K11n135

K11n134

K11n136

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n135 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X12,3,13,4 X14,6,15,5 X7,17,8,16 X22,10,1,9 X11,19,12,18 X2,13,3,14 X15,20,16,21 X17,11,18,10 X19,7,20,6 X8,22,9,21
Gauss code	1, -7, 2, -1, 3, 10, -4, -11, 5, 9, -6, -2, 7, -3, -8, 4, -9, 6, -10, 8, 11, -5
Dowker-Thistlethwaite code	4 12 14 -16 22 -18 2 -20 -10 -6 8

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n135/ThurstonBennequinNumber Hyperbolic Volume 8.03988 A-Polynomial See Data:K11n135/A-polynomial

[edit Notes for K11n135's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n135's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+2 t^2-1+2 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6-4 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 5, 4 }
Jones polynomial	[math]\displaystyle{ q^9-q^8-q^5+q^4-q^3+2 q^2-q+1 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6 a^{-4} +z^4 a^{-2} -5 z^4 a^{-4} +4 z^2 a^{-2} -6 z^2 a^{-4} +z^2 a^{-8} +3 a^{-2} -2 a^{-4} - a^{-6} + a^{-8} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^7 a^{-3} +z^7 a^{-9} +z^6 a^{-2} +2 z^6 a^{-4} +z^6 a^{-10} -4 z^5 a^{-3} +z^5 a^{-5} -z^5 a^{-7} -6 z^5 a^{-9} -5 z^4 a^{-2} -9 z^4 a^{-4} -z^4 a^{-8} -5 z^4 a^{-10} +2 z^3 a^{-3} -4 z^3 a^{-5} +3 z^3 a^{-7} +9 z^3 a^{-9} +7 z^2 a^{-2} +9 z^2 a^{-4} -2 z^2 a^{-6} +z^2 a^{-8} +5 z^2 a^{-10} +2 z a^{-3} +2 z a^{-5} -3 z a^{-7} -3 z a^{-9} -3 a^{-2} -2 a^{-4} + a^{-6} + a^{-8} }[/math]
The A2 invariant	[math]\displaystyle{ 1+ q^{-2} + q^{-4} + q^{-6} + q^{-8} + q^{-10} - q^{-12} -2 q^{-16} - q^{-18} - q^{-20} + q^{-24} + q^{-28} }[/math]
The G2 invariant	Data:K11n135/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n135 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["K11n135"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+2 t^2-1+2 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^6-4 z^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]=

{ 5, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^7 a^{-3} +z^7 a^{-9} +z^6 a^{-2} +2 z^6 a^{-4} +z^6 a^{-10} -4 z^5 a^{-3} +z^5 a^{-5} -z^5 a^{-7} -6 z^5 a^{-9} -5 z^4 a^{-2} -9 z^4 a^{-4} -z^4 a^{-8} -5 z^4 a^{-10} +2 z^3 a^{-3} -4 z^3 a^{-5} +3 z^3 a^{-7} +9 z^3 a^{-9} +7 z^2 a^{-2} +9 z^2 a^{-4} -2 z^2 a^{-6} +z^2 a^{-8} +5 z^2 a^{-10} +2 z a^{-3} +2 z a^{-5} -3 z a^{-7} -3 z a^{-9} -3 a^{-2} -2 a^{-4} + a^{-6} + a^{-8} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n19,}

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["K11n135"];

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^3+2 t^2-1+2 t^{-2} - t^{-3} }[/math], [math]\displaystyle{ q^9-q^8-q^5+q^4-q^3+2 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]=

{K11n19,}

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:

(-1, -4)

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{ -4 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -\frac{158}{3} }[/math] [math]\displaystyle{ \frac{110}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{736}{3} }[/math] [math]\displaystyle{ \frac{352}{3} }[/math] [math]\displaystyle{ 192 }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 512 }[/math] [math]\displaystyle{ \frac{632}{3} }[/math] [math]\displaystyle{ -\frac{440}{3} }[/math] [math]\displaystyle{ \frac{57329}{30} }[/math] [math]\displaystyle{ -\frac{806}{5} }[/math] [math]\displaystyle{ \frac{55378}{45} }[/math] [math]\displaystyle{ \frac{4495}{18} }[/math] [math]\displaystyle{ \frac{1649}{30} }[/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]4 is the signature of K11n135. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 19                     1 1 17                       0 15               1 1 1   -1 13             1 1       0 11           1 1 1       -1 9         1 2 1         0 7       1 1             0 5     1 1 1             1 3   1 2                 1 1                       0 -1 1                     1

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{ i=5 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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.

K11n134 K11n136