K11n67

K11n66

K11n68

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n67 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X14,5,15,6 X2837 X9,19,10,18 X11,17,12,16 X13,20,14,21 X6,15,7,16 X17,11,18,10 X19,1,20,22 X21,12,22,13
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, -5, 9, -6, 11, -7, -3, 8, 6, -9, 5, -10, 7, -11, 10
Dowker-Thistlethwaite code	4 8 14 2 -18 -16 -20 6 -10 -22 -12

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral 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:K11n67/ThurstonBennequinNumber Hyperbolic Volume 10.6503 A-Polynomial See Data:K11n67/A-polynomial

[edit Notes for K11n67's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 0 }[/math] Rasmussen s-Invariant 0

[edit Notes for K11n67's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t+5-2 t^{-1} }[/math]
Conway polynomial	[math]\displaystyle{ 1-2 z^2 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 9, 0 }
Jones polynomial	[math]\displaystyle{ -q^7+2 q^6-2 q^5+3 q^4-2 q^3+q^2-q+ q^{-1} - q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^4 a^{-4} -z^4+a^2 z^2-z^2 a^{-2} +3 z^2 a^{-4} -z^2 a^{-6} -4 z^2+2 a^2- a^{-2} +3 a^{-4} - a^{-6} -2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +z^8 a^{-2} +3 z^8 a^{-4} +2 z^8 a^{-6} +a z^7-6 z^7 a^{-3} -4 z^7 a^{-5} +z^7 a^{-7} +a^2 z^6-7 z^6 a^{-2} -18 z^6 a^{-4} -11 z^6 a^{-6} +z^6-5 a z^5-z^5 a^{-1} +10 z^5 a^{-3} +z^5 a^{-5} -5 z^5 a^{-7} -5 a^2 z^4+12 z^4 a^{-2} +31 z^4 a^{-4} +17 z^4 a^{-6} -7 z^4+5 a z^3-z^3 a^{-1} -9 z^3 a^{-3} +3 z^3 a^{-5} +6 z^3 a^{-7} +6 a^2 z^2-8 z^2 a^{-2} -20 z^2 a^{-4} -9 z^2 a^{-6} +9 z^2-a z+2 z a^{-1} +4 z a^{-3} -z a^{-7} -2 a^2+ a^{-2} +3 a^{-4} + a^{-6} -2 }[/math]
The A2 invariant	Data:K11n67/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n67/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n67 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["K11n67"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -2 t+5-2 t^{-1} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 1-2 z^2 }[/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]=

{ 9, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^7+2 q^6-2 q^5+3 q^4-2 q^3+q^2-q+ q^{-1} - 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{ z^4 a^{-4} -z^4+a^2 z^2-z^2 a^{-2} +3 z^2 a^{-4} -z^2 a^{-6} -4 z^2+2 a^2- a^{-2} +3 a^{-4} - a^{-6} -2 }[/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} +z^8 a^{-2} +3 z^8 a^{-4} +2 z^8 a^{-6} +a z^7-6 z^7 a^{-3} -4 z^7 a^{-5} +z^7 a^{-7} +a^2 z^6-7 z^6 a^{-2} -18 z^6 a^{-4} -11 z^6 a^{-6} +z^6-5 a z^5-z^5 a^{-1} +10 z^5 a^{-3} +z^5 a^{-5} -5 z^5 a^{-7} -5 a^2 z^4+12 z^4 a^{-2} +31 z^4 a^{-4} +17 z^4 a^{-6} -7 z^4+5 a z^3-z^3 a^{-1} -9 z^3 a^{-3} +3 z^3 a^{-5} +6 z^3 a^{-7} +6 a^2 z^2-8 z^2 a^{-2} -20 z^2 a^{-4} -9 z^2 a^{-6} +9 z^2-a z+2 z a^{-1} +4 z a^{-3} -z a^{-7} -2 a^2+ a^{-2} +3 a^{-4} + a^{-6} -2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {6_1, 9_46, K11n97, K11n139,}

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["K11n67"];

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+5-2 t^{-1} }[/math], [math]\displaystyle{ -q^7+2 q^6-2 q^5+3 q^4-2 q^3+q^2-q+ q^{-1} - 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]=

{6_1, 9_46, K11n97, K11n139,}

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:

(-2, 1)

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{ -8 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{212}{3} }[/math] [math]\displaystyle{ \frac{52}{3} }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ -\frac{304}{3} }[/math] [math]\displaystyle{ -\frac{160}{3} }[/math] [math]\displaystyle{ -24 }[/math] [math]\displaystyle{ -\frac{256}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -\frac{1696}{3} }[/math] [math]\displaystyle{ -\frac{416}{3} }[/math] [math]\displaystyle{ -\frac{9751}{15} }[/math] [math]\displaystyle{ -\frac{372}{5} }[/math] [math]\displaystyle{ -\frac{15844}{45} }[/math] [math]\displaystyle{ \frac{343}{9} }[/math] [math]\displaystyle{ -\frac{871}{15} }[/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]0 is the signature of K11n67. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 15                       1 -1 13                     1   1 11                   1 1   0 9                 2 1     1 7             1 1 1       1 5             1 2         -1 3         1 2 1           0 1       1 1 1             -1 -1       1 2               1 -3   1 1                   0 -5                         0 -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=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}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/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=3 }[/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}\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 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/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.

K11n66 K11n68