K11n78

From Knot Atlas
Jump to navigationJump to search

K11n77.gif

K11n77

K11n79.gif

K11n79

K11n78.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n78 at Knotilus!

[edit K11n78 Quick Notes]


[edit K11n78 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X8493 X5,15,6,14 X2837 X9,20,10,21 X11,17,12,16 X13,19,14,18 X15,7,16,6 X17,13,18,12 X19,22,20,1 X21,10,22,11
Gauss code 1, -4, 2, -1, -3, 8, 4, -2, -5, 11, -6, 9, -7, 3, -8, 6, -9, 7, -10, 5, -11, 10
Dowker-Thistlethwaite code 4 8 -14 2 -20 -16 -18 -6 -12 -22 -10
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11n78 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number [math]\displaystyle{ \{2,3\} }[/math]
3-genus 4
Bridge index 4
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n78/ThurstonBennequinNumber
Hyperbolic Volume 12.3125
A-Polynomial See Data:K11n78/A-polynomial

[edit Notes for K11n78's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n78's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^4-3 t^3+6 t^2-8 t+9-8 t^{-1} +6 t^{-2} -3 t^{-3} + t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ z^8+5 z^6+8 z^4+5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \left\{t^2-t+1\right\} }[/math]
Determinant and Signature { 45, 4 }
Jones polynomial [math]\displaystyle{ q^8-4 q^7+5 q^6-7 q^5+8 q^4-6 q^3+7 q^2-4 q+2- q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^8 a^{-4} -z^6 a^{-2} +7 z^6 a^{-4} -z^6 a^{-6} -5 z^4 a^{-2} +19 z^4 a^{-4} -6 z^4 a^{-6} -8 z^2 a^{-2} +25 z^2 a^{-4} -13 z^2 a^{-6} +z^2 a^{-8} -4 a^{-2} +13 a^{-4} -10 a^{-6} +2 a^{-8} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +6 z^8 a^{-4} +4 z^8 a^{-6} +z^7 a^{-1} +z^7 a^{-3} +5 z^7 a^{-5} +5 z^7 a^{-7} -9 z^6 a^{-2} -24 z^6 a^{-4} -13 z^6 a^{-6} +2 z^6 a^{-8} -5 z^5 a^{-1} -18 z^5 a^{-3} -31 z^5 a^{-5} -18 z^5 a^{-7} +13 z^4 a^{-2} +31 z^4 a^{-4} +16 z^4 a^{-6} -2 z^4 a^{-8} +8 z^3 a^{-1} +28 z^3 a^{-3} +42 z^3 a^{-5} +26 z^3 a^{-7} +4 z^3 a^{-9} -9 z^2 a^{-2} -26 z^2 a^{-4} -17 z^2 a^{-6} +z^2 a^{-8} +z^2 a^{-10} -4 z a^{-1} -13 z a^{-3} -21 z a^{-5} -14 z a^{-7} -2 z a^{-9} +4 a^{-2} +13 a^{-4} +10 a^{-6} +2 a^{-8} }[/math]
The A2 invariant [math]\displaystyle{ -q^2-2 q^{-2} + q^{-6} +2 q^{-8} +6 q^{-10} +2 q^{-12} +4 q^{-14} -2 q^{-16} -3 q^{-18} -3 q^{-20} -3 q^{-22} + q^{-24} + q^{-28} }[/math]
The G2 invariant Data:K11n78/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n78 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["K11n78"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^4-3 t^3+6 t^2-8 t+9-8 t^{-1} +6 t^{-2} -3 t^{-3} + t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^8+5 z^6+8 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{ \left\{t^2-t+1\right\} }[/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{ q^8-4 q^7+5 q^6-7 q^5+8 q^4-6 q^3+7 q^2-4 q+2- 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} +7 z^6 a^{-4} -z^6 a^{-6} -5 z^4 a^{-2} +19 z^4 a^{-4} -6 z^4 a^{-6} -8 z^2 a^{-2} +25 z^2 a^{-4} -13 z^2 a^{-6} +z^2 a^{-8} -4 a^{-2} +13 a^{-4} -10 a^{-6} +2 a^{-8} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +6 z^8 a^{-4} +4 z^8 a^{-6} +z^7 a^{-1} +z^7 a^{-3} +5 z^7 a^{-5} +5 z^7 a^{-7} -9 z^6 a^{-2} -24 z^6 a^{-4} -13 z^6 a^{-6} +2 z^6 a^{-8} -5 z^5 a^{-1} -18 z^5 a^{-3} -31 z^5 a^{-5} -18 z^5 a^{-7} +13 z^4 a^{-2} +31 z^4 a^{-4} +16 z^4 a^{-6} -2 z^4 a^{-8} +8 z^3 a^{-1} +28 z^3 a^{-3} +42 z^3 a^{-5} +26 z^3 a^{-7} +4 z^3 a^{-9} -9 z^2 a^{-2} -26 z^2 a^{-4} -17 z^2 a^{-6} +z^2 a^{-8} +z^2 a^{-10} -4 z a^{-1} -13 z a^{-3} -21 z a^{-5} -14 z a^{-7} -2 z a^{-9} +4 a^{-2} +13 a^{-4} +10 a^{-6} +2 a^{-8} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_62, K11n76,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n76,}

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["K11n78"];
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^4-3 t^3+6 t^2-8 t+9-8 t^{-1} +6 t^{-2} -3 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^8-4 q^7+5 q^6-7 q^5+8 q^4-6 q^3+7 q^2-4 q+2- 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]= {10_62, K11n76,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11n76,}

Vassiliev invariants

V2 and V3: (5, 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{ 20 }[/math] [math]\displaystyle{ 56 }[/math] [math]\displaystyle{ 200 }[/math] [math]\displaystyle{ \frac{886}{3} }[/math] [math]\displaystyle{ \frac{98}{3} }[/math] [math]\displaystyle{ 1120 }[/math] [math]\displaystyle{ \frac{4496}{3} }[/math] [math]\displaystyle{ \frac{704}{3} }[/math] [math]\displaystyle{ 152 }[/math] [math]\displaystyle{ \frac{4000}{3} }[/math] [math]\displaystyle{ 1568 }[/math] [math]\displaystyle{ \frac{17720}{3} }[/math] [math]\displaystyle{ \frac{1960}{3} }[/math] [math]\displaystyle{ \frac{50767}{6} }[/math] [math]\displaystyle{ \frac{2090}{3} }[/math] [math]\displaystyle{ \frac{21446}{9} }[/math] [math]\displaystyle{ \frac{853}{18} }[/math] [math]\displaystyle{ \frac{1423}{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 K11n78. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -3-2-10123456χ
17         11
15        3 -3
13       21 1
11      53  -2
9     32   1
7    35    2
5   43     1
3  14      3
1 13       -2
-1 1        1
-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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

K11n77.gif

K11n77

K11n79.gif

K11n79

Retrieved from "https://katlas.org/index.php?title=K11n78&oldid=110753"