K11a164

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K11a163.gif

K11a163

K11a165.gif

K11a165

K11a164.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a164 at Knotilus!

[edit K11a164 Quick Notes]


[edit K11a164 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X18,6,19,5 X14,8,15,7 X16,9,17,10 X2,11,3,12 X20,13,21,14 X22,16,1,15 X12,17,13,18 X6,20,7,19 X8,21,9,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -11, 5, -2, 6, -9, 7, -4, 8, -5, 9, -3, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 10 18 14 16 2 20 22 12 6 8
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
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A Morse Link Presentation K11a164 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:K11a164/ThurstonBennequinNumber
Hyperbolic Volume 18.3585
A-Polynomial See Data:K11a164/A-polynomial

[edit Notes for K11a164's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a164's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^4-7 t^3+20 t^2-35 t+43-35 t^{-1} +20 t^{-2} -7 t^{-3} + t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ z^8+z^6-2 z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 169, 0 }
Jones polynomial [math]\displaystyle{ -q^5+5 q^4-11 q^3+18 q^2-24 q+28-27 q^{-1} +23 q^{-2} -17 q^{-3} +10 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} +4 z^6+a^4 z^4-6 a^2 z^4-2 z^4 a^{-2} +5 z^4+2 a^4 z^2-5 a^2 z^2+z^2+a^4-a^2+ a^{-2} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 3 a^2 z^{10}+3 z^{10}+8 a^3 z^9+18 a z^9+10 z^9 a^{-1} +8 a^4 z^8+16 a^2 z^8+14 z^8 a^{-2} +22 z^8+4 a^5 z^7-10 a^3 z^7-27 a z^7-2 z^7 a^{-1} +11 z^7 a^{-3} +a^6 z^6-17 a^4 z^6-49 a^2 z^6-19 z^6 a^{-2} +5 z^6 a^{-4} -55 z^6-8 a^5 z^5-3 a^3 z^5+a z^5-19 z^5 a^{-1} -14 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+13 a^4 z^4+41 a^2 z^4+6 z^4 a^{-2} -4 z^4 a^{-4} +36 z^4+5 a^5 z^3+5 a^3 z^3+8 a z^3+12 z^3 a^{-1} +4 z^3 a^{-3} +a^6 z^2-6 a^4 z^2-13 a^2 z^2+z^2 a^{-2} -5 z^2-a^5 z-a^3 z-a z-z a^{-1} +a^4+a^2- a^{-2} }[/math]
The A2 invariant [math]\displaystyle{ q^{18}-q^{16}+3 q^{12}-4 q^{10}+4 q^8-2 q^6-2 q^4+4 q^2-5+6 q^{-2} -4 q^{-4} + q^{-6} +3 q^{-8} -3 q^{-10} +3 q^{-12} - q^{-14} }[/math]
The G2 invariant [math]\displaystyle{ q^{94}-3 q^{92}+8 q^{90}-16 q^{88}+23 q^{86}-28 q^{84}+20 q^{82}+10 q^{80}-63 q^{78}+139 q^{76}-210 q^{74}+232 q^{72}-166 q^{70}-19 q^{68}+310 q^{66}-612 q^{64}+808 q^{62}-756 q^{60}+371 q^{58}+267 q^{56}-968 q^{54}+1451 q^{52}-1462 q^{50}+947 q^{48}-31 q^{46}-947 q^{44}+1580 q^{42}-1599 q^{40}+975 q^{38}+18 q^{36}-939 q^{34}+1370 q^{32}-1120 q^{30}+316 q^{28}+702 q^{26}-1454 q^{24}+1593 q^{22}-1032 q^{20}-64 q^{18}+1251 q^{16}-2080 q^{14}+2211 q^{12}-1558 q^{10}+365 q^8+973 q^6-1971 q^4+2260 q^2-1767+683 q^{-2} +538 q^{-4} -1418 q^{-6} +1613 q^{-8} -1076 q^{-10} +123 q^{-12} +820 q^{-14} -1311 q^{-16} +1130 q^{-18} -397 q^{-20} -574 q^{-22} +1335 q^{-24} -1563 q^{-26} +1201 q^{-28} -394 q^{-30} -496 q^{-32} +1148 q^{-34} -1369 q^{-36} +1146 q^{-38} -627 q^{-40} +27 q^{-42} +446 q^{-44} -687 q^{-46} +681 q^{-48} -496 q^{-50} +252 q^{-52} -18 q^{-54} -141 q^{-56} +203 q^{-58} -199 q^{-60} +141 q^{-62} -73 q^{-64} +21 q^{-66} +16 q^{-68} -28 q^{-70} +27 q^{-72} -20 q^{-74} +10 q^{-76} -4 q^{-78} + q^{-80} }[/math]
Further Quantum Invariants
K11a164 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["K11a164"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^4-7 t^3+20 t^2-35 t+43-35 t^{-1} +20 t^{-2} -7 t^{-3} + t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^8+z^6-2 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]= { 169, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^5+5 q^4-11 q^3+18 q^2-24 q+28-27 q^{-1} +23 q^{-2} -17 q^{-3} +10 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} +4 z^6+a^4 z^4-6 a^2 z^4-2 z^4 a^{-2} +5 z^4+2 a^4 z^2-5 a^2 z^2+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{ 3 a^2 z^{10}+3 z^{10}+8 a^3 z^9+18 a z^9+10 z^9 a^{-1} +8 a^4 z^8+16 a^2 z^8+14 z^8 a^{-2} +22 z^8+4 a^5 z^7-10 a^3 z^7-27 a z^7-2 z^7 a^{-1} +11 z^7 a^{-3} +a^6 z^6-17 a^4 z^6-49 a^2 z^6-19 z^6 a^{-2} +5 z^6 a^{-4} -55 z^6-8 a^5 z^5-3 a^3 z^5+a z^5-19 z^5 a^{-1} -14 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+13 a^4 z^4+41 a^2 z^4+6 z^4 a^{-2} -4 z^4 a^{-4} +36 z^4+5 a^5 z^3+5 a^3 z^3+8 a z^3+12 z^3 a^{-1} +4 z^3 a^{-3} +a^6 z^2-6 a^4 z^2-13 a^2 z^2+z^2 a^{-2} -5 z^2-a^5 z-a^3 z-a z-z a^{-1} +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]): {}

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["K11a164"];
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^4-7 t^3+20 t^2-35 t+43-35 t^{-1} +20 t^{-2} -7 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q^5+5 q^4-11 q^3+18 q^2-24 q+28-27 q^{-1} +23 q^{-2} -17 q^{-3} +10 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]= {}

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{164}{3} }[/math] [math]\displaystyle{ \frac{100}{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{1312}{3} }[/math] [math]\displaystyle{ -\frac{800}{3} }[/math] [math]\displaystyle{ -\frac{4111}{15} }[/math] [math]\displaystyle{ \frac{908}{5} }[/math] [math]\displaystyle{ -\frac{21124}{45} }[/math] [math]\displaystyle{ \frac{655}{9} }[/math] [math]\displaystyle{ -\frac{1951}{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 K11a164. 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          4 4
7         71 -6
5        114  7
3       137   -6
1      1511    4
-1     1314     1
-3    1014      -4
-5   713       6
-7  310        -7
-9 17         6
-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}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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.

K11a163.gif

K11a163

K11a165.gif

K11a165

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