K11a80

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K11a79.gif

K11a79

K11a81.gif

K11a81

K11a80.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a80 at Knotilus!

[edit K11a80 Quick Notes]


[edit K11a80 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X22,10,1,9 X2,11,3,12 X18,14,19,13 X20,15,21,16 X6,17,7,18 X8,20,9,19 X16,21,17,22
Gauss code 1, -6, 2, -1, 3, -9, 4, -10, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, -8, 11, -5
Dowker-Thistlethwaite code 4 10 12 14 22 2 18 20 6 8 16
A Braid Representative
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A Morse Link Presentation K11a80 ML.gif

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [math]\displaystyle{ 4 }[/math]
Rasmussen s-Invariant 0

[edit Notes for K11a80's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^4-6 t^3+16 t^2-28 t+35-28 t^{-1} +16 t^{-2} -6 t^{-3} + t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ z^8+2 z^6-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 137, 0 }
Jones polynomial [math]\displaystyle{ -q^5+4 q^4-8 q^3+14 q^2-19 q+22-22 q^{-1} +19 q^{-2} -14 q^{-3} +9 q^{-4} -4 q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-7 a^2 z^4-3 z^4 a^{-2} +9 z^4+2 a^4 z^2-7 a^2 z^2-2 z^2 a^{-2} +5 z^2+a^4-a^2+ a^{-2} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 a^2 z^{10}+2 z^{10}+6 a^3 z^9+12 a z^9+6 z^9 a^{-1} +7 a^4 z^8+12 a^2 z^8+8 z^8 a^{-2} +13 z^8+4 a^5 z^7-7 a^3 z^7-19 a z^7-z^7 a^{-1} +7 z^7 a^{-3} +a^6 z^6-16 a^4 z^6-38 a^2 z^6-9 z^6 a^{-2} +4 z^6 a^{-4} -34 z^6-9 a^5 z^5-7 a^3 z^5+a z^5-12 z^5 a^{-1} -10 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+10 a^4 z^4+33 a^2 z^4-6 z^4 a^{-4} +27 z^4+5 a^5 z^3+8 a^3 z^3+11 a z^3+13 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} +a^6 z^2-4 a^4 z^2-11 a^2 z^2+3 z^2 a^{-2} +2 z^2 a^{-4} -5 z^2-a^5 z-2 a^3 z-4 a z-4 z a^{-1} -z a^{-3} +a^4+a^2- a^{-2} }[/math]
The A2 invariant [math]\displaystyle{ q^{18}-q^{16}+2 q^{12}-3 q^{10}+4 q^8-q^6-q^4+2 q^2-5+4 q^{-2} -3 q^{-4} +2 q^{-6} +3 q^{-8} -2 q^{-10} +2 q^{-12} - q^{-14} }[/math]
The G2 invariant [math]\displaystyle{ q^{94}-3 q^{92}+8 q^{90}-16 q^{88}+22 q^{86}-24 q^{84}+12 q^{82}+20 q^{80}-66 q^{78}+122 q^{76}-163 q^{74}+153 q^{72}-74 q^{70}-81 q^{68}+278 q^{66}-435 q^{64}+489 q^{62}-371 q^{60}+80 q^{58}+297 q^{56}-636 q^{54}+791 q^{52}-674 q^{50}+309 q^{48}+174 q^{46}-588 q^{44}+763 q^{42}-624 q^{40}+243 q^{38}+218 q^{36}-548 q^{34}+593 q^{32}-336 q^{30}-117 q^{28}+568 q^{26}-804 q^{24}+717 q^{22}-313 q^{20}-267 q^{18}+803 q^{16}-1097 q^{14}+1025 q^{12}-605 q^{10}-21 q^8+624 q^6-994 q^4+998 q^2-656+120 q^{-2} +386 q^{-4} -666 q^{-6} +614 q^{-8} -284 q^{-10} -153 q^{-12} +499 q^{-14} -586 q^{-16} +384 q^{-18} +11 q^{-20} -425 q^{-22} +690 q^{-24} -694 q^{-26} +460 q^{-28} -82 q^{-30} -296 q^{-32} +542 q^{-34} -600 q^{-36} +486 q^{-38} -256 q^{-40} +10 q^{-42} +184 q^{-44} -289 q^{-46} +294 q^{-48} -230 q^{-50} +132 q^{-52} -31 q^{-54} -46 q^{-56} +86 q^{-58} -95 q^{-60} +77 q^{-62} -46 q^{-64} +20 q^{-66} +3 q^{-68} -14 q^{-70} +15 q^{-72} -13 q^{-74} +7 q^{-76} -3 q^{-78} + q^{-80} }[/math]
Further Quantum Invariants
K11a80 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["K11a80"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^4-6 t^3+16 t^2-28 t+35-28 t^{-1} +16 t^{-2} -6 t^{-3} + t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^8+2 z^6-2 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]= { 137, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^5+4 q^4-8 q^3+14 q^2-19 q+22-22 q^{-1} +19 q^{-2} -14 q^{-3} +9 q^{-4} -4 q^{-5} + q^{-6} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-7 a^2 z^4-3 z^4 a^{-2} +9 z^4+2 a^4 z^2-7 a^2 z^2-2 z^2 a^{-2} +5 z^2+a^4-a^2+ a^{-2} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 2 a^2 z^{10}+2 z^{10}+6 a^3 z^9+12 a z^9+6 z^9 a^{-1} +7 a^4 z^8+12 a^2 z^8+8 z^8 a^{-2} +13 z^8+4 a^5 z^7-7 a^3 z^7-19 a z^7-z^7 a^{-1} +7 z^7 a^{-3} +a^6 z^6-16 a^4 z^6-38 a^2 z^6-9 z^6 a^{-2} +4 z^6 a^{-4} -34 z^6-9 a^5 z^5-7 a^3 z^5+a z^5-12 z^5 a^{-1} -10 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+10 a^4 z^4+33 a^2 z^4-6 z^4 a^{-4} +27 z^4+5 a^5 z^3+8 a^3 z^3+11 a z^3+13 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} +a^6 z^2-4 a^4 z^2-11 a^2 z^2+3 z^2 a^{-2} +2 z^2 a^{-4} -5 z^2-a^5 z-2 a^3 z-4 a z-4 z a^{-1} -z a^{-3} +a^4+a^2- a^{-2} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a270,}

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["K11a80"];
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+16 t^2-28 t+35-28 t^{-1} +16 t^{-2} -6 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q^5+4 q^4-8 q^3+14 q^2-19 q+22-22 q^{-1} +19 q^{-2} -14 q^{-3} +9 q^{-4} -4 q^{-5} + q^{-6} }[/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]= {K11a270,}

Vassiliev invariants

V2 and V3: (-2, 1)
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{ 8 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{116}{3} }[/math] [math]\displaystyle{ \frac{52}{3} }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ -\frac{400}{3} }[/math] [math]\displaystyle{ -\frac{160}{3} }[/math] [math]\displaystyle{ -24 }[/math] [math]\displaystyle{ -\frac{256}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -\frac{928}{3} }[/math] [math]\displaystyle{ -\frac{416}{3} }[/math] [math]\displaystyle{ -\frac{2311}{15} }[/math] [math]\displaystyle{ -\frac{796}{15} }[/math] [math]\displaystyle{ -\frac{2884}{45} }[/math] [math]\displaystyle{ \frac{199}{9} }[/math] [math]\displaystyle{ -\frac{151}{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]0 is the signature of K11a80. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -6-5-4-3-2-1012345χ
11           1-1
9          3 3
7         51 -4
5        93  6
3       105   -5
1      129    3
-1     1111     0
-3    811      -3
-5   611       5
-7  38        -5
-9 16         5
-11 3          -3
-131           1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=5 }[/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.

K11a79.gif

K11a79

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K11a81

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