T(5,4)

[[Image:T(7,3).{{{ext}}}|80px|link=T(7,3)]] T(7,3)

[[Image:T(15,2).{{{ext}}}|80px|link=T(15,2)]] T(15,2)

Visit T(5,4)'s page at Knotilus!

Visit T(5,4)'s page at the original Knot Atlas!

Knot presentations

Planar diagram presentation	ToString[X[17, 25, 18, 24], FormatType -> HTMLForm]<> <>ToString[X[10, 26, 11, 25], FormatType -> HTMLForm]<> <>ToString[X[3, 27, 4, 26], FormatType -> HTMLForm]<> <>ToString[X[11, 19, 12, 18], FormatType -> HTMLForm]<> <>ToString[X[4, 20, 5, 19], FormatType -> HTMLForm]<> <>ToString[X[27, 21, 28, 20], FormatType -> HTMLForm]<> <>ToString[X[5, 13, 6, 12], FormatType -> HTMLForm]<> <>ToString[X[28, 14, 29, 13], FormatType -> HTMLForm]<> <>ToString[X[21, 15, 22, 14], FormatType -> HTMLForm]<> <>ToString[X[29, 7, 30, 6], FormatType -> HTMLForm]<> <>ToString[X[22, 8, 23, 7], FormatType -> HTMLForm]<> <>ToString[X[15, 9, 16, 8], FormatType -> HTMLForm]<> <>ToString[X[23, 1, 24, 30], FormatType -> HTMLForm]<> <>ToString[X[16, 2, 17, 1], FormatType -> HTMLForm]<> <>ToString[X[9, 3, 10, 2], FormatType -> HTMLForm]<>
Gauss code	-1, 7, -2, 1, -3, 5, -4, 6, -7, 2, -6, 3, -5, 4
Dowker-Thistlethwaite code	4 10 12 14 2 8 6

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	2
Bridge index (super bridge index)	2 (4)
Nakanishi index	1

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^6-t^5+t^2-1+ t^{-2} - t^{-5} + t^{-6} }[/math]
Conway polynomial	[math]\displaystyle{ z^{12}+11 z^{10}+44 z^8+77 z^6+56 z^4+15 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 5, 8 }
Jones polynomial	[math]\displaystyle{ -q^{13}-q^{11}+q^{10}+q^8+q^6 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^{12} a^{-12} +12 z^{10} a^{-12} -z^{10} a^{-14} +55 z^8 a^{-12} -11 z^8 a^{-14} +121 z^6 a^{-12} -45 z^6 a^{-14} +z^6 a^{-16} +133 z^4 a^{-12} -84 z^4 a^{-14} +7 z^4 a^{-16} +70 z^2 a^{-12} -70 z^2 a^{-14} +15 z^2 a^{-16} +14 a^{-12} -21 a^{-14} +9 a^{-16} - a^{-18} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{12} a^{-12} +z^{12} a^{-14} +z^{11} a^{-13} +z^{11} a^{-15} -12 z^{10} a^{-12} -12 z^{10} a^{-14} -11 z^9 a^{-13} -11 z^9 a^{-15} +55 z^8 a^{-12} +56 z^8 a^{-14} +z^8 a^{-16} +45 z^7 a^{-13} +46 z^7 a^{-15} +z^7 a^{-17} -121 z^6 a^{-12} -129 z^6 a^{-14} -8 z^6 a^{-16} -84 z^5 a^{-13} -91 z^5 a^{-15} -7 z^5 a^{-17} +133 z^4 a^{-12} +154 z^4 a^{-14} +21 z^4 a^{-16} +70 z^3 a^{-13} +84 z^3 a^{-15} +14 z^3 a^{-17} -70 z^2 a^{-12} -91 z^2 a^{-14} -22 z^2 a^{-16} -z^2 a^{-18} -21 z a^{-13} -28 z a^{-15} -8 z a^{-17} -z a^{-19} +14 a^{-12} +21 a^{-14} +9 a^{-16} + a^{-18} }[/math]
The A2 invariant	Data:T(5,4)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(5,4)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(5,4) 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["T(5,4)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^6-t^5+t^2-1+ t^{-2} - t^{-5} + t^{-6} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^{12}+11 z^{10}+44 z^8+77 z^6+56 z^4+15 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]=

{ 5, 8 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^{13}-q^{11}+q^{10}+q^8+q^6 }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^{12} a^{-12} +12 z^{10} a^{-12} -z^{10} a^{-14} +55 z^8 a^{-12} -11 z^8 a^{-14} +121 z^6 a^{-12} -45 z^6 a^{-14} +z^6 a^{-16} +133 z^4 a^{-12} -84 z^4 a^{-14} +7 z^4 a^{-16} +70 z^2 a^{-12} -70 z^2 a^{-14} +15 z^2 a^{-16} +14 a^{-12} -21 a^{-14} +9 a^{-16} - a^{-18} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^{12} a^{-12} +z^{12} a^{-14} +z^{11} a^{-13} +z^{11} a^{-15} -12 z^{10} a^{-12} -12 z^{10} a^{-14} -11 z^9 a^{-13} -11 z^9 a^{-15} +55 z^8 a^{-12} +56 z^8 a^{-14} +z^8 a^{-16} +45 z^7 a^{-13} +46 z^7 a^{-15} +z^7 a^{-17} -121 z^6 a^{-12} -129 z^6 a^{-14} -8 z^6 a^{-16} -84 z^5 a^{-13} -91 z^5 a^{-15} -7 z^5 a^{-17} +133 z^4 a^{-12} +154 z^4 a^{-14} +21 z^4 a^{-16} +70 z^3 a^{-13} +84 z^3 a^{-15} +14 z^3 a^{-17} -70 z^2 a^{-12} -91 z^2 a^{-14} -22 z^2 a^{-16} -z^2 a^{-18} -21 z a^{-13} -28 z a^{-15} -8 z a^{-17} -z a^{-19} +14 a^{-12} +21 a^{-14} +9 a^{-16} + a^{-18} }[/math]

Vassiliev invariants

V2 and V3:

(15, 50)

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 Data:T(5,4)/V 2,1 Data:T(5,4)/V 3,1 Data:T(5,4)/V 4,1 Data:T(5,4)/V 4,2 Data:T(5,4)/V 4,3 Data:T(5,4)/V 5,1 Data:T(5,4)/V 5,2 Data:T(5,4)/V 5,3 Data:T(5,4)/V 5,4 Data:T(5,4)/V 6,1 Data:T(5,4)/V 6,2 Data:T(5,4)/V 6,3 Data:T(5,4)/V 6,4 Data:T(5,4)/V 6,5 Data:T(5,4)/V 6,6 Data:T(5,4)/V 6,7 Data:T(5,4)/V 6,8 Data:T(5,4)/V 6,9

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.