K11a74

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K11a73.gif

K11a73

K11a75.gif

K11a75

K11a74.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a74 at Knotilus!

[edit K11a74 Quick Notes]


[edit K11a74 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X14,8,15,7 X18,9,19,10 X2,11,3,12 X6,14,7,13 X20,16,21,15 X22,18,1,17 X8,19,9,20 X16,22,17,21
Gauss code 1, -6, 2, -1, 3, -7, 4, -10, 5, -2, 6, -3, 7, -4, 8, -11, 9, -5, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 10 12 14 18 2 6 20 22 8 16
A Braid Representative
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A Morse Link Presentation K11a74 ML.gif

Four dimensional invariants

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

[edit Notes for K11a74's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^4-5 t^3+10 t^2-13 t+15-13 t^{-1} +10 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ z^8+3 z^6-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 73, 4 }
Jones polynomial [math]\displaystyle{ -q^9+3 q^8-5 q^7+8 q^6-10 q^5+11 q^4-11 q^3+9 q^2-7 q+5-2 q^{-1} + q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^8 a^{-4} -2 z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -10 z^4 a^{-2} +13 z^4 a^{-4} -4 z^4 a^{-6} +z^4-15 z^2 a^{-2} +13 z^2 a^{-4} -4 z^2 a^{-6} +4 z^2-7 a^{-2} +5 a^{-4} - a^{-6} +4 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +6 z^9 a^{-3} +4 z^9 a^{-5} +z^8 a^{-2} +7 z^8 a^{-4} +7 z^8 a^{-6} +z^8-10 z^7 a^{-1} -23 z^7 a^{-3} -5 z^7 a^{-5} +8 z^7 a^{-7} -23 z^6 a^{-2} -40 z^6 a^{-4} -16 z^6 a^{-6} +7 z^6 a^{-8} -6 z^6+15 z^5 a^{-1} +18 z^5 a^{-3} -18 z^5 a^{-5} -16 z^5 a^{-7} +5 z^5 a^{-9} +47 z^4 a^{-2} +51 z^4 a^{-4} +5 z^4 a^{-6} -9 z^4 a^{-8} +3 z^4 a^{-10} +13 z^4-6 z^3 a^{-1} +8 z^3 a^{-3} +26 z^3 a^{-5} +8 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} -32 z^2 a^{-2} -23 z^2 a^{-4} +2 z^2 a^{-8} -z^2 a^{-10} -12 z^2-z a^{-1} -7 z a^{-3} -9 z a^{-5} -3 z a^{-7} +7 a^{-2} +5 a^{-4} + a^{-6} +4 }[/math]
The A2 invariant [math]\displaystyle{ q^6+q^4+q^2+2- q^{-2} -2 q^{-6} -2 q^{-8} + q^{-10} -2 q^{-12} +3 q^{-14} + q^{-18} + q^{-20} - q^{-22} + q^{-24} - q^{-26} }[/math]
The G2 invariant [math]\displaystyle{ q^{26}-q^{24}+5 q^{22}-7 q^{20}+10 q^{18}-9 q^{16}+2 q^{14}+15 q^{12}-31 q^{10}+46 q^8-42 q^6+23 q^4+15 q^2-53+84 q^{-2} -81 q^{-4} +53 q^{-6} -2 q^{-8} -52 q^{-10} +82 q^{-12} -81 q^{-14} +50 q^{-16} -4 q^{-18} -40 q^{-20} +57 q^{-22} -50 q^{-24} +11 q^{-26} +25 q^{-28} -58 q^{-30} +60 q^{-32} -40 q^{-34} -7 q^{-36} +51 q^{-38} -87 q^{-40} +94 q^{-42} -70 q^{-44} +20 q^{-46} +40 q^{-48} -85 q^{-50} +103 q^{-52} -82 q^{-54} +41 q^{-56} +18 q^{-58} -54 q^{-60} +67 q^{-62} -48 q^{-64} +15 q^{-66} +24 q^{-68} -40 q^{-70} +35 q^{-72} -9 q^{-74} -21 q^{-76} +43 q^{-78} -46 q^{-80} +32 q^{-82} -8 q^{-84} -19 q^{-86} +35 q^{-88} -43 q^{-90} +38 q^{-92} -25 q^{-94} +9 q^{-96} +5 q^{-98} -20 q^{-100} +27 q^{-102} -31 q^{-104} +28 q^{-106} -18 q^{-108} +7 q^{-110} +7 q^{-112} -19 q^{-114} +23 q^{-116} -23 q^{-118} +18 q^{-120} -8 q^{-122} +7 q^{-126} -12 q^{-128} +13 q^{-130} -10 q^{-132} +7 q^{-134} -2 q^{-136} - q^{-138} +2 q^{-140} -4 q^{-142} +3 q^{-144} -2 q^{-146} + q^{-148} }[/math]
Further Quantum Invariants
K11a74 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["K11a74"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^4-5 t^3+10 t^2-13 t+15-13 t^{-1} +10 t^{-2} -5 t^{-3} + t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^8+3 z^6-2 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]= { 73, 4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^9+3 q^8-5 q^7+8 q^6-10 q^5+11 q^4-11 q^3+9 q^2-7 q+5-2 q^{-1} + q^{-2} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^8 a^{-4} -2 z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -10 z^4 a^{-2} +13 z^4 a^{-4} -4 z^4 a^{-6} +z^4-15 z^2 a^{-2} +13 z^2 a^{-4} -4 z^2 a^{-6} +4 z^2-7 a^{-2} +5 a^{-4} - a^{-6} +4 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +6 z^9 a^{-3} +4 z^9 a^{-5} +z^8 a^{-2} +7 z^8 a^{-4} +7 z^8 a^{-6} +z^8-10 z^7 a^{-1} -23 z^7 a^{-3} -5 z^7 a^{-5} +8 z^7 a^{-7} -23 z^6 a^{-2} -40 z^6 a^{-4} -16 z^6 a^{-6} +7 z^6 a^{-8} -6 z^6+15 z^5 a^{-1} +18 z^5 a^{-3} -18 z^5 a^{-5} -16 z^5 a^{-7} +5 z^5 a^{-9} +47 z^4 a^{-2} +51 z^4 a^{-4} +5 z^4 a^{-6} -9 z^4 a^{-8} +3 z^4 a^{-10} +13 z^4-6 z^3 a^{-1} +8 z^3 a^{-3} +26 z^3 a^{-5} +8 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} -32 z^2 a^{-2} -23 z^2 a^{-4} +2 z^2 a^{-8} -z^2 a^{-10} -12 z^2-z a^{-1} -7 z a^{-3} -9 z a^{-5} -3 z a^{-7} +7 a^{-2} +5 a^{-4} + a^{-6} +4 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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["K11a74"];
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+10 t^2-13 t+15-13 t^{-1} +10 t^{-2} -5 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q^9+3 q^8-5 q^7+8 q^6-10 q^5+11 q^4-11 q^3+9 q^2-7 q+5-2 q^{-1} + q^{-2} }[/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]= {}

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{164}{3} }[/math] [math]\displaystyle{ \frac{52}{3} }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ \frac{208}{3} }[/math] [math]\displaystyle{ -\frac{128}{3} }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ -\frac{256}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -\frac{1312}{3} }[/math] [math]\displaystyle{ -\frac{416}{3} }[/math] [math]\displaystyle{ -\frac{8911}{15} }[/math] [math]\displaystyle{ -\frac{5276}{15} }[/math] [math]\displaystyle{ -\frac{5044}{45} }[/math] [math]\displaystyle{ \frac{127}{9} }[/math] [math]\displaystyle{ \frac{209}{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]4 is the signature of K11a74. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-101234567χ
19           1-1
17          2 2
15         31 -2
13        52  3
11       53   -2
9      65    1
7     55     0
5    46      -2
3   46       2
1  13        -2
-1 14         3
-3 1          -1
-51           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=5 }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=7 }[/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.

K11a73.gif

K11a73

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K11a75

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