K11a131

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K11a130.gif

K11a130

K11a132.gif

K11a132

K11a131.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a131 at Knotilus!

[edit K11a131 Quick Notes]


[edit K11a131 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X14,6,15,5 X22,8,1,7 X18,9,19,10 X2,11,3,12 X20,13,21,14 X6,16,7,15 X8,17,9,18 X12,19,13,20 X16,21,17,22
Gauss code 1, -6, 2, -1, 3, -8, 4, -9, 5, -2, 6, -10, 7, -3, 8, -11, 9, -5, 10, -7, 11, -4
Dowker-Thistlethwaite code 4 10 14 22 18 2 20 6 8 12 16
A Braid Representative
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A Morse Link Presentation K11a131 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:K11a131/ThurstonBennequinNumber
Hyperbolic Volume 16.9146
A-Polynomial See Data:K11a131/A-polynomial

[edit Notes for K11a131'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 K11a131's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^4+6 t^3-16 t^2+27 t-31+27 t^{-1} -16 t^{-2} +6 t^{-3} - t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ -z^8-2 z^6+z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 131, -2 }
Jones polynomial [math]\displaystyle{ q^3-4 q^2+9 q-14+19 q^{-1} -21 q^{-2} +21 q^{-3} -18 q^{-4} +13 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^4-10 a^2 z^4+3 z^4-3 a^6 z^2+11 a^4 z^2-10 a^2 z^2+3 z^2-2 a^6+5 a^4-4 a^2+2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 a^4 z^{10}+2 a^2 z^{10}+6 a^5 z^9+12 a^3 z^9+6 a z^9+7 a^6 z^8+12 a^4 z^8+12 a^2 z^8+7 z^8+5 a^7 z^7-7 a^5 z^7-23 a^3 z^7-7 a z^7+4 z^7 a^{-1} +3 a^8 z^6-12 a^6 z^6-40 a^4 z^6-42 a^2 z^6+z^6 a^{-2} -16 z^6+a^9 z^5-6 a^7 z^5+3 a^5 z^5+11 a^3 z^5-8 a z^5-9 z^5 a^{-1} -5 a^8 z^4+13 a^6 z^4+50 a^4 z^4+44 a^2 z^4-2 z^4 a^{-2} +10 z^4-2 a^9 z^3+a^7 z^3+3 a^5 z^3+3 a^3 z^3+8 a z^3+5 z^3 a^{-1} +2 a^8 z^2-8 a^6 z^2-26 a^4 z^2-22 a^2 z^2+z^2 a^{-2} -5 z^2+a^9 z-2 a^5 z-2 a^3 z-2 a z-z a^{-1} +2 a^6+5 a^4+4 a^2+2 }[/math]
The A2 invariant [math]\displaystyle{ -q^{24}-2 q^{18}+4 q^{16}-2 q^{14}+2 q^{12}+2 q^{10}-3 q^8+4 q^6-5 q^4+3 q^2- q^{-2} +3 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}+12 q^{114}-27 q^{112}+44 q^{110}-56 q^{108}+54 q^{106}-36 q^{104}-6 q^{102}+67 q^{100}-134 q^{98}+189 q^{96}-215 q^{94}+175 q^{92}-65 q^{90}-116 q^{88}+328 q^{86}-483 q^{84}+513 q^{82}-369 q^{80}+57 q^{78}+313 q^{76}-618 q^{74}+723 q^{72}-554 q^{70}+171 q^{68}+276 q^{66}-586 q^{64}+623 q^{62}-357 q^{60}-81 q^{58}+486 q^{56}-662 q^{54}+512 q^{52}-88 q^{50}-432 q^{48}+841 q^{46}-941 q^{44}+692 q^{42}-174 q^{40}-429 q^{38}+894 q^{36}-1061 q^{34}+873 q^{32}-404 q^{30}-173 q^{28}+650 q^{26}-849 q^{24}+707 q^{22}-295 q^{20}-203 q^{18}+556 q^{16}-628 q^{14}+376 q^{12}+72 q^{10}-503 q^8+729 q^6-637 q^4+269 q^2+211-607 q^{-2} +766 q^{-4} -646 q^{-6} +328 q^{-8} +51 q^{-10} -350 q^{-12} +482 q^{-14} -435 q^{-16} +280 q^{-18} -84 q^{-20} -71 q^{-22} +150 q^{-24} -163 q^{-26} +123 q^{-28} -66 q^{-30} +20 q^{-32} +12 q^{-34} -24 q^{-36} +22 q^{-38} -16 q^{-40} +8 q^{-42} -3 q^{-44} + q^{-46} }[/math]
Further Quantum Invariants
K11a131 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["K11a131"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^4+6 t^3-16 t^2+27 t-31+27 t^{-1} -16 t^{-2} +6 t^{-3} - t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^8-2 z^6+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]= { 131, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^3-4 q^2+9 q-14+19 q^{-1} -21 q^{-2} +21 q^{-3} -18 q^{-4} +13 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^4-10 a^2 z^4+3 z^4-3 a^6 z^2+11 a^4 z^2-10 a^2 z^2+3 z^2-2 a^6+5 a^4-4 a^2+2 }[/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}+6 a^5 z^9+12 a^3 z^9+6 a z^9+7 a^6 z^8+12 a^4 z^8+12 a^2 z^8+7 z^8+5 a^7 z^7-7 a^5 z^7-23 a^3 z^7-7 a z^7+4 z^7 a^{-1} +3 a^8 z^6-12 a^6 z^6-40 a^4 z^6-42 a^2 z^6+z^6 a^{-2} -16 z^6+a^9 z^5-6 a^7 z^5+3 a^5 z^5+11 a^3 z^5-8 a z^5-9 z^5 a^{-1} -5 a^8 z^4+13 a^6 z^4+50 a^4 z^4+44 a^2 z^4-2 z^4 a^{-2} +10 z^4-2 a^9 z^3+a^7 z^3+3 a^5 z^3+3 a^3 z^3+8 a z^3+5 z^3 a^{-1} +2 a^8 z^2-8 a^6 z^2-26 a^4 z^2-22 a^2 z^2+z^2 a^{-2} -5 z^2+a^9 z-2 a^5 z-2 a^3 z-2 a z-z a^{-1} +2 a^6+5 a^4+4 a^2+2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a252, K11a254,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a218,}

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["K11a131"];
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^3-16 t^2+27 t-31+27 t^{-1} -16 t^{-2} +6 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^3-4 q^2+9 q-14+19 q^{-1} -21 q^{-2} +21 q^{-3} -18 q^{-4} +13 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]= {K11a252, K11a254,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11a218,}

Vassiliev invariants

V2 and V3: (1, -3)
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{ -24 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{206}{3} }[/math] [math]\displaystyle{ \frac{10}{3} }[/math] [math]\displaystyle{ -96 }[/math] [math]\displaystyle{ -176 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ \frac{824}{3} }[/math] [math]\displaystyle{ \frac{40}{3} }[/math] [math]\displaystyle{ \frac{24031}{30} }[/math] [math]\displaystyle{ \frac{4138}{15} }[/math] [math]\displaystyle{ -\frac{4858}{45} }[/math] [math]\displaystyle{ -\frac{127}{18} }[/math] [math]\displaystyle{ -\frac{1889}{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 K11a131. 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        83  -5
-1       116   5
-3      119    -2
-5     1010     0
-7    811      3
-9   510       -5
-11  28        6
-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}^{8}\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^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\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^{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.

K11a130.gif

K11a130

K11a132.gif

K11a132

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