K11a184

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K11a183.gif

K11a183

K11a185.gif

K11a185

K11a184.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a184 at Knotilus!

[edit K11a184 Quick Notes]


[edit K11a184 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X14,6,15,5 X16,7,17,8 X18,9,19,10 X22,12,1,11 X2,13,3,14 X20,15,21,16 X8,17,9,18 X10,19,11,20 X6,22,7,21
Gauss code 1, -7, 2, -1, 3, -11, 4, -9, 5, -10, 6, -2, 7, -3, 8, -4, 9, -5, 10, -8, 11, -6
Dowker-Thistlethwaite code 4 12 14 16 18 22 2 20 8 10 6
A Braid Representative
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A Morse Link Presentation K11a184 ML.gif

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [math]\displaystyle{ 4 }[/math]
Rasmussen s-Invariant 2

[edit Notes for K11a184's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^4+5 t^3-11 t^2+17 t-19+17 t^{-1} -11 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ -z^8-3 z^6-z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 87, -2 }
Jones polynomial [math]\displaystyle{ q^3-3 q^2+5 q-8+12 q^{-1} -13 q^{-2} +14 q^{-3} -12 q^{-4} +9 q^{-5} -6 q^{-6} +3 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -a^2 z^8+2 a^4 z^6-6 a^2 z^6+z^6-a^6 z^4+9 a^4 z^4-13 a^2 z^4+4 z^4-3 a^6 z^2+12 a^4 z^2-11 a^2 z^2+4 z^2-2 a^6+4 a^4-2 a^2+1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a^4 z^{10}+a^2 z^{10}+3 a^5 z^9+6 a^3 z^9+3 a z^9+4 a^6 z^8+5 a^4 z^8+5 a^2 z^8+4 z^8+4 a^7 z^7-4 a^5 z^7-17 a^3 z^7-6 a z^7+3 z^7 a^{-1} +3 a^8 z^6-6 a^6 z^6-21 a^4 z^6-26 a^2 z^6+z^6 a^{-2} -13 z^6+a^9 z^5-6 a^7 z^5+3 a^5 z^5+18 a^3 z^5-2 a z^5-10 z^5 a^{-1} -6 a^8 z^4+4 a^6 z^4+34 a^4 z^4+39 a^2 z^4-3 z^4 a^{-2} +12 z^4-2 a^9 z^3-2 a^3 z^3+7 a z^3+7 z^3 a^{-1} +2 a^8 z^2-5 a^6 z^2-20 a^4 z^2-20 a^2 z^2+z^2 a^{-2} -6 z^2+a^9 z-a^5 z-a^3 z-2 a z-z a^{-1} +2 a^6+4 a^4+2 a^2+1 }[/math]
The A2 invariant [math]\displaystyle{ -q^{24}-q^{18}+2 q^{16}-2 q^{14}+q^{12}+q^{10}+4 q^6-2 q^4+2 q^2-1- q^{-2} + q^{-4} - q^{-6} + q^{-8} }[/math]
The G2 invariant [math]\displaystyle{ q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+8 q^{120}-5 q^{118}-3 q^{116}+16 q^{114}-27 q^{112}+36 q^{110}-37 q^{108}+21 q^{106}+2 q^{104}-35 q^{102}+65 q^{100}-82 q^{98}+80 q^{96}-56 q^{94}+11 q^{92}+43 q^{90}-94 q^{88}+126 q^{86}-123 q^{84}+82 q^{82}-19 q^{80}-56 q^{78}+110 q^{76}-126 q^{74}+101 q^{72}-35 q^{70}-37 q^{68}+85 q^{66}-88 q^{64}+40 q^{62}+34 q^{60}-101 q^{58}+126 q^{56}-95 q^{54}+15 q^{52}+89 q^{50}-172 q^{48}+207 q^{46}-167 q^{44}+71 q^{42}+48 q^{40}-152 q^{38}+207 q^{36}-190 q^{34}+121 q^{32}-15 q^{30}-80 q^{28}+140 q^{26}-134 q^{24}+74 q^{22}+9 q^{20}-79 q^{18}+102 q^{16}-75 q^{14}+4 q^{12}+79 q^{10}-129 q^8+135 q^6-89 q^4-q^2+85-143 q^{-2} +151 q^{-4} -113 q^{-6} +48 q^{-8} +25 q^{-10} -78 q^{-12} +103 q^{-14} -94 q^{-16} +64 q^{-18} -24 q^{-20} -12 q^{-22} +33 q^{-24} -41 q^{-26} +36 q^{-28} -23 q^{-30} +12 q^{-32} + q^{-34} -7 q^{-36} +7 q^{-38} -7 q^{-40} +4 q^{-42} -2 q^{-44} + q^{-46} }[/math]
Further Quantum Invariants
K11a184 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["K11a184"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^4+5 t^3-11 t^2+17 t-19+17 t^{-1} -11 t^{-2} +5 t^{-3} - t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^8-3 z^6-z^4+2 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]= { 87, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^3-3 q^2+5 q-8+12 q^{-1} -13 q^{-2} +14 q^{-3} -12 q^{-4} +9 q^{-5} -6 q^{-6} +3 q^{-7} - q^{-8} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -a^2 z^8+2 a^4 z^6-6 a^2 z^6+z^6-a^6 z^4+9 a^4 z^4-13 a^2 z^4+4 z^4-3 a^6 z^2+12 a^4 z^2-11 a^2 z^2+4 z^2-2 a^6+4 a^4-2 a^2+1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a^4 z^{10}+a^2 z^{10}+3 a^5 z^9+6 a^3 z^9+3 a z^9+4 a^6 z^8+5 a^4 z^8+5 a^2 z^8+4 z^8+4 a^7 z^7-4 a^5 z^7-17 a^3 z^7-6 a z^7+3 z^7 a^{-1} +3 a^8 z^6-6 a^6 z^6-21 a^4 z^6-26 a^2 z^6+z^6 a^{-2} -13 z^6+a^9 z^5-6 a^7 z^5+3 a^5 z^5+18 a^3 z^5-2 a z^5-10 z^5 a^{-1} -6 a^8 z^4+4 a^6 z^4+34 a^4 z^4+39 a^2 z^4-3 z^4 a^{-2} +12 z^4-2 a^9 z^3-2 a^3 z^3+7 a z^3+7 z^3 a^{-1} +2 a^8 z^2-5 a^6 z^2-20 a^4 z^2-20 a^2 z^2+z^2 a^{-2} -6 z^2+a^9 z-a^5 z-a^3 z-2 a z-z a^{-1} +2 a^6+4 a^4+2 a^2+1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_112,}

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["K11a184"];
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^3-11 t^2+17 t-19+17 t^{-1} -11 t^{-2} +5 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^3-3 q^2+5 q-8+12 q^{-1} -13 q^{-2} +14 q^{-3} -12 q^{-4} +9 q^{-5} -6 q^{-6} +3 q^{-7} - q^{-8} }[/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]= {10_112,}
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: (2, -4)
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{ 8 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{364}{3} }[/math] [math]\displaystyle{ \frac{68}{3} }[/math] [math]\displaystyle{ -256 }[/math] [math]\displaystyle{ -\frac{1472}{3} }[/math] [math]\displaystyle{ -\frac{128}{3} }[/math] [math]\displaystyle{ -96 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 512 }[/math] [math]\displaystyle{ \frac{2912}{3} }[/math] [math]\displaystyle{ \frac{544}{3} }[/math] [math]\displaystyle{ \frac{33511}{15} }[/math] [math]\displaystyle{ \frac{596}{15} }[/math] [math]\displaystyle{ \frac{36844}{45} }[/math] [math]\displaystyle{ \frac{473}{9} }[/math] [math]\displaystyle{ \frac{1351}{15} }[/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]-2 is the signature of K11a184. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -7-6-5-4-3-2-101234χ
7           11
5          2 -2
3         31 2
1        52  -3
-1       73   4
-3      76    -1
-5     76     1
-7    57      2
-9   47       -3
-11  25        3
-13 14         -3
-15 2          2
-171           -1
Integral Khovanov Homology

(db, data source)

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

K11a183.gif

K11a183

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K11a185

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