K11a68

From Knot Atlas
Revision as of 16:18, 1 September 2005 by ScottTestRobot (talk | contribs)
Jump to navigationJump to search

K11a67.gif

K11a67

K11a69.gif

K11a69

K11a68.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a68 at Knotilus!

[edit K11a68 Quick Notes]


[edit K11a68 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X8394 X16,5,17,6 X14,8,15,7 X2,9,3,10 X18,12,19,11 X20,14,21,13 X22,15,1,16 X10,18,11,17 X12,20,13,19 X6,21,7,22
Gauss code 1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -10, 7, -4, 8, -3, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 8 16 14 2 18 20 22 10 12 6
A Braid Representative {{{braid_table}}}
A Morse Link Presentation K11a68 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:K11a68/ThurstonBennequinNumber
Hyperbolic Volume 14.7003
A-Polynomial See Data:K11a68/A-polynomial

[edit Notes for K11a68'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 K11a68's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^4+6 t^3-14 t^2+20 t-21+20 t^{-1} -14 t^{-2} +6 t^{-3} - t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ -z^8-2 z^6+2 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 103, 2 }
Jones polynomial [math]\displaystyle{ q^7-4 q^6+7 q^5-11 q^4+15 q^3-16 q^2+16 q-13+10 q^{-1} -6 q^{-2} +3 q^{-3} - q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^8 a^{-2} -5 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-8 z^4 a^{-2} +3 z^4 a^{-4} +8 z^4-3 a^2 z^2-4 z^2 a^{-2} +z^2 a^{-4} +8 z^2-a^2+ a^{-2} - a^{-4} +2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10}+4 a z^9+10 z^9 a^{-1} +6 z^9 a^{-3} +3 a^2 z^8+6 z^8 a^{-2} +8 z^8 a^{-4} +z^8+a^3 z^7-14 a z^7-33 z^7 a^{-1} -10 z^7 a^{-3} +8 z^7 a^{-5} -12 a^2 z^6-30 z^6 a^{-2} -11 z^6 a^{-4} +7 z^6 a^{-6} -24 z^6-4 a^3 z^5+12 a z^5+32 z^5 a^{-1} +5 z^5 a^{-3} -7 z^5 a^{-5} +4 z^5 a^{-7} +14 a^2 z^4+30 z^4 a^{-2} +2 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +34 z^4+4 a^3 z^3-2 a z^3-10 z^3 a^{-1} -4 z^3 a^{-3} -3 z^3 a^{-5} -3 z^3 a^{-7} -7 a^2 z^2-7 z^2 a^{-2} +2 z^2 a^{-4} +z^2 a^{-6} -15 z^2-a^3 z-a z+z a^{-1} +3 z a^{-3} +2 z a^{-5} +a^2- a^{-2} - a^{-4} +2 }[/math]
The A2 invariant [math]\displaystyle{ -q^{12}+q^8-q^6+2 q^4-2 q^2+1+2 q^{-2} - q^{-4} +5 q^{-6} -2 q^{-8} +2 q^{-10} - q^{-12} -2 q^{-14} + q^{-16} -2 q^{-18} + q^{-20} }[/math]
The G2 invariant [math]\displaystyle{ q^{60}-2 q^{58}+5 q^{56}-9 q^{54}+11 q^{52}-13 q^{50}+6 q^{48}+11 q^{46}-36 q^{44}+65 q^{42}-85 q^{40}+75 q^{38}-31 q^{36}-52 q^{34}+146 q^{32}-211 q^{30}+217 q^{28}-139 q^{26}-7 q^{24}+171 q^{22}-287 q^{20}+306 q^{18}-209 q^{16}+35 q^{14}+141 q^{12}-251 q^{10}+246 q^8-135 q^6-24 q^4+167 q^2-218+159 q^{-2} -20 q^{-4} -145 q^{-6} +262 q^{-8} -285 q^{-10} +203 q^{-12} -34 q^{-14} -159 q^{-16} +317 q^{-18} -370 q^{-20} +309 q^{-22} -137 q^{-24} -74 q^{-26} +250 q^{-28} -329 q^{-30} +286 q^{-32} -138 q^{-34} -38 q^{-36} +174 q^{-38} -207 q^{-40} +137 q^{-42} -7 q^{-44} -119 q^{-46} +176 q^{-48} -148 q^{-50} +46 q^{-52} +72 q^{-54} -166 q^{-56} +201 q^{-58} -166 q^{-60} +85 q^{-62} +10 q^{-64} -97 q^{-66} +141 q^{-68} -155 q^{-70} +134 q^{-72} -86 q^{-74} +29 q^{-76} +33 q^{-78} -81 q^{-80} +104 q^{-82} -100 q^{-84} +75 q^{-86} -37 q^{-88} -3 q^{-90} +32 q^{-92} -49 q^{-94} +47 q^{-96} -32 q^{-98} +18 q^{-100} -2 q^{-102} -6 q^{-104} +9 q^{-106} -10 q^{-108} +6 q^{-110} -3 q^{-112} + q^{-114} }[/math]
Further Quantum Invariants
K11a68 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["K11a68"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^4+6 t^3-14 t^2+20 t-21+20 t^{-1} -14 t^{-2} +6 t^{-3} - t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^8-2 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]= { 103, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^7-4 q^6+7 q^5-11 q^4+15 q^3-16 q^2+16 q-13+10 q^{-1} -6 q^{-2} +3 q^{-3} - q^{-4} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -z^8 a^{-2} -5 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-8 z^4 a^{-2} +3 z^4 a^{-4} +8 z^4-3 a^2 z^2-4 z^2 a^{-2} +z^2 a^{-4} +8 z^2-a^2+ a^{-2} - a^{-4} +2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10}+4 a z^9+10 z^9 a^{-1} +6 z^9 a^{-3} +3 a^2 z^8+6 z^8 a^{-2} +8 z^8 a^{-4} +z^8+a^3 z^7-14 a z^7-33 z^7 a^{-1} -10 z^7 a^{-3} +8 z^7 a^{-5} -12 a^2 z^6-30 z^6 a^{-2} -11 z^6 a^{-4} +7 z^6 a^{-6} -24 z^6-4 a^3 z^5+12 a z^5+32 z^5 a^{-1} +5 z^5 a^{-3} -7 z^5 a^{-5} +4 z^5 a^{-7} +14 a^2 z^4+30 z^4 a^{-2} +2 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +34 z^4+4 a^3 z^3-2 a z^3-10 z^3 a^{-1} -4 z^3 a^{-3} -3 z^3 a^{-5} -3 z^3 a^{-7} -7 a^2 z^2-7 z^2 a^{-2} +2 z^2 a^{-4} +z^2 a^{-6} -15 z^2-a^3 z-a z+z a^{-1} +3 z a^{-3} +2 z a^{-5} +a^2- a^{-2} - a^{-4} +2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a111,}

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["K11a68"];
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^3-14 t^2+20 t-21+20 t^{-1} -14 t^{-2} +6 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^7-4 q^6+7 q^5-11 q^4+15 q^3-16 q^2+16 q-13+10 q^{-1} -6 q^{-2} +3 q^{-3} - q^{-4} }[/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]= {K11a111,}

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{124}{3} }[/math] [math]\displaystyle{ -\frac{4}{3} }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ \frac{272}{3} }[/math] [math]\displaystyle{ \frac{224}{3} }[/math] [math]\displaystyle{ -56 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{992}{3} }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ \frac{6271}{15} }[/math] [math]\displaystyle{ \frac{3916}{15} }[/math] [math]\displaystyle{ -\frac{8156}{45} }[/math] [math]\displaystyle{ \frac{401}{9} }[/math] [math]\displaystyle{ -\frac{929}{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]2 is the signature of K11a68. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-10123456χ
15           11
13          3 -3
11         41 3
9        73  -4
7       84   4
5      87    -1
3     88     0
1    69      3
-1   47       -3
-3  26        4
-5 14         -3
-7 2          2
-91           -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=3 }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=6 }[/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.

K11a67.gif

K11a67

K11a69.gif

K11a69

Retrieved from "https://katlas.org/index.php?title=K11a68&oldid=44313"