K11a196

From Knot Atlas
Jump to navigationJump to search

K11a195.gif

K11a195

K11a197.gif

K11a197

K11a196.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a196 at Knotilus!

[edit K11a196 Quick Notes]


[edit K11a196 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X14,6,15,5 X20,7,21,8 X22,10,1,9 X18,11,19,12 X2,13,3,14 X8,16,9,15 X10,17,11,18 X6,19,7,20 X16,21,17,22
Gauss code 1, -7, 2, -1, 3, -10, 4, -8, 5, -9, 6, -2, 7, -3, 8, -11, 9, -6, 10, -4, 11, -5
Dowker-Thistlethwaite code 4 12 14 20 22 18 2 8 10 6 16
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11a196 ML.gif

Three dimensional invariants

Symmetry type Reversible
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:K11a196/ThurstonBennequinNumber
Hyperbolic Volume 17.2855
A-Polynomial See Data:K11a196/A-polynomial

[edit Notes for K11a196's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [math]\displaystyle{ 1 }[/math]
Rasmussen s-Invariant 2

[edit Notes for K11a196's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^4+6 t^3-17 t^2+31 t-37+31 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ -z^8-2 z^6-z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{7,t+1\} }[/math]
Determinant and Signature { 147, -2 }
Jones polynomial [math]\displaystyle{ q^3-4 q^2+9 q-15+21 q^{-1} -23 q^{-2} +24 q^{-3} -21 q^{-4} +15 q^{-5} -9 q^{-6} +4 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -a^2 z^8+2 a^4 z^6-5 a^2 z^6+z^6-a^6 z^4+7 a^4 z^4-10 a^2 z^4+3 z^4-2 a^6 z^2+8 a^4 z^2-8 a^2 z^2+3 z^2-a^6+2 a^4-a^2+1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 a^4 z^{10}+2 a^2 z^{10}+7 a^5 z^9+13 a^3 z^9+6 a z^9+10 a^6 z^8+18 a^4 z^8+15 a^2 z^8+7 z^8+8 a^7 z^7-16 a^3 z^7-4 a z^7+4 z^7 a^{-1} +4 a^8 z^6-15 a^6 z^6-47 a^4 z^6-44 a^2 z^6+z^6 a^{-2} -15 z^6+a^9 z^5-12 a^7 z^5-18 a^5 z^5-9 a^3 z^5-13 a z^5-9 z^5 a^{-1} -5 a^8 z^4+10 a^6 z^4+41 a^4 z^4+38 a^2 z^4-2 z^4 a^{-2} +10 z^4-a^9 z^3+7 a^7 z^3+18 a^5 z^3+16 a^3 z^3+12 a z^3+6 z^3 a^{-1} +a^8 z^2-4 a^6 z^2-15 a^4 z^2-15 a^2 z^2+z^2 a^{-2} -4 z^2-2 a^7 z-4 a^5 z-4 a^3 z-3 a z-z a^{-1} +a^6+2 a^4+a^2+1 }[/math]
The A2 invariant [math]\displaystyle{ -q^{24}+q^{22}-2 q^{18}+4 q^{16}-4 q^{14}+q^{12}+q^{10}-2 q^8+6 q^6-4 q^4+4 q^2-1-2 q^{-2} +3 q^{-4} -2 q^{-6} + q^{-8} }[/math]
The G2 invariant [math]\displaystyle{ q^{128}-3 q^{126}+7 q^{124}-13 q^{122}+16 q^{120}-16 q^{118}+7 q^{116}+16 q^{114}-45 q^{112}+83 q^{110}-114 q^{108}+116 q^{106}-81 q^{104}-10 q^{102}+143 q^{100}-285 q^{98}+395 q^{96}-412 q^{94}+288 q^{92}-19 q^{90}-340 q^{88}+676 q^{86}-847 q^{84}+752 q^{82}-393 q^{80}-154 q^{78}+682 q^{76}-983 q^{74}+934 q^{72}-512 q^{70}-90 q^{68}+619 q^{66}-839 q^{64}+636 q^{62}-113 q^{60}-501 q^{58}+913 q^{56}-917 q^{54}+476 q^{52}+253 q^{50}-968 q^{48}+1385 q^{46}-1320 q^{44}+767 q^{42}+61 q^{40}-878 q^{38}+1394 q^{36}-1426 q^{34}+992 q^{32}-239 q^{30}-517 q^{28}+1002 q^{26}-1043 q^{24}+647 q^{22}-12 q^{20}-573 q^{18}+842 q^{16}-688 q^{14}+185 q^{12}+451 q^{10}-920 q^8+1023 q^6-718 q^4+123 q^2+493-913 q^{-2} +997 q^{-4} -744 q^{-6} +304 q^{-8} +158 q^{-10} -480 q^{-12} +587 q^{-14} -495 q^{-16} +292 q^{-18} -70 q^{-20} -97 q^{-22} +174 q^{-24} -178 q^{-26} +130 q^{-28} -66 q^{-30} +18 q^{-32} +14 q^{-34} -25 q^{-36} +22 q^{-38} -16 q^{-40} +8 q^{-42} -3 q^{-44} + q^{-46} }[/math]
Further Quantum Invariants
K11a196 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["K11a196"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^4+6 t^3-17 t^2+31 t-37+31 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^8-2 z^6-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{ \{7,t+1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 147, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^3-4 q^2+9 q-15+21 q^{-1} -23 q^{-2} +24 q^{-3} -21 q^{-4} +15 q^{-5} -9 q^{-6} +4 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^6-5 a^2 z^6+z^6-a^6 z^4+7 a^4 z^4-10 a^2 z^4+3 z^4-2 a^6 z^2+8 a^4 z^2-8 a^2 z^2+3 z^2-a^6+2 a^4-a^2+1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 2 a^4 z^{10}+2 a^2 z^{10}+7 a^5 z^9+13 a^3 z^9+6 a z^9+10 a^6 z^8+18 a^4 z^8+15 a^2 z^8+7 z^8+8 a^7 z^7-16 a^3 z^7-4 a z^7+4 z^7 a^{-1} +4 a^8 z^6-15 a^6 z^6-47 a^4 z^6-44 a^2 z^6+z^6 a^{-2} -15 z^6+a^9 z^5-12 a^7 z^5-18 a^5 z^5-9 a^3 z^5-13 a z^5-9 z^5 a^{-1} -5 a^8 z^4+10 a^6 z^4+41 a^4 z^4+38 a^2 z^4-2 z^4 a^{-2} +10 z^4-a^9 z^3+7 a^7 z^3+18 a^5 z^3+16 a^3 z^3+12 a z^3+6 z^3 a^{-1} +a^8 z^2-4 a^6 z^2-15 a^4 z^2-15 a^2 z^2+z^2 a^{-2} -4 z^2-2 a^7 z-4 a^5 z-4 a^3 z-3 a z-z a^{-1} +a^6+2 a^4+a^2+1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a216, K11a286,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a216,}

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["K11a196"];
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^3-17 t^2+31 t-37+31 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^3-4 q^2+9 q-15+21 q^{-1} -23 q^{-2} +24 q^{-3} -21 q^{-4} +15 q^{-5} -9 q^{-6} +4 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]= {K11a216, K11a286,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11a216,}

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{158}{3} }[/math] [math]\displaystyle{ \frac{34}{3} }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ -\frac{448}{3} }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ -48 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{632}{3} }[/math] [math]\displaystyle{ \frac{136}{3} }[/math] [math]\displaystyle{ \frac{16591}{30} }[/math] [math]\displaystyle{ -\frac{234}{5} }[/math] [math]\displaystyle{ \frac{10142}{45} }[/math] [math]\displaystyle{ \frac{689}{18} }[/math] [math]\displaystyle{ \frac{271}{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]-2 is the signature of K11a196. 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          3 -3
3         61 5
1        93  -6
-1       126   6
-3      1210    -2
-5     1211     1
-7    912      3
-9   612       -6
-11  39        6
-13 16         -5
-15 3          3
-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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

K11a195.gif

K11a195

K11a197.gif

K11a197

Retrieved from "https://katlas.org/index.php?title=K11a196&oldid=110749"