K11n117

K11n116

K11n118

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n117 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X5,14,6,15 X7,19,8,18 X9,17,10,16 X2,11,3,12 X13,21,14,20 X15,22,16,1 X17,9,18,8 X19,13,20,12 X21,7,22,6
Gauss code	1, -6, 2, -1, -3, 11, -4, 9, -5, -2, 6, 10, -7, 3, -8, 5, -9, 4, -10, 7, -11, 8
Dowker-Thistlethwaite code	4 10 -14 -18 -16 2 -20 -22 -8 -12 -6

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:K11n117/ThurstonBennequinNumber Hyperbolic Volume 11.3952 A-Polynomial See Data:K11n117/A-polynomial

[edit Notes for K11n117'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 K11n117's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -3 t^2+9 t-11+9 t^{-1} -3 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -3 z^4-3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 35, 2 }
Jones polynomial	[math]\displaystyle{ 2 q^5-4 q^4+5 q^3-6 q^2+6 q-5+4 q^{-1} -2 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -2 z^4 a^{-2} -z^4+a^2 z^2-5 z^2 a^{-2} +2 z^2 a^{-4} -z^2+a^2-3 a^{-2} +2 a^{-4} +1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^9 a^{-1} +z^9 a^{-3} +3 z^8 a^{-2} +z^8 a^{-4} +2 z^8+2 a z^7-2 z^7 a^{-1} -4 z^7 a^{-3} +a^2 z^6-11 z^6 a^{-2} -4 z^6 a^{-4} -6 z^6-7 a z^5+z^5 a^{-1} +9 z^5 a^{-3} +z^5 a^{-5} -4 a^2 z^4+17 z^4 a^{-2} +10 z^4 a^{-4} +3 z^4+5 a z^3-4 z^3 a^{-1} -7 z^3 a^{-3} +2 z^3 a^{-5} +4 a^2 z^2-14 z^2 a^{-2} -7 z^2 a^{-4} +2 z^2 a^{-6} -z^2+2 z a^{-1} +z a^{-3} -z a^{-5} -a^2+3 a^{-2} +2 a^{-4} +1 }[/math]
The A2 invariant	Data:K11n117/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n117/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n117 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["K11n117"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -3 t^2+9 t-11+9 t^{-1} -3 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -3 z^4-3 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]=

{ 35, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -2 z^4 a^{-2} -z^4+a^2 z^2-5 z^2 a^{-2} +2 z^2 a^{-4} -z^2+a^2-3 a^{-2} +2 a^{-4} +1 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^9 a^{-1} +z^9 a^{-3} +3 z^8 a^{-2} +z^8 a^{-4} +2 z^8+2 a z^7-2 z^7 a^{-1} -4 z^7 a^{-3} +a^2 z^6-11 z^6 a^{-2} -4 z^6 a^{-4} -6 z^6-7 a z^5+z^5 a^{-1} +9 z^5 a^{-3} +z^5 a^{-5} -4 a^2 z^4+17 z^4 a^{-2} +10 z^4 a^{-4} +3 z^4+5 a z^3-4 z^3 a^{-1} -7 z^3 a^{-3} +2 z^3 a^{-5} +4 a^2 z^2-14 z^2 a^{-2} -7 z^2 a^{-4} +2 z^2 a^{-6} -z^2+2 z a^{-1} +z a^{-3} -z a^{-5} -a^2+3 a^{-2} +2 a^{-4} +1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_20, 10_162,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {10_138,}

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["K11n117"];

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{ -3 t^2+9 t-11+9 t^{-1} -3 t^{-2} }[/math], [math]\displaystyle{ 2 q^5-4 q^4+5 q^3-6 q^2+6 q-5+4 q^{-1} -2 q^{-2} + q^{-3} }[/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_20, 10_162,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{10_138,}

Vassiliev invariants

V2 and V3:

(-3, -2)

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{ -12 }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 114 }[/math] [math]\displaystyle{ 62 }[/math] [math]\displaystyle{ 192 }[/math] [math]\displaystyle{ \frac{1088}{3} }[/math] [math]\displaystyle{ \frac{224}{3} }[/math] [math]\displaystyle{ 112 }[/math] [math]\displaystyle{ -288 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ -1368 }[/math] [math]\displaystyle{ -744 }[/math] [math]\displaystyle{ -\frac{8831}{10} }[/math] [math]\displaystyle{ \frac{6106}{15} }[/math] [math]\displaystyle{ -\frac{19982}{15} }[/math] [math]\displaystyle{ \frac{1567}{6} }[/math] [math]\displaystyle{ -\frac{3551}{10} }[/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 K11n117. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 χ 11                 2 2 9               2   -2 7             3 2   1 5           3 2     -1 3         3 3       0 1       3 4         1 -1     1 2           -1 -3   1 3             2 -5   1               -1 -7 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{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

K11n116 K11n118