K11a113

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K11a112.gif

K11a112

K11a114.gif

K11a114

K11a113.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a113 at Knotilus!

[edit K11a113 Quick Notes]


[edit K11a113 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X14,6,15,5 X16,7,17,8 X18,9,19,10 X2,11,3,12 X20,13,21,14 X22,16,1,15 X8,17,9,18 X12,19,13,20 X6,21,7,22
Gauss code 1, -6, 2, -1, 3, -11, 4, -9, 5, -2, 6, -10, 7, -3, 8, -4, 9, -5, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 10 14 16 18 2 20 22 8 12 6
A Braid Representative
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A Morse Link Presentation K11a113 ML.gif

Four dimensional invariants

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

[edit Notes for K11a113's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^4-6 t^3+15 t^2-21 t+23-21 t^{-1} +15 t^{-2} -6 t^{-3} + t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ z^8+2 z^6-z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 109, -4 }
Jones polynomial [math]\displaystyle{ -q+4-7 q^{-1} +12 q^{-2} -15 q^{-3} +17 q^{-4} -17 q^{-5} +15 q^{-6} -11 q^{-7} +6 q^{-8} -3 q^{-9} + q^{-10} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^4 a^8+3 z^2 a^8+a^8-2 z^6 a^6-8 z^4 a^6-9 z^2 a^6-3 a^6+z^8 a^4+5 z^6 a^4+9 z^4 a^4+8 z^2 a^4+2 a^4-z^6 a^2-3 z^4 a^2-z^2 a^2+a^2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-3 z^3 a^{11}+z a^{11}+5 z^6 a^{10}-4 z^4 a^{10}+z^2 a^{10}+7 z^7 a^9-8 z^5 a^9+3 z^3 a^9+z a^9+8 z^8 a^8-14 z^6 a^8+11 z^4 a^8-4 z^2 a^8+a^8+6 z^9 a^7-8 z^7 a^7-2 z^5 a^7+4 z^3 a^7-z a^7+2 z^{10} a^6+9 z^8 a^6-40 z^6 a^6+43 z^4 a^6-19 z^2 a^6+3 a^6+11 z^9 a^5-32 z^7 a^5+24 z^5 a^5-6 z^3 a^5+2 z^{10} a^4+5 z^8 a^4-36 z^6 a^4+43 z^4 a^4-17 z^2 a^4+2 a^4+5 z^9 a^3-16 z^7 a^3+12 z^5 a^3-2 z^3 a^3+z a^3+4 z^8 a^2-15 z^6 a^2+16 z^4 a^2-4 z^2 a^2-a^2+z^7 a-3 z^5 a+2 z^3 a }[/math]
The A2 invariant [math]\displaystyle{ q^{30}-q^{26}+q^{24}-3 q^{22}+2 q^{20}-q^{18}-q^{16}+2 q^{14}-4 q^{12}+4 q^{10}-q^8+2 q^6+2 q^4-q^2+2- q^{-2} }[/math]
The G2 invariant [math]\displaystyle{ q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+5 q^{154}-4 q^{152}-2 q^{150}+10 q^{148}-18 q^{146}+26 q^{144}-29 q^{142}+24 q^{140}-9 q^{138}-12 q^{136}+39 q^{134}-62 q^{132}+75 q^{130}-75 q^{128}+50 q^{126}-10 q^{124}-37 q^{122}+93 q^{120}-130 q^{118}+154 q^{116}-145 q^{114}+86 q^{112}+10 q^{110}-131 q^{108}+232 q^{106}-267 q^{104}+218 q^{102}-83 q^{100}-96 q^{98}+246 q^{96}-295 q^{94}+211 q^{92}-33 q^{90}-171 q^{88}+281 q^{86}-247 q^{84}+79 q^{82}+156 q^{80}-341 q^{78}+390 q^{76}-277 q^{74}+32 q^{72}+239 q^{70}-441 q^{68}+487 q^{66}-364 q^{64}+129 q^{62}+143 q^{60}-350 q^{58}+432 q^{56}-363 q^{54}+171 q^{52}+63 q^{50}-261 q^{48}+328 q^{46}-237 q^{44}+42 q^{42}+182 q^{40}-318 q^{38}+305 q^{36}-145 q^{34}-96 q^{32}+312 q^{30}-404 q^{28}+340 q^{26}-146 q^{24}-86 q^{22}+266 q^{20}-326 q^{18}+274 q^{16}-141 q^{14}-3 q^{12}+105 q^{10}-145 q^8+126 q^6-74 q^4+26 q^2+11-26 q^{-2} +23 q^{-4} -17 q^{-6} +8 q^{-8} -3 q^{-10} + q^{-12} }[/math]
Further Quantum Invariants
K11a113 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["K11a113"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^4-6 t^3+15 t^2-21 t+23-21 t^{-1} +15 t^{-2} -6 t^{-3} + t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^8+2 z^6-z^4+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]= { 109, -4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q+4-7 q^{-1} +12 q^{-2} -15 q^{-3} +17 q^{-4} -17 q^{-5} +15 q^{-6} -11 q^{-7} +6 q^{-8} -3 q^{-9} + q^{-10} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^4 a^8+3 z^2 a^8+a^8-2 z^6 a^6-8 z^4 a^6-9 z^2 a^6-3 a^6+z^8 a^4+5 z^6 a^4+9 z^4 a^4+8 z^2 a^4+2 a^4-z^6 a^2-3 z^4 a^2-z^2 a^2+a^2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-3 z^3 a^{11}+z a^{11}+5 z^6 a^{10}-4 z^4 a^{10}+z^2 a^{10}+7 z^7 a^9-8 z^5 a^9+3 z^3 a^9+z a^9+8 z^8 a^8-14 z^6 a^8+11 z^4 a^8-4 z^2 a^8+a^8+6 z^9 a^7-8 z^7 a^7-2 z^5 a^7+4 z^3 a^7-z a^7+2 z^{10} a^6+9 z^8 a^6-40 z^6 a^6+43 z^4 a^6-19 z^2 a^6+3 a^6+11 z^9 a^5-32 z^7 a^5+24 z^5 a^5-6 z^3 a^5+2 z^{10} a^4+5 z^8 a^4-36 z^6 a^4+43 z^4 a^4-17 z^2 a^4+2 a^4+5 z^9 a^3-16 z^7 a^3+12 z^5 a^3-2 z^3 a^3+z a^3+4 z^8 a^2-15 z^6 a^2+16 z^4 a^2-4 z^2 a^2-a^2+z^7 a-3 z^5 a+2 z^3 a }[/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["K11a113"];
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^4-6 t^3+15 t^2-21 t+23-21 t^{-1} +15 t^{-2} -6 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q+4-7 q^{-1} +12 q^{-2} -15 q^{-3} +17 q^{-4} -17 q^{-5} +15 q^{-6} -11 q^{-7} +6 q^{-8} -3 q^{-9} + q^{-10} }[/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: (1, 0)
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{ 4 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -\frac{34}{3} }[/math] [math]\displaystyle{ \frac{34}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{136}{3} }[/math] [math]\displaystyle{ \frac{136}{3} }[/math] [math]\displaystyle{ \frac{5071}{30} }[/math] [math]\displaystyle{ -\frac{1742}{15} }[/math] [math]\displaystyle{ \frac{25982}{45} }[/math] [math]\displaystyle{ -\frac{175}{18} }[/math] [math]\displaystyle{ \frac{2671}{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]-4 is the signature of K11a113. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -8-7-6-5-4-3-2-10123χ
3           1-1
1          3 3
-1         41 -3
-3        83  5
-5       85   -3
-7      97    2
-9     88     0
-11    79      -2
-13   48       4
-15  27        -5
-17 14         3
-19 2          -2
-211           1
Integral Khovanov Homology

(db, data source)

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

K11a112.gif

K11a112

K11a114.gif

K11a114

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