K11n140

K11n139

K11n141

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n140 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X12,3,13,4 X5,16,6,17 X7,14,8,15 X20,9,21,10 X18,11,19,12 X2,13,3,14 X15,6,16,7 X22,18,1,17 X10,19,11,20 X8,21,9,22
Gauss code	1, -7, 2, -1, -3, 8, -4, -11, 5, -10, 6, -2, 7, 4, -8, 3, 9, -6, 10, -5, 11, -9
Dowker-Thistlethwaite code	4 12 -16 -14 20 18 2 -6 22 10 8

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number [math]\displaystyle{ \{1,2\} }[/math] 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n140/ThurstonBennequinNumber Hyperbolic Volume 11.4116 A-Polynomial See Data:K11n140/A-polynomial

[edit Notes for K11n140's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n140's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^2+13 t-21+13 t^{-1} -2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -2 z^4+5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 51, -2 }
Jones polynomial	[math]\displaystyle{ q-3+5 q^{-1} -7 q^{-2} +9 q^{-3} -8 q^{-4} +8 q^{-5} -5 q^{-6} +3 q^{-7} -2 q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -2 a^8+3 z^2 a^6+2 a^6-z^4 a^4+z^2 a^4+a^4-z^4 a^2+z^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 3 z^5 a^9-10 z^3 a^9+7 z a^9+z^8 a^8-2 z^6 a^8+z^2 a^8-2 a^8+z^9 a^7-3 z^7 a^7+9 z^5 a^7-17 z^3 a^7+9 z a^7+4 z^8 a^6-12 z^6 a^6+15 z^4 a^6-6 z^2 a^6-2 a^6+z^9 a^5+z^7 a^5-3 z^5 a^5+z^3 a^5+2 z a^5+3 z^8 a^4-6 z^6 a^4+9 z^4 a^4-5 z^2 a^4+a^4+4 z^7 a^3-6 z^5 a^3+4 z^3 a^3+4 z^6 a^2-5 z^4 a^2+z^2 a^2+3 z^5 a-4 z^3 a+z^4-z^2 }[/math]
The A2 invariant	Data:K11n140/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n140/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n140 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["K11n140"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ -2 t^2+13 t-21+13 t^{-1} -2 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -2 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]=

{ 51, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ q-3+5 q^{-1} -7 q^{-2} +9 q^{-3} -8 q^{-4} +8 q^{-5} -5 q^{-6} +3 q^{-7} -2 q^{-8} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ -2 a^8+3 z^2 a^6+2 a^6-z^4 a^4+z^2 a^4+a^4-z^4 a^2+z^2 }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ 3 z^5 a^9-10 z^3 a^9+7 z a^9+z^8 a^8-2 z^6 a^8+z^2 a^8-2 a^8+z^9 a^7-3 z^7 a^7+9 z^5 a^7-17 z^3 a^7+9 z a^7+4 z^8 a^6-12 z^6 a^6+15 z^4 a^6-6 z^2 a^6-2 a^6+z^9 a^5+z^7 a^5-3 z^5 a^5+z^3 a^5+2 z a^5+3 z^8 a^4-6 z^6 a^4+9 z^4 a^4-5 z^2 a^4+a^4+4 z^7 a^3-6 z^5 a^3+4 z^3 a^3+4 z^6 a^2-5 z^4 a^2+z^2 a^2+3 z^5 a-4 z^3 a+z^4-z^2 }[/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["K11n140"];

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^2+13 t-21+13 t^{-1} -2 t^{-2} }[/math], [math]\displaystyle{ q-3+5 q^{-1} -7 q^{-2} +9 q^{-3} -8 q^{-4} +8 q^{-5} -5 q^{-6} +3 q^{-7} -2 q^{-8} }[/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, -11)

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{ -88 }[/math] [math]\displaystyle{ 200 }[/math] [math]\displaystyle{ \frac{1606}{3} }[/math] [math]\displaystyle{ \frac{338}{3} }[/math] [math]\displaystyle{ -1760 }[/math] [math]\displaystyle{ -\frac{10288}{3} }[/math] [math]\displaystyle{ -\frac{1696}{3} }[/math] [math]\displaystyle{ -696 }[/math] [math]\displaystyle{ \frac{4000}{3} }[/math] [math]\displaystyle{ 3872 }[/math] [math]\displaystyle{ \frac{32120}{3} }[/math] [math]\displaystyle{ \frac{6760}{3} }[/math] [math]\displaystyle{ \frac{131935}{6} }[/math] [math]\displaystyle{ -\frac{5030}{3} }[/math] [math]\displaystyle{ \frac{106310}{9} }[/math] [math]\displaystyle{ \frac{4837}{18} }[/math] [math]\displaystyle{ \frac{11071}{6} }[/math]

V_2,1 through V_6,9 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 K11n140. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                   1 1 1                 2   -2 -1               3 1   2 -3             5 3     -2 -5           4 2       2 -7         4 5         1 -9       4 4           0 -11     1 4             3 -13   2 4               -2 -15   1                 1 -17 2                   -2

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=-1 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/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}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/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.

K11n139 K11n141