K11n178

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K11n177.gif

K11n177

K11n179.gif

K11n179

K11n178.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n178 at Knotilus!

[edit K11n178 Quick Notes]


[edit K11n178 Further Notes and Views]


Knot presentations

Planar diagram presentation X6271 X3,11,4,10 X16,6,17,5 X18,7,19,8 X20,10,21,9 X11,5,12,4 X8,14,9,13 X2,16,3,15 X22,17,1,18 X14,19,15,20 X12,22,13,21
Gauss code 1, -8, -2, 6, 3, -1, 4, -7, 5, 2, -6, -11, 7, -10, 8, -3, 9, -4, 10, -5, 11, -9
Dowker-Thistlethwaite code 6 -10 16 18 20 -4 8 2 22 14 12
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11n178 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number [math]\displaystyle{ \{1,2\} }[/math]
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n178/ThurstonBennequinNumber
Hyperbolic Volume 16.9983
A-Polynomial See Data:K11n178/A-polynomial

[edit Notes for K11n178's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant -2

[edit Notes for K11n178's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 2 t^3-9 t^2+22 t-29+22 t^{-1} -9 t^{-2} +2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ 2 z^6+3 z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 95, 2 }
Jones polynomial [math]\displaystyle{ q^9-5 q^8+9 q^7-13 q^6+16 q^5-16 q^4+15 q^3-11 q^2+7 q-2 }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ 2 z^6 a^{-4} -2 z^4 a^{-2} +8 z^4 a^{-4} -3 z^4 a^{-6} -3 z^2 a^{-2} +12 z^2 a^{-4} -6 z^2 a^{-6} +z^2 a^{-8} - a^{-2} +5 a^{-4} -3 a^{-6} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 4 z^9 a^{-5} +4 z^9 a^{-7} +9 z^8 a^{-4} +17 z^8 a^{-6} +8 z^8 a^{-8} +6 z^7 a^{-3} +3 z^7 a^{-5} +2 z^7 a^{-7} +5 z^7 a^{-9} +z^6 a^{-2} -22 z^6 a^{-4} -44 z^6 a^{-6} -20 z^6 a^{-8} +z^6 a^{-10} -7 z^5 a^{-3} -20 z^5 a^{-5} -24 z^5 a^{-7} -11 z^5 a^{-9} +8 z^4 a^{-2} +31 z^4 a^{-4} +35 z^4 a^{-6} +11 z^4 a^{-8} -z^4 a^{-10} +3 z^3 a^{-1} +9 z^3 a^{-3} +16 z^3 a^{-5} +14 z^3 a^{-7} +4 z^3 a^{-9} -6 z^2 a^{-2} -20 z^2 a^{-4} -15 z^2 a^{-6} -z^2 a^{-8} -z a^{-1} -3 z a^{-3} -5 z a^{-5} -z a^{-7} +2 z a^{-9} + a^{-2} +5 a^{-4} +3 a^{-6} }[/math]
The A2 invariant [math]\displaystyle{ -2+3 q^{-2} -2 q^{-4} + q^{-6} +4 q^{-8} -2 q^{-10} +4 q^{-12} -2 q^{-14} +2 q^{-16} + q^{-18} -3 q^{-20} +2 q^{-22} -3 q^{-24} - q^{-26} + q^{-28} }[/math]
The G2 invariant Data:K11n178/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n178 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["K11n178"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 2 t^3-9 t^2+22 t-29+22 t^{-1} -9 t^{-2} +2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 2 z^6+3 z^4+4 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]= { 95, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^9-5 q^8+9 q^7-13 q^6+16 q^5-16 q^4+15 q^3-11 q^2+7 q-2 }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ 2 z^6 a^{-4} -2 z^4 a^{-2} +8 z^4 a^{-4} -3 z^4 a^{-6} -3 z^2 a^{-2} +12 z^2 a^{-4} -6 z^2 a^{-6} +z^2 a^{-8} - a^{-2} +5 a^{-4} -3 a^{-6} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 4 z^9 a^{-5} +4 z^9 a^{-7} +9 z^8 a^{-4} +17 z^8 a^{-6} +8 z^8 a^{-8} +6 z^7 a^{-3} +3 z^7 a^{-5} +2 z^7 a^{-7} +5 z^7 a^{-9} +z^6 a^{-2} -22 z^6 a^{-4} -44 z^6 a^{-6} -20 z^6 a^{-8} +z^6 a^{-10} -7 z^5 a^{-3} -20 z^5 a^{-5} -24 z^5 a^{-7} -11 z^5 a^{-9} +8 z^4 a^{-2} +31 z^4 a^{-4} +35 z^4 a^{-6} +11 z^4 a^{-8} -z^4 a^{-10} +3 z^3 a^{-1} +9 z^3 a^{-3} +16 z^3 a^{-5} +14 z^3 a^{-7} +4 z^3 a^{-9} -6 z^2 a^{-2} -20 z^2 a^{-4} -15 z^2 a^{-6} -z^2 a^{-8} -z a^{-1} -3 z a^{-3} -5 z a^{-5} -z a^{-7} +2 z a^{-9} + a^{-2} +5 a^{-4} +3 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["K11n178"];
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{ 2 t^3-9 t^2+22 t-29+22 t^{-1} -9 t^{-2} +2 t^{-3} }[/math], [math]\displaystyle{ q^9-5 q^8+9 q^7-13 q^6+16 q^5-16 q^4+15 q^3-11 q^2+7 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: (4, 7)
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{ 16 }[/math] [math]\displaystyle{ 56 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{824}{3} }[/math] [math]\displaystyle{ \frac{112}{3} }[/math] [math]\displaystyle{ 896 }[/math] [math]\displaystyle{ \frac{4496}{3} }[/math] [math]\displaystyle{ \frac{800}{3} }[/math] [math]\displaystyle{ 184 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 1568 }[/math] [math]\displaystyle{ \frac{13184}{3} }[/math] [math]\displaystyle{ \frac{1792}{3} }[/math] [math]\displaystyle{ \frac{124142}{15} }[/math] [math]\displaystyle{ \frac{4192}{15} }[/math] [math]\displaystyle{ \frac{138608}{45} }[/math] [math]\displaystyle{ \frac{610}{9} }[/math] [math]\displaystyle{ \frac{6062}{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]2 is the signature of K11n178. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -1012345678χ
19         11
17        4 -4
15       51 4
13      84  -4
11     85   3
9    88    0
7   78     -1
5  48      4
3 37       -4
1 5        5
-12         -2
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=3 }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=8 }[/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.

K11n177.gif

K11n177

K11n179.gif

K11n179

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