K11n59

K11n58

K11n60

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n59 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,15,6,14 X2837 X16,9,17,10 X18,12,19,11 X20,14,21,13 X15,7,16,6 X22,17,1,18 X12,20,13,19 X10,22,11,21
Gauss code	1, -4, 2, -1, -3, 8, 4, -2, 5, -11, 6, -10, 7, 3, -8, -5, 9, -6, 10, -7, 11, -9
Dowker-Thistlethwaite code	4 8 -14 2 16 18 20 -6 22 12 10

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:K11n59/ThurstonBennequinNumber Hyperbolic Volume 11.895 A-Polynomial See Data:K11n59/A-polynomial

[edit Notes for K11n59'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 K11n59's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+6 t^2-12 t+15-12 t^{-1} +6 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 53, 4 }
Jones polynomial	[math]\displaystyle{ -q^{11}+3 q^{10}-5 q^9+7 q^8-9 q^7+9 q^6-8 q^5+6 q^4-3 q^3+2 q^2 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6 a^{-6} +2 z^4 a^{-4} -4 z^4 a^{-6} +2 z^4 a^{-8} +6 z^2 a^{-4} -7 z^2 a^{-6} +5 z^2 a^{-8} -z^2 a^{-10} +4 a^{-4} -5 a^{-6} +3 a^{-8} - a^{-10} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^9 a^{-7} +z^9 a^{-9} +2 z^8 a^{-6} +5 z^8 a^{-8} +3 z^8 a^{-10} +z^7 a^{-5} +3 z^7 a^{-9} +4 z^7 a^{-11} -7 z^6 a^{-6} -16 z^6 a^{-8} -6 z^6 a^{-10} +3 z^6 a^{-12} -z^5 a^{-5} -4 z^5 a^{-7} -14 z^5 a^{-9} -10 z^5 a^{-11} +z^5 a^{-13} +3 z^4 a^{-4} +16 z^4 a^{-6} +24 z^4 a^{-8} +4 z^4 a^{-10} -7 z^4 a^{-12} +z^3 a^{-5} +11 z^3 a^{-7} +19 z^3 a^{-9} +7 z^3 a^{-11} -2 z^3 a^{-13} -8 z^2 a^{-4} -16 z^2 a^{-6} -12 z^2 a^{-8} -2 z^2 a^{-10} +2 z^2 a^{-12} -3 z a^{-5} -6 z a^{-7} -6 z a^{-9} -3 z a^{-11} +4 a^{-4} +5 a^{-6} +3 a^{-8} + a^{-10} }[/math]
The A2 invariant	[math]\displaystyle{ 2 q^{-6} +3 q^{-10} + q^{-12} - q^{-14} + q^{-16} -3 q^{-18} - q^{-22} +2 q^{-26} - q^{-28} + q^{-30} - q^{-34} }[/math]
The G2 invariant	Data:K11n59/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n59 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["K11n59"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+6 t^2-12 t+15-12 t^{-1} +6 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 53, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^9 a^{-7} +z^9 a^{-9} +2 z^8 a^{-6} +5 z^8 a^{-8} +3 z^8 a^{-10} +z^7 a^{-5} +3 z^7 a^{-9} +4 z^7 a^{-11} -7 z^6 a^{-6} -16 z^6 a^{-8} -6 z^6 a^{-10} +3 z^6 a^{-12} -z^5 a^{-5} -4 z^5 a^{-7} -14 z^5 a^{-9} -10 z^5 a^{-11} +z^5 a^{-13} +3 z^4 a^{-4} +16 z^4 a^{-6} +24 z^4 a^{-8} +4 z^4 a^{-10} -7 z^4 a^{-12} +z^3 a^{-5} +11 z^3 a^{-7} +19 z^3 a^{-9} +7 z^3 a^{-11} -2 z^3 a^{-13} -8 z^2 a^{-4} -16 z^2 a^{-6} -12 z^2 a^{-8} -2 z^2 a^{-10} +2 z^2 a^{-12} -3 z a^{-5} -6 z a^{-7} -6 z a^{-9} -3 z a^{-11} +4 a^{-4} +5 a^{-6} +3 a^{-8} + a^{-10} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n8,}

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["K11n59"];

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^2-12 t+15-12 t^{-1} +6 t^{-2} - t^{-3} }[/math], [math]\displaystyle{ -q^{11}+3 q^{10}-5 q^9+7 q^8-9 q^7+9 q^6-8 q^5+6 q^4-3 q^3+2 q^2 }[/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]=

{K11n8,}

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:

(3, 6)

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{ 48 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 238 }[/math] [math]\displaystyle{ 34 }[/math] [math]\displaystyle{ 576 }[/math] [math]\displaystyle{ 1248 }[/math] [math]\displaystyle{ 192 }[/math] [math]\displaystyle{ 176 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 1152 }[/math] [math]\displaystyle{ 2856 }[/math] [math]\displaystyle{ 408 }[/math] [math]\displaystyle{ \frac{66591}{10} }[/math] [math]\displaystyle{ \frac{214}{15} }[/math] [math]\displaystyle{ \frac{39662}{15} }[/math] [math]\displaystyle{ \frac{545}{6} }[/math] [math]\displaystyle{ \frac{3231}{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]4 is the signature of K11n59. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 χ 23                   1 -1 21                 2   2 19               3 1   -2 17             4 2     2 15           5 3       -2 13         4 4         0 11       4 5           1 9     2 4             -2 7   1 4               3 5 1 2                 -1 3 2                   2

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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=9 }[/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.

K11n58 K11n60