K11n173

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K11n172.gif

K11n172

K11n174.gif

K11n174

K11n173.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n173 at Knotilus!

[edit K11n173 Quick Notes]


[edit K11n173 Further Notes and Views]


Knot presentations

Planar diagram presentation X6271 X3,11,4,10 X16,6,17,5 X12,8,13,7 X20,10,21,9 X11,3,12,2 X18,13,19,14 X4,16,5,15 X22,17,1,18 X8,20,9,19 X14,21,15,22
Gauss code 1, 6, -2, -8, 3, -1, 4, -10, 5, 2, -6, -4, 7, -11, 8, -3, 9, -7, 10, -5, 11, -9
Dowker-Thistlethwaite code 6 -10 16 12 20 -2 18 4 22 8 14
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gif
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A Morse Link Presentation K11n173 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number [math]\displaystyle{ \{2,3\} }[/math]
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n173/ThurstonBennequinNumber
Hyperbolic Volume 13.5804
A-Polynomial See Data:K11n173/A-polynomial

[edit Notes for K11n173's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n173's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^4-4 t^3+8 t^2-7 t+5-7 t^{-1} +8 t^{-2} -4 t^{-3} + t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ z^8+4 z^6+4 z^4+5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 45, 4 }
Jones polynomial [math]\displaystyle{ -3 q^7+5 q^6-6 q^5+8 q^4-7 q^3+7 q^2-5 q+3- q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^8 a^{-4} -z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -4 z^4 a^{-2} +13 z^4 a^{-4} -5 z^4 a^{-6} -4 z^2 a^{-2} +15 z^2 a^{-4} -7 z^2 a^{-6} +z^2 a^{-8} -2 a^{-2} +7 a^{-4} -4 a^{-6} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 z^9 a^{-3} +2 z^9 a^{-5} +3 z^8 a^{-2} +8 z^8 a^{-4} +5 z^8 a^{-6} +z^7 a^{-1} -4 z^7 a^{-3} -2 z^7 a^{-5} +3 z^7 a^{-7} -13 z^6 a^{-2} -34 z^6 a^{-4} -21 z^6 a^{-6} -4 z^5 a^{-1} -9 z^5 a^{-3} -14 z^5 a^{-5} -9 z^5 a^{-7} +16 z^4 a^{-2} +43 z^4 a^{-4} +30 z^4 a^{-6} +3 z^4 a^{-8} +5 z^3 a^{-1} +17 z^3 a^{-3} +20 z^3 a^{-5} +8 z^3 a^{-7} -7 z^2 a^{-2} -25 z^2 a^{-4} -20 z^2 a^{-6} -2 z^2 a^{-8} -2 z a^{-1} -6 z a^{-3} -8 z a^{-5} -z a^{-7} +3 z a^{-9} +2 a^{-2} +7 a^{-4} +4 a^{-6} }[/math]
The A2 invariant [math]\displaystyle{ -q^2+1- q^{-2} + q^{-8} +4 q^{-10} +4 q^{-14} - q^{-16} - q^{-20} -2 q^{-22} -2 q^{-26} + q^{-28} }[/math]
The G2 invariant Data:K11n173/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n173 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["K11n173"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^4-4 t^3+8 t^2-7 t+5-7 t^{-1} +8 t^{-2} -4 t^{-3} + t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^8+4 z^6+4 z^4+5 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]= { 45, 4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -3 q^7+5 q^6-6 q^5+8 q^4-7 q^3+7 q^2-5 q+3- q^{-1} }[/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} -z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -4 z^4 a^{-2} +13 z^4 a^{-4} -5 z^4 a^{-6} -4 z^2 a^{-2} +15 z^2 a^{-4} -7 z^2 a^{-6} +z^2 a^{-8} -2 a^{-2} +7 a^{-4} -4 a^{-6} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 2 z^9 a^{-3} +2 z^9 a^{-5} +3 z^8 a^{-2} +8 z^8 a^{-4} +5 z^8 a^{-6} +z^7 a^{-1} -4 z^7 a^{-3} -2 z^7 a^{-5} +3 z^7 a^{-7} -13 z^6 a^{-2} -34 z^6 a^{-4} -21 z^6 a^{-6} -4 z^5 a^{-1} -9 z^5 a^{-3} -14 z^5 a^{-5} -9 z^5 a^{-7} +16 z^4 a^{-2} +43 z^4 a^{-4} +30 z^4 a^{-6} +3 z^4 a^{-8} +5 z^3 a^{-1} +17 z^3 a^{-3} +20 z^3 a^{-5} +8 z^3 a^{-7} -7 z^2 a^{-2} -25 z^2 a^{-4} -20 z^2 a^{-6} -2 z^2 a^{-8} -2 z a^{-1} -6 z a^{-3} -8 z a^{-5} -z a^{-7} +3 z a^{-9} +2 a^{-2} +7 a^{-4} +4 a^{-6} }[/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["K11n173"];
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^4-4 t^3+8 t^2-7 t+5-7 t^{-1} +8 t^{-2} -4 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -3 q^7+5 q^6-6 q^5+8 q^4-7 q^3+7 q^2-5 q+3- q^{-1} }[/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: (5, 9)
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{ 20 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 200 }[/math] [math]\displaystyle{ \frac{1222}{3} }[/math] [math]\displaystyle{ \frac{194}{3} }[/math] [math]\displaystyle{ 1440 }[/math] [math]\displaystyle{ 2416 }[/math] [math]\displaystyle{ 416 }[/math] [math]\displaystyle{ 360 }[/math] [math]\displaystyle{ \frac{4000}{3} }[/math] [math]\displaystyle{ 2592 }[/math] [math]\displaystyle{ \frac{24440}{3} }[/math] [math]\displaystyle{ \frac{3880}{3} }[/math] [math]\displaystyle{ \frac{87967}{6} }[/math] [math]\displaystyle{ \frac{170}{3} }[/math] [math]\displaystyle{ \frac{55766}{9} }[/math] [math]\displaystyle{ \frac{1381}{18} }[/math] [math]\displaystyle{ \frac{5119}{6} }[/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 K11n173. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -3-2-1012345χ
15        3-3
13       2 2
11      43 -1
9     42  2
7    34   1
5   44    0
3  24     2
1 13      -2
-1 2       2
-31        -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=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

K11n172.gif

K11n172

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K11n174

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