K11a6

	Visit K11a6's page at Knotilus! Visit K11a6's page at the original Knot Atlas!
	K11a6 Quick Notes

K11a6 Further Notes and Views

Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₃₉₄ X_10,6,11,5 X_16,8,17,7 X_2,9,3,10 X_18,12,19,11 X_20,13,21,14 X_6,16,7,15 X_14,18,15,17 X_22,20,1,19 X_12,21,13,22
Gauss code	1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -11, 7, -9, 8, -4, 9, -6, 10, -7, 11, -10
Dowker-Thistlethwaite code	4 8 10 16 2 18 20 6 14 22 12
Conway Notation	[211,211,2+]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	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 \{1,2\}}
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a6/ThurstonBennequinNumber
Hyperbolic Volume	15.844
A-Polynomial	See Data:K11a6/A-polynomial

[edit Notes for K11a6's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	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 [2,3]}
Rasmussen s-Invariant	-2

[edit Notes for K11a6's four dimensional invariants]

Polynomial invariants

Alexander polynomial	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 2 t^3-13 t^2+32 t-41+32 t^{-1} -13 t^{-2} +2 t^{-3} }
Conway polynomial	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 2 z^6-z^4-2 z^2+1}
2nd Alexander ideal (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 \{1\}}
Determinant and Signature	{ 135, 2 }
Jones polynomial	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^8+5 q^7-10 q^6+15 q^5-20 q^4+22 q^3-21 q^2+18 q-12+7 q^{-1} -3 q^{-2} + q^{-3} }
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^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2+z^2 a^{-2} -z^2 a^{-4} -3 z^2+a^2+2 a^{-2} -2 a^{-4} + a^{-6} -1}
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 z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +8 z^9 a^{-3} +5 z^9 a^{-5} +12 z^8 a^{-2} +18 z^8 a^{-4} +10 z^8 a^{-6} +4 z^8+3 a z^7+4 z^7 a^{-1} +2 z^7 a^{-3} +11 z^7 a^{-5} +10 z^7 a^{-7} +a^2 z^6-26 z^6 a^{-2} -34 z^6 a^{-4} -10 z^6 a^{-6} +5 z^6 a^{-8} -6 z^6-8 a z^5-21 z^5 a^{-1} -33 z^5 a^{-3} -36 z^5 a^{-5} -15 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+14 z^4 a^{-2} +13 z^4 a^{-4} -4 z^4 a^{-6} -5 z^4 a^{-8} -z^4+7 a z^3+20 z^3 a^{-1} +32 z^3 a^{-3} +24 z^3 a^{-5} +5 z^3 a^{-7} +3 a^2 z^2+z^2 a^{-2} +4 z^2 a^{-4} +4 z^2 a^{-6} +4 z^2-2 a z-6 z a^{-1} -8 z a^{-3} -4 z a^{-5} -a^2-2 a^{-2} -2 a^{-4} - a^{-6} -1}
The A2 invariant	Data:K11a6/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a6/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a6 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["K11a6"];

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

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

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 2 t^3-13 t^2+32 t-41+32 t^{-1} -13 t^{-2} +2 t^{-3} }

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

Out[5]= 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 2 z^6-z^4-2 z^2+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]= 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 \{1\}}

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

Out[7]= { 135, 2 }

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

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

Out[8]= 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^8+5 q^7-10 q^6+15 q^5-20 q^4+22 q^3-21 q^2+18 q-12+7 q^{-1} -3 q^{-2} + q^{-3} }

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^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2+z^2 a^{-2} -z^2 a^{-4} -3 z^2+a^2+2 a^{-2} -2 a^{-4} + a^{-6} -1}

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 z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +8 z^9 a^{-3} +5 z^9 a^{-5} +12 z^8 a^{-2} +18 z^8 a^{-4} +10 z^8 a^{-6} +4 z^8+3 a z^7+4 z^7 a^{-1} +2 z^7 a^{-3} +11 z^7 a^{-5} +10 z^7 a^{-7} +a^2 z^6-26 z^6 a^{-2} -34 z^6 a^{-4} -10 z^6 a^{-6} +5 z^6 a^{-8} -6 z^6-8 a z^5-21 z^5 a^{-1} -33 z^5 a^{-3} -36 z^5 a^{-5} -15 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+14 z^4 a^{-2} +13 z^4 a^{-4} -4 z^4 a^{-6} -5 z^4 a^{-8} -z^4+7 a z^3+20 z^3 a^{-1} +32 z^3 a^{-3} +24 z^3 a^{-5} +5 z^3 a^{-7} +3 a^2 z^2+z^2 a^{-2} +4 z^2 a^{-4} +4 z^2 a^{-6} +4 z^2-2 a z-6 z a^{-1} -8 z a^{-3} -4 z a^{-5} -a^2-2 a^{-2} -2 a^{-4} - a^{-6} -1}

Vassiliev invariants

V₂ and V₃:

(-2, -2)

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

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 -8}

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 -16}

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 32}

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 \frac{68}{3}}

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 \frac{76}{3}}

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 128}

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 \frac{512}{3}}

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 \frac{128}{3}}

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 48}

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 -\frac{256}{3}}

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 128}

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 -\frac{544}{3}}

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 -\frac{608}{3}}

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 \frac{3089}{15}}

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 \frac{844}{15}}

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 -\frac{3724}{45}}

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 \frac{463}{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 -\frac{751}{15}}

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 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^rq^j} are shown, along with their alternating sums 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 \chi} (fixed 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 j} , alternation over 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 r} ). The squares with yellow highlighting are those on the "critical diagonals", where 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 j-2r=s+1} or 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 j-2r=s-1} , where 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 s=} 2 is the signature of K11a6. 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

17

1

-1

15

4

13

6

1

-5

11

9

4

5

9

11

6

-5

7

11

9

2

5

10

11

1

3

8

11

-3

1

5

11

6

-1

2

7

-5

-3

1

5

4

-5

2

-2

-7

1

Integral Khovanov Homology

(db, data source)

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 \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	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 i=1}	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 i=3}
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 r=-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 {\mathbb Z}}
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 r=-3}	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 {\mathbb Z}^{2}\oplus{\mathbb Z}_2}	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 {\mathbb Z}}
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 r=-2}	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 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}}	${\mathbb {Z} }^{2}$
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 r=-1}	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 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}}	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 {\mathbb Z}^{5}}
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 r=0}	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 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}}	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 {\mathbb Z}^{8}}
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 r=1}	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 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=2}	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 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}}	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 {\mathbb Z}^{11}}
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 r=3}	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 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}}	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 {\mathbb Z}^{11}}
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 r=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 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=5}	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 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}}	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 {\mathbb Z}^{6}}
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 r=6}	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 {\mathbb Z}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=7}	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 {\mathbb Z}_2}	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 {\mathbb Z}}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

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 \textrm{Include}(\textrm{ColouredJonesM.mhtml})}

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[11, Alternating, 6]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 6]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[16, 8, 17, 7], X[2, 9, 3, 10], X[18, 12, 19, 11], X[20, 13, 21, 14], X[6, 16, 7, 15], X[14, 18, 15, 17], X[22, 20, 1, 19], X[12, 21, 13, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 6]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -11, 7, -9, 8, -4, 9, -6, 10, -7, 11, -10]
In[5]:=	BR[Knot[11, Alternating, 6]]
Out[5]=	BR[Knot[11, Alternating, 6]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 6]][t]
Out[6]=	2 13 32 2 3 -41 + -- - -- + -- + 32 t - 13 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 6]][z]
Out[7]=	2 4 6 1 - 2 z - z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 6], Knot[11, Alternating, 132], Knot[11, Alternating, 352]}
In[9]:=	{KnotDet[Knot[11, Alternating, 6]], KnotSignature[Knot[11, Alternating, 6]]}
Out[9]=	{135, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 6]][q]
Out[10]=	-3 3 7 2 3 4 5 6 -12 + q - -- + - + 18 q - 21 q + 22 q - 20 q + 15 q - 10 q + 2 q q 7 8 5 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 6]}
In[12]:=	A2Invariant[Knot[11, Alternating, 6]][q]
Out[12]=	-10 -6 3 2 2 4 6 8 10 12 -1 + q - q + -- - -- + 4 q - 3 q + 4 q - q - q + q - 4 2 q q 14 16 18 20 22 24 5 q + 4 q - q - q + 3 q - q
In[13]:=	Kauffman[Knot[11, Alternating, 6]][a, z]
Out[13]=	2 -6 2 2 2 4 z 8 z 6 z 2 4 z -1 - a - -- - -- - a - --- - --- - --- - 2 a z + 4 z + ---- + 4 2 5 3 a 6 a a a a a 2 2 3 3 3 3 4 z z 2 2 5 z 24 z 32 z 20 z 3 4 ---- + -- + 3 a z + ---- + ----- + ----- + ----- + 7 a z - z - 4 2 7 5 3 a a a a a a 4 4 4 4 5 5 5 5 5 z 4 z 13 z 14 z 2 4 z 15 z 36 z 33 z ---- - ---- + ----- + ----- - 3 a z + -- - ----- - ----- - ----- - 8 6 4 2 9 7 5 3 a a a a a a a a 5 6 6 6 6 21 z 5 6 5 z 10 z 34 z 26 z 2 6 ----- - 8 a z - 6 z + ---- - ----- - ----- - ----- + a z + a 8 6 4 2 a a a a 7 7 7 7 8 8 8 10 z 11 z 2 z 4 z 7 8 10 z 18 z 12 z ----- + ----- + ---- + ---- + 3 a z + 4 z + ----- + ----- + ----- + 7 5 3 a 6 4 2 a a a a a a 9 9 9 10 10 5 z 8 z 3 z z z ---- + ---- + ---- + --- + --- 5 3 a 4 2 a a a a
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 6]], Vassiliev[3][Knot[11, Alternating, 6]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[11, Alternating, 6]][q, t]
Out[15]=	3 1 2 1 5 2 7 5 q 11 q + 8 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 11 q t + 10 q t + 11 q t + 11 q t + 9 q t + 11 q t + 9 4 11 4 11 5 13 5 13 6 15 6 17 7 6 q t + 9 q t + 4 q t + 6 q t + q t + 4 q t + q t

K11a6

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