K11a146

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K11a145.gif

K11a145

K11a147.gif

K11a147

K11a146.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a146 at Knotilus!

[edit K11a146 Quick Notes]


[edit K11a146 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X16,5,17,6 X18,7,19,8 X14,10,15,9 X2,11,3,12 X20,14,21,13 X22,15,1,16 X6,17,7,18 X12,20,13,19 X8,21,9,22
Gauss code 1, -6, 2, -1, 3, -9, 4, -11, 5, -2, 6, -10, 7, -5, 8, -3, 9, -4, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 10 16 18 14 2 20 22 6 12 8
A Braid Representative
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A Morse Link Presentation K11a146 ML.gif

Four dimensional invariants

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

[edit Notes for K11a146's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^4+6 t^3-15 t^2+25 t-29+25 t^{-1} -15 t^{-2} +6 t^{-3} - t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ -z^8-2 z^6+z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 123, -2 }
Jones polynomial [math]\displaystyle{ q^3-4 q^2+8 q-13+18 q^{-1} -19 q^{-2} +20 q^{-3} -17 q^{-4} +12 q^{-5} -7 q^{-6} +3 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+8 a^4 z^4-9 a^2 z^4+3 z^4-3 a^6 z^2+10 a^4 z^2-6 a^2 z^2+2 z^2-2 a^6+3 a^4 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 a^4 z^{10}+2 a^2 z^{10}+5 a^5 z^9+11 a^3 z^9+6 a z^9+6 a^6 z^8+8 a^4 z^8+9 a^2 z^8+7 z^8+5 a^7 z^7-4 a^5 z^7-24 a^3 z^7-11 a z^7+4 z^7 a^{-1} +3 a^8 z^6-8 a^6 z^6-26 a^4 z^6-35 a^2 z^6+z^6 a^{-2} -19 z^6+a^9 z^5-7 a^7 z^5-a^5 z^5+18 a^3 z^5+a z^5-10 z^5 a^{-1} -5 a^8 z^4+6 a^6 z^4+31 a^4 z^4+37 a^2 z^4-2 z^4 a^{-2} +15 z^4-2 a^9 z^3+3 a^7 z^3+4 a^5 z^3-3 a^3 z^3+3 a z^3+5 z^3 a^{-1} +2 a^8 z^2-5 a^6 z^2-16 a^4 z^2-14 a^2 z^2-5 z^2+a^9 z-a^7 z-2 a^5 z+2 a^6+3 a^4 }[/math]
The A2 invariant [math]\displaystyle{ -q^{24}-2 q^{18}+3 q^{16}-3 q^{14}+q^{12}+2 q^{10}-q^8+6 q^6-3 q^4+3 q^2-1-2 q^{-2} +2 q^{-4} -2 q^{-6} + q^{-8} }[/math]
The G2 invariant [math]\displaystyle{ q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+9 q^{120}-8 q^{118}+q^{116}+13 q^{114}-29 q^{112}+47 q^{110}-58 q^{108}+51 q^{106}-27 q^{104}-22 q^{102}+82 q^{100}-138 q^{98}+174 q^{96}-174 q^{94}+120 q^{92}-12 q^{90}-135 q^{88}+288 q^{86}-385 q^{84}+378 q^{82}-253 q^{80}+10 q^{78}+262 q^{76}-476 q^{74}+541 q^{72}-397 q^{70}+105 q^{68}+222 q^{66}-443 q^{64}+447 q^{62}-240 q^{60}-93 q^{58}+389 q^{56}-504 q^{54}+369 q^{52}-22 q^{50}-379 q^{48}+680 q^{46}-732 q^{44}+507 q^{42}-92 q^{40}-371 q^{38}+716 q^{36}-819 q^{34}+660 q^{32}-282 q^{30}-157 q^{28}+513 q^{26}-649 q^{24}+524 q^{22}-206 q^{20}-166 q^{18}+428 q^{16}-470 q^{14}+276 q^{12}+73 q^{10}-400 q^8+570 q^6-493 q^4+193 q^2+180-489 q^{-2} +609 q^{-4} -510 q^{-6} +255 q^{-8} +49 q^{-10} -289 q^{-12} +393 q^{-14} -355 q^{-16} +223 q^{-18} -65 q^{-20} -62 q^{-22} +125 q^{-24} -135 q^{-26} +103 q^{-28} -54 q^{-30} +17 q^{-32} +11 q^{-34} -20 q^{-36} +18 q^{-38} -14 q^{-40} +7 q^{-42} -3 q^{-44} + q^{-46} }[/math]
Further Quantum Invariants
K11a146 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["K11a146"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^4+6 t^3-15 t^2+25 t-29+25 t^{-1} -15 t^{-2} +6 t^{-3} - t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^8-2 z^6+z^4+3 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]= { 123, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^3-4 q^2+8 q-13+18 q^{-1} -19 q^{-2} +20 q^{-3} -17 q^{-4} +12 q^{-5} -7 q^{-6} +3 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+8 a^4 z^4-9 a^2 z^4+3 z^4-3 a^6 z^2+10 a^4 z^2-6 a^2 z^2+2 z^2-2 a^6+3 a^4 }[/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}+5 a^5 z^9+11 a^3 z^9+6 a z^9+6 a^6 z^8+8 a^4 z^8+9 a^2 z^8+7 z^8+5 a^7 z^7-4 a^5 z^7-24 a^3 z^7-11 a z^7+4 z^7 a^{-1} +3 a^8 z^6-8 a^6 z^6-26 a^4 z^6-35 a^2 z^6+z^6 a^{-2} -19 z^6+a^9 z^5-7 a^7 z^5-a^5 z^5+18 a^3 z^5+a z^5-10 z^5 a^{-1} -5 a^8 z^4+6 a^6 z^4+31 a^4 z^4+37 a^2 z^4-2 z^4 a^{-2} +15 z^4-2 a^9 z^3+3 a^7 z^3+4 a^5 z^3-3 a^3 z^3+3 a z^3+5 z^3 a^{-1} +2 a^8 z^2-5 a^6 z^2-16 a^4 z^2-14 a^2 z^2-5 z^2+a^9 z-a^7 z-2 a^5 z+2 a^6+3 a^4 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a294,}

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["K11a146"];
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^3-15 t^2+25 t-29+25 t^{-1} -15 t^{-2} +6 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^3-4 q^2+8 q-13+18 q^{-1} -19 q^{-2} +20 q^{-3} -17 q^{-4} +12 q^{-5} -7 q^{-6} +3 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]= {}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11a294,}

Vassiliev invariants

V2 and V3: (3, -5)
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{ 12 }[/math] [math]\displaystyle{ -40 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 174 }[/math] [math]\displaystyle{ 26 }[/math] [math]\displaystyle{ -480 }[/math] [math]\displaystyle{ -\frac{2512}{3} }[/math] [math]\displaystyle{ -\frac{256}{3} }[/math] [math]\displaystyle{ -168 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 800 }[/math] [math]\displaystyle{ 2088 }[/math] [math]\displaystyle{ 312 }[/math] [math]\displaystyle{ \frac{41151}{10} }[/math] [math]\displaystyle{ -\frac{1826}{15} }[/math] [math]\displaystyle{ \frac{8594}{5} }[/math] [math]\displaystyle{ \frac{235}{2} }[/math] [math]\displaystyle{ \frac{2271}{10} }[/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 K11a146. 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         51 4
1        83  -5
-1       105   5
-3      109    -1
-5     109     1
-7    710      3
-9   510       -5
-11  27        5
-13 15         -4
-15 2          2
-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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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.

K11a145.gif

K11a145

K11a147.gif

K11a147

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