^7 a^{11}-11 z^5 a^{11}+7 z^3 a^{11}-2 z a^{11}+5 z^8 a^{10}-15 z^6 a^{10}+15 z^4 a^{10}-11 z^2 a^{10}+3 a^{10}+3 z^9 a^9-6 z^7 a^9-6 z^5 a^9+10 z^3 a^9-3 z a^9+z^{10} a^8+3 z^8 a^8-26 z^6 a^8+38 z^4 a^8-21 z^2 a^8+5 a^8+5 z^9 a^7-22 z^7 a^7+27 z^5 a^7-11 z^3 a^7+2 z a^7+z^{10} a^6-z^8 a^6-13 z^6 a^6+28 z^4 a^6-16 z^2 a^6+3 a^6+2 z^9 a^5-11 z^7 a^5+19 z^5 a^5-11 z^3 a^5+z a^5+z^8 a^4-6 z^6 a^4+12 z^4 a^4-9 z^2 a^4+2 a^4 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

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.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["K11a142"];
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]= { [math]\displaystyle{ -t^4+5 t^3-8 t^2+10 t-11+10 t^{-1} -8 t^{-2} +5 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^{-1} -2 q^{-2} +4 q^{-3} -6 q^{-4} +8 q^{-5} -8 q^{-6} +9 q^{-7} -8 q^{-8} +6 q^{-9} -4 q^{-10} +2 q^{-11} - q^{-12} }[/math] }
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]= {}

Vassiliev invariants

V2 and V3: (7, -20)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
[math]\displaystyle{ 28 }[/math] [math]\displaystyle{ -160 }[/math] [math]\displaystyle{ 392 }[/math] [math]\displaystyle{ \frac{3314}{3} }[/math] [math]\displaystyle{ \frac{526}{3} }[/math] [math]\displaystyle{ -4480 }[/math] [math]\displaystyle{ -\frac{25408}{3} }[/math] [math]\displaystyle{ -\frac{4384}{3} }[/math] [math]\displaystyle{ -1216 }[/math] [math]\displaystyle{ \frac{10976}{3} }[/math] [math]\displaystyle{ 12800 }[/math] [math]\displaystyle{ \frac{92792}{3} }[/math] [math]\displaystyle{ \frac{14728}{3} }[/math] [math]\displaystyle{ \frac{1979977}{30} }[/math] [math]\displaystyle{ \frac{12206}{15} }[/math] [math]\displaystyle{ \frac{1211474}{45} }[/math] [math]\displaystyle{ \frac{10295}{18} }[/math] [math]\displaystyle{ \frac{107017}{30} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-6 is the signature of K11a142. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -9-8-7-6-5-4-3-2-1012χ
-1           11
-3          1 -1
-5         31 2
-7        42  -2
-9       42   2
-11      44    0
-13     54     1
-15    34      1
-17   35       -2
-19  13        2
-21 13         -2
-23 1          1
-251           -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-7 }[/math] [math]\displaystyle{ i=-5 }[/math]
[math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

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.

K11a141.gif

K11a141

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K11a143

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