K11a109

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K11a108.gif

K11a108

K11a110.gif

K11a110

K11a109.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a109 at Knotilus!

[edit K11a109 Quick Notes]


[edit K11a109 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,5,15,6 X16,8,17,7 X2,10,3,9 X20,11,21,12 X22,13,1,14 X18,16,19,15 X8,18,9,17 X6,19,7,20 X12,21,13,22
Gauss code 1, -5, 2, -1, 3, -10, 4, -9, 5, -2, 6, -11, 7, -3, 8, -4, 9, -8, 10, -6, 11, -7
Dowker-Thistlethwaite code 4 10 14 16 2 20 22 18 8 6 12
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation K11a109 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:K11a109/ThurstonBennequinNumber
Hyperbolic Volume 15.1191
A-Polynomial See Data:K11a109/A-polynomial

[edit Notes for K11a109's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [math]\displaystyle{ [2,4] }[/math]
Rasmussen s-Invariant 0

[edit Notes for K11a109's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^4-5 t^3+14 t^2-24 t+29-24 t^{-1} +14 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ z^8+3 z^6+4 z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 117, 0 }
Jones polynomial [math]\displaystyle{ q^6-4 q^5+7 q^4-12 q^3+17 q^2-18 q+19-16 q^{-1} +12 q^{-2} -7 q^{-3} +3 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^8-a^2 z^6-2 z^6 a^{-2} +6 z^6-4 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +15 z^4-6 a^2 z^2-10 z^2 a^{-2} +2 z^2 a^{-4} +17 z^2-3 a^2-3 a^{-2} +7 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^{10} a^{-2} +z^{10}+4 a z^9+8 z^9 a^{-1} +4 z^9 a^{-3} +6 a^2 z^8+13 z^8 a^{-2} +6 z^8 a^{-4} +13 z^8+5 a^3 z^7+a z^7-9 z^7 a^{-1} -z^7 a^{-3} +4 z^7 a^{-5} +3 a^4 z^6-9 a^2 z^6-42 z^6 a^{-2} -16 z^6 a^{-4} +z^6 a^{-6} -37 z^6+a^5 z^5-7 a^3 z^5-11 a z^5-10 z^5 a^{-1} -18 z^5 a^{-3} -11 z^5 a^{-5} -5 a^4 z^4+9 a^2 z^4+46 z^4 a^{-2} +13 z^4 a^{-4} -2 z^4 a^{-6} +45 z^4-2 a^5 z^3+3 a^3 z^3+13 a z^3+20 z^3 a^{-1} +19 z^3 a^{-3} +7 z^3 a^{-5} +2 a^4 z^2-8 a^2 z^2-22 z^2 a^{-2} -5 z^2 a^{-4} -27 z^2+a^5 z-a^3 z-6 a z-8 z a^{-1} -4 z a^{-3} +3 a^2+3 a^{-2} +7 }[/math]
The A2 invariant [math]\displaystyle{ -q^{14}+q^{12}-3 q^{10}+q^8+q^6-2 q^4+5 q^2-2+4 q^{-2} + q^{-4} +3 q^{-8} -4 q^{-10} - q^{-14} - q^{-16} + q^{-18} }[/math]
The G2 invariant [math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+9 q^{72}-8 q^{70}+q^{68}+13 q^{66}-29 q^{64}+47 q^{62}-59 q^{60}+53 q^{58}-30 q^{56}-20 q^{54}+87 q^{52}-151 q^{50}+190 q^{48}-184 q^{46}+111 q^{44}+17 q^{42}-178 q^{40}+321 q^{38}-383 q^{36}+329 q^{34}-167 q^{32}-75 q^{30}+296 q^{28}-427 q^{26}+416 q^{24}-248 q^{22}-q^{20}+233 q^{18}-340 q^{16}+281 q^{14}-81 q^{12}-170 q^{10}+352 q^8-375 q^6+215 q^4+84 q^2-392+602 q^{-2} -592 q^{-4} +372 q^{-6} -11 q^{-8} -369 q^{-10} +626 q^{-12} -669 q^{-14} +501 q^{-16} -172 q^{-18} -178 q^{-20} +435 q^{-22} -489 q^{-24} +346 q^{-26} -84 q^{-28} -187 q^{-30} +334 q^{-32} -310 q^{-34} +123 q^{-36} +138 q^{-38} -351 q^{-40} +438 q^{-42} -346 q^{-44} +100 q^{-46} +172 q^{-48} -392 q^{-50} +469 q^{-52} -393 q^{-54} +203 q^{-56} +24 q^{-58} -208 q^{-60} +303 q^{-62} -291 q^{-64} +199 q^{-66} -76 q^{-68} -33 q^{-70} +95 q^{-72} -115 q^{-74} +96 q^{-76} -55 q^{-78} +22 q^{-80} +7 q^{-82} -18 q^{-84} +17 q^{-86} -14 q^{-88} +7 q^{-90} -3 q^{-92} + q^{-94} }[/math]
Further Quantum Invariants
K11a109 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["K11a109"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^4-5 t^3+14 t^2-24 t+29-24 t^{-1} +14 t^{-2} -5 t^{-3} + t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^8+3 z^6+4 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]= { 117, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^6-4 q^5+7 q^4-12 q^3+17 q^2-18 q+19-16 q^{-1} +12 q^{-2} -7 q^{-3} +3 q^{-4} - q^{-5} }[/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 z^6-2 z^6 a^{-2} +6 z^6-4 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +15 z^4-6 a^2 z^2-10 z^2 a^{-2} +2 z^2 a^{-4} +17 z^2-3 a^2-3 a^{-2} +7 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^{10} a^{-2} +z^{10}+4 a z^9+8 z^9 a^{-1} +4 z^9 a^{-3} +6 a^2 z^8+13 z^8 a^{-2} +6 z^8 a^{-4} +13 z^8+5 a^3 z^7+a z^7-9 z^7 a^{-1} -z^7 a^{-3} +4 z^7 a^{-5} +3 a^4 z^6-9 a^2 z^6-42 z^6 a^{-2} -16 z^6 a^{-4} +z^6 a^{-6} -37 z^6+a^5 z^5-7 a^3 z^5-11 a z^5-10 z^5 a^{-1} -18 z^5 a^{-3} -11 z^5 a^{-5} -5 a^4 z^4+9 a^2 z^4+46 z^4 a^{-2} +13 z^4 a^{-4} -2 z^4 a^{-6} +45 z^4-2 a^5 z^3+3 a^3 z^3+13 a z^3+20 z^3 a^{-1} +19 z^3 a^{-3} +7 z^3 a^{-5} +2 a^4 z^2-8 a^2 z^2-22 z^2 a^{-2} -5 z^2 a^{-4} -27 z^2+a^5 z-a^3 z-6 a z-8 z a^{-1} -4 z a^{-3} +3 a^2+3 a^{-2} +7 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a44, K11a47,}

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

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["K11a109"];
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+14 t^2-24 t+29-24 t^{-1} +14 t^{-2} -5 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^6-4 q^5+7 q^4-12 q^3+17 q^2-18 q+19-16 q^{-1} +12 q^{-2} -7 q^{-3} +3 q^{-4} - q^{-5} }[/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]= {K11a44, K11a47,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {}

Vassiliev invariants

V2 and V3: (3, 0)
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{ 0 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 78 }[/math] [math]\displaystyle{ 2 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 936 }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ \frac{9151}{10} }[/math] [math]\displaystyle{ \frac{1894}{15} }[/math] [math]\displaystyle{ \frac{794}{5} }[/math] [math]\displaystyle{ \frac{11}{2} }[/math] [math]\displaystyle{ \frac{31}{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]0 is the signature of K11a109. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-10123456χ
13           11
11          3 -3
9         41 3
7        83  -5
5       94   5
3      98    -1
1     109     1
-1    710      3
-3   59       -4
-5  27        5
-7 15         -4
-9 2          2
-111           -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=-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}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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.

K11a108.gif

K11a108

K11a110.gif

K11a110

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