K11n159

K11n158

K11n160

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n159 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X10,3,11,4 X12,6,13,5 X7,18,8,19 X9,20,10,21 X16,11,17,12 X22,13,1,14 X4,16,5,15 X2,17,3,18 X19,8,20,9 X14,21,15,22
Gauss code	1, -9, 2, -8, 3, -1, -4, 10, -5, -2, 6, -3, 7, -11, 8, -6, 9, 4, -10, 5, 11, -7
Dowker-Thistlethwaite code	6 10 12 -18 -20 16 22 4 2 -8 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n159/ThurstonBennequinNumber Hyperbolic Volume 15.181 A-Polynomial See Data:K11n159/A-polynomial

[edit Notes for K11n159's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 3 }[/math] Rasmussen s-Invariant 2

[edit Notes for K11n159's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-6 t^2+17 t-23+17 t^{-1} -6 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 71, -2 }
Jones polynomial	[math]\displaystyle{ -1+5 q^{-1} -8 q^{-2} +11 q^{-3} -12 q^{-4} +12 q^{-5} -10 q^{-6} +7 q^{-7} -4 q^{-8} + q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^8-2 z^4 a^6-3 z^2 a^6-a^6+z^6 a^4+3 z^4 a^4+4 z^2 a^4+a^4-z^4 a^2+a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{10}-2 z^4 a^{10}+z^2 a^{10}+4 z^7 a^9-11 z^5 a^9+7 z^3 a^9+5 z^8 a^8-12 z^6 a^8+4 z^4 a^8+z^2 a^8+2 z^9 a^7+4 z^7 a^7-21 z^5 a^7+13 z^3 a^7-2 z a^7+9 z^8 a^6-21 z^6 a^6+13 z^4 a^6-5 z^2 a^6+a^6+2 z^9 a^5+2 z^7 a^5-9 z^5 a^5+6 z^3 a^5-2 z a^5+4 z^8 a^4-8 z^6 a^4+12 z^4 a^4-6 z^2 a^4+a^4+2 z^7 a^3+z^5 a^3+z^3 a^3+5 z^4 a^2-z^2 a^2-a^2+z^3 a }[/math]
The A2 invariant	[math]\displaystyle{ q^{28}-q^{26}-2 q^{24}+2 q^{22}-2 q^{20}+q^{18}+q^{16}-2 q^{14}+2 q^{12}-2 q^{10}+3 q^8+q^6-q^4+3 q^2-1 }[/math]
The G2 invariant	Data:K11n159/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n159 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["K11n159"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^3-6 t^2+17 t-23+17 t^{-1} -6 t^{-2} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^6+2 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]=

{ 71, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^6 a^{10}-2 z^4 a^{10}+z^2 a^{10}+4 z^7 a^9-11 z^5 a^9+7 z^3 a^9+5 z^8 a^8-12 z^6 a^8+4 z^4 a^8+z^2 a^8+2 z^9 a^7+4 z^7 a^7-21 z^5 a^7+13 z^3 a^7-2 z a^7+9 z^8 a^6-21 z^6 a^6+13 z^4 a^6-5 z^2 a^6+a^6+2 z^9 a^5+2 z^7 a^5-9 z^5 a^5+6 z^3 a^5-2 z a^5+4 z^8 a^4-8 z^6 a^4+12 z^4 a^4-6 z^2 a^4+a^4+2 z^7 a^3+z^5 a^3+z^3 a^3+5 z^4 a^2-z^2 a^2-a^2+z^3 a }[/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["K11n159"];

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^3-6 t^2+17 t-23+17 t^{-1} -6 t^{-2} + t^{-3} }[/math], [math]\displaystyle{ -1+5 q^{-1} -8 q^{-2} +11 q^{-3} -12 q^{-4} +12 q^{-5} -10 q^{-6} +7 q^{-7} -4 q^{-8} + q^{-9} }[/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:

(2, -3)

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{ -24 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{268}{3} }[/math] [math]\displaystyle{ \frac{44}{3} }[/math] [math]\displaystyle{ -192 }[/math] [math]\displaystyle{ -368 }[/math] [math]\displaystyle{ -96 }[/math] [math]\displaystyle{ -24 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ \frac{2144}{3} }[/math] [math]\displaystyle{ \frac{352}{3} }[/math] [math]\displaystyle{ \frac{22951}{15} }[/math] [math]\displaystyle{ \frac{1252}{5} }[/math] [math]\displaystyle{ \frac{21124}{45} }[/math] [math]\displaystyle{ -\frac{439}{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]-2 is the signature of K11n159. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 1                   1 -1 -1                 4   4 -3               5 2   -3 -5             6 3     3 -7           6 5       -1 -9         6 6         0 -11       4 6           2 -13     3 6             -3 -15   1 4               3 -17   3                 -3 -19 1                   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=-1 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/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.

K11n158 K11n160