K11n158

K11n157

K11n159

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n158 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X10,3,11,4 X12,6,13,5 X18,8,19,7 X20,10,21,9 X16,11,17,12 X13,1,14,22 X4,16,5,15 X2,17,3,18 X8,20,9,19 X21,15,22,14
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 Reversible Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n158/ThurstonBennequinNumber Hyperbolic Volume 12.7254 A-Polynomial See Data:K11n158/A-polynomial

[edit Notes for K11n158's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n158's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-4 t^3+7 t^2-7 t+7-7 t^{-1} +7 t^{-2} -4 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+4 z^6+3 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 45, 4 }
Jones polynomial	[math]\displaystyle{ q^8-3 q^7+5 q^6-7 q^5+7 q^4-7 q^3+7 q^2-4 q+3- q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-4} -z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -4 z^4 a^{-2} +12 z^4 a^{-4} -5 z^4 a^{-6} -3 z^2 a^{-2} +10 z^2 a^{-4} -7 z^2 a^{-6} +z^2 a^{-8} + a^{-2} +2 a^{-4} -3 a^{-6} + a^{-8} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^9 a^{-3} +2 z^9 a^{-5} +3 z^8 a^{-2} +7 z^8 a^{-4} +4 z^8 a^{-6} +z^7 a^{-1} -6 z^7 a^{-3} -4 z^7 a^{-5} +3 z^7 a^{-7} -14 z^6 a^{-2} -29 z^6 a^{-4} -14 z^6 a^{-6} +z^6 a^{-8} -4 z^5 a^{-1} +z^5 a^{-3} -2 z^5 a^{-5} -7 z^5 a^{-7} +19 z^4 a^{-2} +35 z^4 a^{-4} +17 z^4 a^{-6} +z^4 a^{-8} +4 z^3 a^{-1} +2 z^3 a^{-3} +5 z^3 a^{-7} +3 z^3 a^{-9} -7 z^2 a^{-2} -17 z^2 a^{-4} -12 z^2 a^{-6} -z^2 a^{-8} +z^2 a^{-10} +2 z a^{-3} +2 z a^{-5} -z a^{-7} -z a^{-9} - a^{-2} +2 a^{-4} +3 a^{-6} + a^{-8} }[/math]
The A2 invariant	[math]\displaystyle{ -q^2+1+2 q^{-4} +2 q^{-6} + q^{-8} +2 q^{-10} -3 q^{-12} + q^{-14} -2 q^{-16} - q^{-22} + q^{-24} - q^{-26} + q^{-28} }[/math]
The G2 invariant	Data:K11n158/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n158 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["K11n158"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-4 t^3+7 t^2-7 t+7-7 t^{-1} +7 t^{-2} -4 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+4 z^6+3 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]=

{ 45, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^8 a^{-4} -z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -4 z^4 a^{-2} +12 z^4 a^{-4} -5 z^4 a^{-6} -3 z^2 a^{-2} +10 z^2 a^{-4} -7 z^2 a^{-6} +z^2 a^{-8} + a^{-2} +2 a^{-4} -3 a^{-6} + a^{-8} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 z^9 a^{-3} +2 z^9 a^{-5} +3 z^8 a^{-2} +7 z^8 a^{-4} +4 z^8 a^{-6} +z^7 a^{-1} -6 z^7 a^{-3} -4 z^7 a^{-5} +3 z^7 a^{-7} -14 z^6 a^{-2} -29 z^6 a^{-4} -14 z^6 a^{-6} +z^6 a^{-8} -4 z^5 a^{-1} +z^5 a^{-3} -2 z^5 a^{-5} -7 z^5 a^{-7} +19 z^4 a^{-2} +35 z^4 a^{-4} +17 z^4 a^{-6} +z^4 a^{-8} +4 z^3 a^{-1} +2 z^3 a^{-3} +5 z^3 a^{-7} +3 z^3 a^{-9} -7 z^2 a^{-2} -17 z^2 a^{-4} -12 z^2 a^{-6} -z^2 a^{-8} +z^2 a^{-10} +2 z a^{-3} +2 z a^{-5} -z a^{-7} -z a^{-9} - a^{-2} +2 a^{-4} +3 a^{-6} + a^{-8} }[/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["K11n158"];

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

{}

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, 0)

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{ 0 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -\frac{130}{3} }[/math] [math]\displaystyle{ -\frac{62}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -224 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ -96 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{520}{3} }[/math] [math]\displaystyle{ -\frac{248}{3} }[/math] [math]\displaystyle{ -\frac{23249}{30} }[/math] [math]\displaystyle{ \frac{2978}{15} }[/math] [math]\displaystyle{ -\frac{25378}{45} }[/math] [math]\displaystyle{ -\frac{2287}{18} }[/math] [math]\displaystyle{ -\frac{2129}{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 K11n158. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

K11n157 K11n159