K11a156

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K11a155.gif

K11a155

K11a157.gif

K11a157

K11a156.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a156 at Knotilus!

[edit K11a156 Quick Notes]


[edit K11a156 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X18,5,19,6 X12,8,13,7 X14,10,15,9 X2,11,3,12 X8,14,9,13 X20,15,21,16 X22,17,1,18 X6,19,7,20 X16,21,17,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -7, 5, -2, 6, -4, 7, -5, 8, -11, 9, -3, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 10 18 12 14 2 8 20 22 6 16
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gif
A Morse Link Presentation K11a156 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number [math]\displaystyle{ \{1,2\} }[/math]
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a156/ThurstonBennequinNumber
Hyperbolic Volume 14.0856
A-Polynomial See Data:K11a156/A-polynomial

[edit Notes for K11a156's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a156's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^4+5 t^3-11 t^2+18 t-21+18 t^{-1} -11 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ -z^8-3 z^6-z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 91, -2 }
Jones polynomial [math]\displaystyle{ q^3-3 q^2+6 q-10+13 q^{-1} -14 q^{-2} +15 q^{-3} -12 q^{-4} +9 q^{-5} -5 q^{-6} +2 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+10 a^4 z^4-14 a^2 z^4+4 z^4-4 a^6 z^2+17 a^4 z^2-15 a^2 z^2+5 z^2-4 a^6+9 a^4-6 a^2+2 }[/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+6 a^4 z^8+6 a^2 z^8+4 z^8+3 a^7 z^7-4 a^5 z^7-13 a^3 z^7-3 a z^7+3 z^7 a^{-1} +2 a^8 z^6-10 a^6 z^6-25 a^4 z^6-24 a^2 z^6+z^6 a^{-2} -10 z^6+a^9 z^5-5 a^7 z^5+7 a^3 z^5-8 a z^5-9 z^5 a^{-1} -4 a^8 z^4+16 a^6 z^4+43 a^4 z^4+32 a^2 z^4-3 z^4 a^{-2} +6 z^4-3 a^9 z^3+2 a^7 z^3+10 a^5 z^3+8 a^3 z^3+10 a z^3+7 z^3 a^{-1} +a^8 z^2-13 a^6 z^2-30 a^4 z^2-22 a^2 z^2+2 z^2 a^{-2} -4 z^2+2 a^9 z-a^7 z-6 a^5 z-6 a^3 z-5 a z-2 z a^{-1} +4 a^6+9 a^4+6 a^2+2 }[/math]
The A2 invariant [math]\displaystyle{ -q^{24}-q^{22}-q^{20}-2 q^{18}+3 q^{16}+3 q^{12}+3 q^{10}-q^8+3 q^6-4 q^4+q^2-1- q^{-2} +2 q^{-4} - q^{-6} + q^{-8} }[/math]
The G2 invariant [math]\displaystyle{ q^{128}-q^{126}+3 q^{124}-4 q^{122}+4 q^{120}-3 q^{118}-q^{116}+7 q^{114}-13 q^{112}+19 q^{110}-22 q^{108}+18 q^{106}-9 q^{104}-9 q^{102}+32 q^{100}-54 q^{98}+66 q^{96}-69 q^{94}+44 q^{92}-5 q^{90}-54 q^{88}+112 q^{86}-150 q^{84}+147 q^{82}-100 q^{80}+4 q^{78}+100 q^{76}-183 q^{74}+210 q^{72}-158 q^{70}+50 q^{68}+73 q^{66}-160 q^{64}+172 q^{62}-100 q^{60}-16 q^{58}+133 q^{56}-179 q^{54}+140 q^{52}-14 q^{50}-128 q^{48}+248 q^{46}-272 q^{44}+200 q^{42}-47 q^{40}-126 q^{38}+268 q^{36}-318 q^{34}+269 q^{32}-130 q^{30}-44 q^{28}+185 q^{26}-253 q^{24}+216 q^{22}-105 q^{20}-41 q^{18}+145 q^{16}-177 q^{14}+114 q^{12}+8 q^{10}-135 q^8+203 q^6-185 q^4+81 q^2+55-173 q^{-2} +231 q^{-4} -201 q^{-6} +115 q^{-8} -100 q^{-12} +152 q^{-14} -149 q^{-16} +107 q^{-18} -44 q^{-20} -11 q^{-22} +47 q^{-24} -60 q^{-26} +53 q^{-28} -33 q^{-30} +15 q^{-32} + q^{-34} -10 q^{-36} +10 q^{-38} -9 q^{-40} +5 q^{-42} -2 q^{-44} + q^{-46} }[/math]
Further Quantum Invariants
K11a156 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["K11a156"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^4+5 t^3-11 t^2+18 t-21+18 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+3 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]= { 91, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^3-3 q^2+6 q-10+13 q^{-1} -14 q^{-2} +15 q^{-3} -12 q^{-4} +9 q^{-5} -5 q^{-6} +2 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+10 a^4 z^4-14 a^2 z^4+4 z^4-4 a^6 z^2+17 a^4 z^2-15 a^2 z^2+5 z^2-4 a^6+9 a^4-6 a^2+2 }[/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+6 a^4 z^8+6 a^2 z^8+4 z^8+3 a^7 z^7-4 a^5 z^7-13 a^3 z^7-3 a z^7+3 z^7 a^{-1} +2 a^8 z^6-10 a^6 z^6-25 a^4 z^6-24 a^2 z^6+z^6 a^{-2} -10 z^6+a^9 z^5-5 a^7 z^5+7 a^3 z^5-8 a z^5-9 z^5 a^{-1} -4 a^8 z^4+16 a^6 z^4+43 a^4 z^4+32 a^2 z^4-3 z^4 a^{-2} +6 z^4-3 a^9 z^3+2 a^7 z^3+10 a^5 z^3+8 a^3 z^3+10 a z^3+7 z^3 a^{-1} +a^8 z^2-13 a^6 z^2-30 a^4 z^2-22 a^2 z^2+2 z^2 a^{-2} -4 z^2+2 a^9 z-a^7 z-6 a^5 z-6 a^3 z-5 a z-2 z a^{-1} +4 a^6+9 a^4+6 a^2+2 }[/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["K11a156"];
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+18 t-21+18 t^{-1} -11 t^{-2} +5 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^3-3 q^2+6 q-10+13 q^{-1} -14 q^{-2} +15 q^{-3} -12 q^{-4} +9 q^{-5} -5 q^{-6} +2 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]= {}
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: (3, -7)
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{ 12 }[/math] [math]\displaystyle{ -56 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 238 }[/math] [math]\displaystyle{ 42 }[/math] [math]\displaystyle{ -672 }[/math] [math]\displaystyle{ -\frac{3728}{3} }[/math] [math]\displaystyle{ -\frac{512}{3} }[/math] [math]\displaystyle{ -216 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 1568 }[/math] [math]\displaystyle{ 2856 }[/math] [math]\displaystyle{ 504 }[/math] [math]\displaystyle{ \frac{67711}{10} }[/math] [math]\displaystyle{ -\frac{1186}{15} }[/math] [math]\displaystyle{ \frac{43462}{15} }[/math] [math]\displaystyle{ \frac{673}{6} }[/math] [math]\displaystyle{ \frac{3711}{10} }[/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 K11a156. 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         41 3
1        62  -4
-1       74   3
-3      87    -1
-5     76     1
-7    58      3
-9   47       -3
-11  15        4
-13 14         -3
-15 1          1
-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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{8}\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^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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.

K11a155.gif

K11a155

K11a157.gif

K11a157

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