K11a88

From Knot Atlas
Jump to navigationJump to search

K11a87.gif

K11a87

K11a89.gif

K11a89

K11a88.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a88 at Knotilus!

[edit K11a88 Quick Notes]


[edit K11a88 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X16,7,17,8 X18,9,19,10 X2,11,3,12 X20,14,21,13 X22,16,1,15 X8,17,9,18 X6,19,7,20 X14,22,15,21
Gauss code 1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 10 12 16 18 2 20 22 8 6 14
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11a88 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 K11a88's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^4-5 t^3+12 t^2-20 t+25-20 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ z^8+3 z^6+2 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 101, 0 }
Jones polynomial [math]\displaystyle{ -q^5+3 q^4-6 q^3+11 q^2-14 q+16-16 q^{-1} +14 q^{-2} -10 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +6 z^6+a^4 z^4-9 a^2 z^4-4 z^4 a^{-2} +14 z^4+3 a^4 z^2-13 a^2 z^2-5 z^2 a^{-2} +14 z^2+2 a^4-5 a^2- a^{-2} +5 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a^2 z^{10}+z^{10}+3 a^3 z^9+7 a z^9+4 z^9 a^{-1} +4 a^4 z^8+7 a^2 z^8+6 z^8 a^{-2} +9 z^8+3 a^5 z^7-3 a^3 z^7-16 a z^7-5 z^7 a^{-1} +5 z^7 a^{-3} +a^6 z^6-10 a^4 z^6-25 a^2 z^6-14 z^6 a^{-2} +3 z^6 a^{-4} -31 z^6-9 a^5 z^5-4 a^3 z^5+18 a z^5+2 z^5 a^{-1} -10 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+7 a^4 z^4+33 a^2 z^4+16 z^4 a^{-2} -6 z^4 a^{-4} +45 z^4+7 a^5 z^3+2 a^3 z^3-10 a z^3+3 z^3 a^{-1} +6 z^3 a^{-3} -2 z^3 a^{-5} +2 a^6 z^2-4 a^4 z^2-23 a^2 z^2-7 z^2 a^{-2} +2 z^2 a^{-4} -26 z^2-a^5 z+a z-z a^{-1} -z a^{-3} +2 a^4+5 a^2+ a^{-2} +5 }[/math]
The A2 invariant [math]\displaystyle{ q^{18}+q^{12}-3 q^{10}+2 q^8-q^6-q^4+2 q^2-3+4 q^{-2} - q^{-4} +2 q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} - q^{-14} }[/math]
The G2 invariant [math]\displaystyle{ q^{94}-2 q^{92}+5 q^{90}-9 q^{88}+10 q^{86}-10 q^{84}+2 q^{82}+15 q^{80}-34 q^{78}+55 q^{76}-63 q^{74}+49 q^{72}-15 q^{70}-43 q^{68}+110 q^{66}-157 q^{64}+169 q^{62}-122 q^{60}+26 q^{58}+100 q^{56}-212 q^{54}+273 q^{52}-249 q^{50}+137 q^{48}+23 q^{46}-180 q^{44}+267 q^{42}-248 q^{40}+141 q^{38}+20 q^{36}-159 q^{34}+209 q^{32}-161 q^{30}+8 q^{28}+159 q^{26}-270 q^{24}+270 q^{22}-147 q^{20}-56 q^{18}+264 q^{16}-395 q^{14}+397 q^{12}-269 q^{10}+41 q^8+194 q^6-357 q^4+399 q^2-297+117 q^{-2} +85 q^{-4} -224 q^{-6} +254 q^{-8} -171 q^{-10} +19 q^{-12} +132 q^{-14} -205 q^{-16} +178 q^{-18} -50 q^{-20} -109 q^{-22} +236 q^{-24} -274 q^{-26} +216 q^{-28} -84 q^{-30} -82 q^{-32} +206 q^{-34} -259 q^{-36} +232 q^{-38} -136 q^{-40} +24 q^{-42} +70 q^{-44} -129 q^{-46} +139 q^{-48} -114 q^{-50} +67 q^{-52} -16 q^{-54} -22 q^{-56} +41 q^{-58} -45 q^{-60} +37 q^{-62} -23 q^{-64} +11 q^{-66} + q^{-68} -7 q^{-70} +7 q^{-72} -7 q^{-74} +4 q^{-76} -2 q^{-78} + q^{-80} }[/math]
Further Quantum Invariants
K11a88 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["K11a88"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^4-5 t^3+12 t^2-20 t+25-20 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^8+3 z^6+2 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]= { 101, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^5+3 q^4-6 q^3+11 q^2-14 q+16-16 q^{-1} +14 q^{-2} -10 q^{-3} +6 q^{-4} -3 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} +6 z^6+a^4 z^4-9 a^2 z^4-4 z^4 a^{-2} +14 z^4+3 a^4 z^2-13 a^2 z^2-5 z^2 a^{-2} +14 z^2+2 a^4-5 a^2- a^{-2} +5 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a^2 z^{10}+z^{10}+3 a^3 z^9+7 a z^9+4 z^9 a^{-1} +4 a^4 z^8+7 a^2 z^8+6 z^8 a^{-2} +9 z^8+3 a^5 z^7-3 a^3 z^7-16 a z^7-5 z^7 a^{-1} +5 z^7 a^{-3} +a^6 z^6-10 a^4 z^6-25 a^2 z^6-14 z^6 a^{-2} +3 z^6 a^{-4} -31 z^6-9 a^5 z^5-4 a^3 z^5+18 a z^5+2 z^5 a^{-1} -10 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+7 a^4 z^4+33 a^2 z^4+16 z^4 a^{-2} -6 z^4 a^{-4} +45 z^4+7 a^5 z^3+2 a^3 z^3-10 a z^3+3 z^3 a^{-1} +6 z^3 a^{-3} -2 z^3 a^{-5} +2 a^6 z^2-4 a^4 z^2-23 a^2 z^2-7 z^2 a^{-2} +2 z^2 a^{-4} -26 z^2-a^5 z+a z-z a^{-1} -z a^{-3} +2 a^4+5 a^2+ a^{-2} +5 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a84,}

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["K11a88"];
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+12 t^2-20 t+25-20 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q^5+3 q^4-6 q^3+11 q^2-14 q+16-16 q^{-1} +14 q^{-2} -10 q^{-3} +6 q^{-4} -3 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]= {K11a84,}

Vassiliev invariants

V2 and V3: (-1, 2)
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{ 16 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -\frac{62}{3} }[/math] [math]\displaystyle{ -\frac{34}{3} }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ -\frac{320}{3} }[/math] [math]\displaystyle{ -\frac{320}{3} }[/math] [math]\displaystyle{ 48 }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{248}{3} }[/math] [math]\displaystyle{ \frac{136}{3} }[/math] [math]\displaystyle{ \frac{11729}{30} }[/math] [math]\displaystyle{ -\frac{1778}{15} }[/math] [math]\displaystyle{ \frac{18658}{45} }[/math] [math]\displaystyle{ -\frac{1265}{18} }[/math] [math]\displaystyle{ \frac{2609}{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]0 is the signature of K11a88. 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          2 2
7         41 -3
5        72  5
3       74   -3
1      97    2
-1     88     0
-3    68      -2
-5   48       4
-7  26        -4
-9 14         3
-11 2          -2
-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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

K11a87.gif

K11a87

K11a89.gif

K11a89

Retrieved from "https://katlas.org/index.php?title=K11a88&oldid=110328"