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See other torus knots
Visit T(5,2) at Knotilus!
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| Edit T(5,2) Quick Notes
An interlaced pentagram, this is known variously as the "Cinquefoil Knot", after certain herbs and shrubs of the rose family which have 5-lobed leaves and 5-petaled flowers (see e.g. [4]),
as the "Pentafoil Knot" (visit Bert Jagers' pentafoil page),
as the "Double Overhand Knot", as 5_1, or finally as the torus knot T(5,2).
When taken off the post the strangle knot (hitch) of practical knot tying deforms to 5_1
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Edit T(5,2) Further Notes and Views
 A kolam of a 2x3 dot array |
 The VISA Interlink Logo [1] |
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 A pentagonal table by Bob Mackay [2] |
 The Utah State Parks logo |
 As impossible object ("Penrose" pentagram) |
 Folded ribbon which is single-sided (more complex version of Möbius Strip). |
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 Alternate pentagram of intersecting circles. |
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 Partial view of US bicentennial logo on a shirt seen in Lisboa [3] |
 Non-prime knot with two 5_1 configurations on a closed loop. |
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 Sum of two 5_1s, Vienna, orthodox church |
This sentence was last edited by Dror.
Sometime later, Scott added this sentence.
Knot presentations
| Planar diagram presentation
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X3948 X9,5,10,4 X5,1,6,10 X1726 X7382
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| Gauss code
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-4, 5, -1, 2, -3, 4, -5, 1, -2, 3
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| Dowker-Thistlethwaite code
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6 8 10 2 4
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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 t^2-t+1- t^{-1} + t^{-2} }
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| 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 z^4+3 z^2+1}
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| 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\}}
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| Determinant and Signature |
{ 5, 4 } |
| 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^7+q^6-q^5+q^4+q^2}
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| 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^4 a^{-4} +4 z^2 a^{-4} -z^2 a^{-6} +3 a^{-4} -2 a^{-6} }
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| 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^4 a^{-4} +z^4 a^{-6} +z^3 a^{-5} +z^3 a^{-7} -4 z^2 a^{-4} -3 z^2 a^{-6} +z^2 a^{-8} -2 z a^{-5} -z a^{-7} +z a^{-9} +3 a^{-4} +2 a^{-6} }
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| The A2 invariant |
Data:T(5,2)/QuantumInvariant/A2/1,0 |
| The G2 invariant |
Data:T(5,2)/QuantumInvariant/G2/1,0 |
Further 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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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In[3]:=
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K = Knot["T(5,2)"];
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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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^2-t+1- t^{-1} + t^{-2} }
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Out[5]=
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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^4+3 z^2+1}
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In[6]:=
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Alexander[K, 2][t]
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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.
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Out[6]=
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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\}}
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In[7]:=
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{KnotDet[K], KnotSignature[K]}
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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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^7+q^6-q^5+q^4+q^2}
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In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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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^4 a^{-4} +4 z^2 a^{-4} -z^2 a^{-6} +3 a^{-4} -2 a^{-6} }
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In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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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^4 a^{-4} +z^4 a^{-6} +z^3 a^{-5} +z^3 a^{-7} -4 z^2 a^{-4} -3 z^2 a^{-6} +z^2 a^{-8} -2 z a^{-5} -z a^{-7} +z a^{-9} +3 a^{-4} +2 a^{-6} }
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"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial:
{5_1, 10_132,}
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}}
):
{5_1, 10_132,}
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`
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In[3]:=
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K = Knot["T(5,2)"];
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In[4]:=
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{A = Alexander[K][t], J = Jones[K][q]}
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[4]=
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{ 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^2-t+1- t^{-1} + t^{-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 -q^7+q^6-q^5+q^4+q^2}
}
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In[5]:=
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DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
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KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
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KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
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In[6]:=
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DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
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|
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KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
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| V2,1 through V6,9:
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| V2,1
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V3,1
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V4,1
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V4,2
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V4,3
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V5,1
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V5,2
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V5,3
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V5,4
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V6,1
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V6,2
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V6,3
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V6,4
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V6,5
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V6,6
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V6,7
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V6,8
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V6,9
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| Data:T(5,2)/V 2,1
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Data:T(5,2)/V 3,1
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Data:T(5,2)/V 4,1
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Data:T(5,2)/V 4,2
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Data:T(5,2)/V 4,3
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Data:T(5,2)/V 5,1
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Data:T(5,2)/V 5,2
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Data:T(5,2)/V 5,3
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Data:T(5,2)/V 5,4
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Data:T(5,2)/V 6,1
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Data:T(5,2)/V 6,2
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Data:T(5,2)/V 6,3
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Data:T(5,2)/V 6,4
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Data:T(5,2)/V 6,5
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Data:T(5,2)/V 6,6
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Data:T(5,2)/V 6,7
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Data:T(5,2)/V 6,8
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Data:T(5,2)/V 6,9
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V2,1 through V6,9 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.
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