K11a62

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K11a61.gif

K11a61

K11a63.gif

K11a63

K11a62.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a62 at Knotilus!

[edit K11a62 Quick Notes]


[edit K11a62 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X8394 X16,5,17,6 X10,8,11,7 X2,9,3,10 X18,11,19,12 X20,13,21,14 X22,15,1,16 X6,17,7,18 X12,19,13,20 X14,21,15,22
Gauss code 1, -5, 2, -1, 3, -9, 4, -2, 5, -4, 6, -10, 7, -11, 8, -3, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 8 16 10 2 18 20 22 6 12 14
A Braid Representative
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BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gif
A Morse Link Presentation K11a62 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 K11a62's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^4+5 t^3-8 t^2+9 t-9+9 t^{-1} -8 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ -z^8-3 z^6+2 z^4+6 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 55, -6 }
Jones polynomial [math]\displaystyle{ q^{-1} -2 q^{-2} +4 q^{-3} -5 q^{-4} +7 q^{-5} -8 q^{-6} +8 q^{-7} -7 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+6 a^8-z^8 a^6-6 z^6 a^6-12 z^4 a^6-11 z^2 a^6-5 a^6+z^6 a^4+5 z^4 a^4+7 z^2 a^4+3 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}-6 z^4 a^{12}+4 z^2 a^{12}+4 z^7 a^{11}-7 z^5 a^{11}+2 z^3 a^{11}+4 z^8 a^{10}-11 z^6 a^{10}+10 z^4 a^{10}-9 z^2 a^{10}+3 a^{10}+3 z^9 a^9-9 z^7 a^9+6 z^5 a^9-3 z^3 a^9+z^{10} a^8+2 z^8 a^8-24 z^6 a^8+41 z^4 a^8-27 z^2 a^8+6 a^8+5 z^9 a^7-24 z^7 a^7+34 z^5 a^7-17 z^3 a^7+3 z a^7+z^{10} a^6-z^8 a^6-15 z^6 a^6+35 z^4 a^6-23 z^2 a^6+5 a^6+2 z^9 a^5-11 z^7 a^5+18 z^5 a^5-9 z^3 a^5+z a^5+z^8 a^4-6 z^6 a^4+12 z^4 a^4-10 z^2 a^4+3 a^4 }[/math]
The A2 invariant [math]\displaystyle{ -q^{36}-q^{34}-q^{30}+q^{28}+2 q^{24}+q^{22}-q^{20}+q^{18}-2 q^{16}+q^{14}+q^{10}+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}-6 q^{176}+3 q^{174}+2 q^{172}-6 q^{170}+11 q^{168}-12 q^{166}+10 q^{164}-8 q^{162}-q^{160}+7 q^{158}-14 q^{156}+15 q^{154}-12 q^{152}+5 q^{150}-6 q^{146}+8 q^{144}-8 q^{142}+6 q^{140}-6 q^{138}+4 q^{136}-5 q^{134}+3 q^{132}+2 q^{130}-8 q^{128}+14 q^{126}-13 q^{124}+7 q^{122}+q^{120}-10 q^{118}+11 q^{116}-4 q^{114}-5 q^{112}+14 q^{110}-14 q^{108}+7 q^{106}+11 q^{104}-24 q^{102}+31 q^{100}-26 q^{98}+12 q^{96}+7 q^{94}-21 q^{92}+33 q^{90}-30 q^{88}+23 q^{86}-9 q^{84}-7 q^{82}+19 q^{80}-24 q^{78}+18 q^{76}-9 q^{74}-6 q^{72}+16 q^{70}-18 q^{68}+7 q^{66}+10 q^{64}-25 q^{62}+29 q^{60}-22 q^{58}+q^{56}+20 q^{54}-34 q^{52}+38 q^{50}-26 q^{48}+9 q^{46}+11 q^{44}-22 q^{42}+25 q^{40}-17 q^{38}+10 q^{36}-4 q^{32}+6 q^{30}-5 q^{28}+4 q^{26}-q^{24}+q^{22} }[/math]
Further Quantum Invariants
K11a62 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["K11a62"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^4+5 t^3-8 t^2+9 t-9+9 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+6 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]= { 55, -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} -5 q^{-4} +7 q^{-5} -8 q^{-6} +8 q^{-7} -7 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+6 a^8-z^8 a^6-6 z^6 a^6-12 z^4 a^6-11 z^2 a^6-5 a^6+z^6 a^4+5 z^4 a^4+7 z^2 a^4+3 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}-6 z^4 a^{12}+4 z^2 a^{12}+4 z^7 a^{11}-7 z^5 a^{11}+2 z^3 a^{11}+4 z^8 a^{10}-11 z^6 a^{10}+10 z^4 a^{10}-9 z^2 a^{10}+3 a^{10}+3 z^9 a^9-9 z^7 a^9+6 z^5 a^9-3 z^3 a^9+z^{10} a^8+2 z^8 a^8-24 z^6 a^8+41 z^4 a^8-27 z^2 a^8+6 a^8+5 z^9 a^7-24 z^7 a^7+34 z^5 a^7-17 z^3 a^7+3 z a^7+z^{10} a^6-z^8 a^6-15 z^6 a^6+35 z^4 a^6-23 z^2 a^6+5 a^6+2 z^9 a^5-11 z^7 a^5+18 z^5 a^5-9 z^3 a^5+z a^5+z^8 a^4-6 z^6 a^4+12 z^4 a^4-10 z^2 a^4+3 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["K11a62"];
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+9 t-9+9 t^{-1} -8 t^{-2} +5 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^{-1} -2 q^{-2} +4 q^{-3} -5 q^{-4} +7 q^{-5} -8 q^{-6} +8 q^{-7} -7 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: (6, -17)
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{ 24 }[/math] [math]\displaystyle{ -136 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 876 }[/math] [math]\displaystyle{ 124 }[/math] [math]\displaystyle{ -3264 }[/math] [math]\displaystyle{ -\frac{18832}{3} }[/math] [math]\displaystyle{ -\frac{3232}{3} }[/math] [math]\displaystyle{ -776 }[/math] [math]\displaystyle{ 2304 }[/math] [math]\displaystyle{ 9248 }[/math] [math]\displaystyle{ 21024 }[/math] [math]\displaystyle{ 2976 }[/math] [math]\displaystyle{ \frac{231671}{5} }[/math] [math]\displaystyle{ \frac{28148}{15} }[/math] [math]\displaystyle{ \frac{84188}{5} }[/math] [math]\displaystyle{ 339 }[/math] [math]\displaystyle{ \frac{10391}{5} }[/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 K11a62. 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        32  -1
-9       42   2
-11      43    -1
-13     44     0
-15    34      1
-17   34       -1
-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}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

K11a61.gif

K11a61

K11a63.gif

K11a63

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