K11a266

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a266 at Knotilus!
	[edit K11a266 Quick Notes]

[edit K11a266 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₆₂₇₁ X_10,3,11,4 X_12,6,13,5 X_14,7,15,8 X_16,10,17,9 X_18,11,19,12 X_22,13,1,14 X_20,16,21,15 X_4,18,5,17 X_2,19,3,20 X_8,21,9,22
Gauss code	1, -10, 2, -9, 3, -1, 4, -11, 5, -2, 6, -3, 7, -4, 8, -5, 9, -6, 10, -8, 11, -7
Dowker-Thistlethwaite code	6 10 12 14 16 18 22 20 4 2 8

A Braid Representative

A Morse Link Presentation

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	4
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a266/ThurstonBennequinNumber
Hyperbolic Volume	20.2286
A-Polynomial	See Data:K11a266/A-polynomial

[edit Notes for K11a266's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$4$
Rasmussen s-Invariant	0

[edit Notes for K11a266's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-7t^{3}+23t^{2}-45t+57-45t^{-1}+23t^{-2}-7t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+z^{6}+z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 209, 0 }
Jones polynomial	$-q^{5}+5q^{4}-12q^{3}+21q^{2}-29q+34-34q^{-1}+30q^{-2}-22q^{-3}+14q^{-4}-6q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8-2 a^2 z^6-z^6 a^{-2} +4 z^6+a^4 z^4-4 a^2 z^4-2 z^4 a^{-2} +6 z^4-z^2 a^{-2} +z^2-a^4+3 a^2+ a^{-2} -2}
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6 a^2 z^{10}+6 z^{10}+15 a^3 z^9+31 a z^9+16 z^9 a^{-1} +14 a^4 z^8+20 a^2 z^8+18 z^8 a^{-2} +24 z^8+6 a^5 z^7-23 a^3 z^7-57 a z^7-16 z^7 a^{-1} +12 z^7 a^{-3} +a^6 z^6-26 a^4 z^6-65 a^2 z^6-26 z^6 a^{-2} +5 z^6 a^{-4} -69 z^6-7 a^5 z^5+2 a^3 z^5+19 a z^5-4 z^5 a^{-1} -13 z^5 a^{-3} +z^5 a^{-5} +10 a^4 z^4+37 a^2 z^4+14 z^4 a^{-2} -3 z^4 a^{-4} +44 z^4+2 a^3 z^3+5 a z^3+7 z^3 a^{-1} +4 z^3 a^{-3} +2 a^4 z^2+a^2 z^2-2 z^2 a^{-2} -3 z^2+a^5 z+a^3 z-a z-z a^{-1} -a^4-3 a^2- a^{-2} -2}
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{18}-3 q^{16}+3 q^{12}-5 q^{10}+7 q^8-q^6+4 q^2-7+6 q^{-2} -6 q^{-4} +2 q^{-6} +4 q^{-8} -4 q^{-10} +3 q^{-12} - q^{-14} }
The G2 invariant	Data:K11a266/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a266 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.

In[3]:= K = Knot["K11a266"];

In[4]:= Alexander[K][t]

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

Out[4]= $t^{4}-7t^{3}+23t^{2}-45t+57-45t^{-1}+23t^{-2}-7t^{-3}+t^{-4}$

In[5]:= Conway[K][z]

Out[5]= $z^{8}+z^{6}+z^{4}+1$

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]= $\{1\}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 209, 0 }

In[8]:= Jones[K][q]

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

Out[8]= $-q^{5}+5q^{4}-12q^{3}+21q^{2}-29q+34-34q^{-1}+30q^{-2}-22q^{-3}+14q^{-4}-6q^{-5}+q^{-6}$

In[9]:= HOMFLYPT[K][a, z]

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

Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8-2 a^2 z^6-z^6 a^{-2} +4 z^6+a^4 z^4-4 a^2 z^4-2 z^4 a^{-2} +6 z^4-z^2 a^{-2} +z^2-a^4+3 a^2+ a^{-2} -2}

In[10]:= Kauffman[K][a, z]

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

Out[10]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6 a^2 z^{10}+6 z^{10}+15 a^3 z^9+31 a z^9+16 z^9 a^{-1} +14 a^4 z^8+20 a^2 z^8+18 z^8 a^{-2} +24 z^8+6 a^5 z^7-23 a^3 z^7-57 a z^7-16 z^7 a^{-1} +12 z^7 a^{-3} +a^6 z^6-26 a^4 z^6-65 a^2 z^6-26 z^6 a^{-2} +5 z^6 a^{-4} -69 z^6-7 a^5 z^5+2 a^3 z^5+19 a z^5-4 z^5 a^{-1} -13 z^5 a^{-3} +z^5 a^{-5} +10 a^4 z^4+37 a^2 z^4+14 z^4 a^{-2} -3 z^4 a^{-4} +44 z^4+2 a^3 z^3+5 a z^3+7 z^3 a^{-1} +4 z^3 a^{-3} +2 a^4 z^2+a^2 z^2-2 z^2 a^{-2} -3 z^2+a^5 z+a^3 z-a z-z a^{-1} -a^4-3 a^2- a^{-2} -2}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {}

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.

In[3]:= K = Knot["K11a266"];

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]= { Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^4-7 t^3+23 t^2-45 t+57-45 t^{-1} +23 t^{-2} -7 t^{-3} + t^{-4} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^5+5 q^4-12 q^3+21 q^2-29 q+34-34 q^{-1} +30 q^{-2} -22 q^{-3} +14 q^{-4} -6 q^{-5} + q^{-6} } }

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:

(0, -1)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9