T(7,2)

T(5,2)

T(4,3)

	Visit [[[:Template:KnotilusURL]] T(7,2)'s page] at Knotilus! Visit T(7,2)'s page at the original Knot Atlas!
	See also 7_1.

T(7,2) Further Notes and Views

Knot presentations

Planar diagram presentation	X5,13,6,12 X13,7,14,6 X7,1,8,14 X1928 X9,3,10,2 X3,11,4,10 X11,5,12,4
Gauss code	-4, 5, -6, 7, -1, 2, -3, 4, -5, 6, -7, 1, -2, 3
Dowker-Thistlethwaite code	8 10 12 14 2 4 6
Conway Notation	Data:T(7,2)/Conway Notation

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-t^2+t-1+ t^{-1} - t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+5 z^4+6 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 7, 6 }
Jones polynomial	[math]\displaystyle{ -q^{10}+q^9-q^8+q^7-q^6+q^5+q^3 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-6} +6 z^4 a^{-6} -z^4 a^{-8} +10 z^2 a^{-6} -4 z^2 a^{-8} +4 a^{-6} -3 a^{-8} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-6} +z^6 a^{-8} +z^5 a^{-7} +z^5 a^{-9} -6 z^4 a^{-6} -5 z^4 a^{-8} +z^4 a^{-10} -4 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} +10 z^2 a^{-6} +7 z^2 a^{-8} -2 z^2 a^{-10} +z^2 a^{-12} +3 z a^{-7} +z a^{-9} -z a^{-11} +z a^{-13} -4 a^{-6} -3 a^{-8} }[/math]
The A2 invariant	Data:T(7,2)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(7,2)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(7,2) 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(7,2)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^6+5 z^4+6 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]=

{ 7, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

Vassiliev invariants

V2 and V3:

(6, 14)

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

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]6 is the signature of T(7,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ i=7 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/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.

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math]

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[TorusKnot[7, 2]]
Out[2]=	7
In[3]:=	PD[TorusKnot[7, 2]]
Out[3]=	PD[X[5, 13, 6, 12], X[13, 7, 14, 6], X[7, 1, 8, 14], X[1, 9, 2, 8], X[9, 3, 10, 2], X[3, 11, 4, 10], X[11, 5, 12, 4]]
In[4]:=	GaussCode[TorusKnot[7, 2]]
Out[4]=	GaussCode[-4, 5, -6, 7, -1, 2, -3, 4, -5, 6, -7, 1, -2, 3]
In[5]:=	BR[TorusKnot[7, 2]]
Out[5]=	BR[2, {1, 1, 1, 1, 1, 1, 1}]
In[6]:=	alex = Alexander[TorusKnot[7, 2]][t]
Out[6]=	-3 -2 1 -1 + Alternating - Alternating + ----------- + Alternating - Alternating 2 3 Alternating + Alternating
In[7]:=	Conway[TorusKnot[7, 2]][z]
Out[7]=	2 4 6 1 + 6 z + 5 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[7, 1]}
In[9]:=	{KnotDet[TorusKnot[7, 2]], KnotSignature[TorusKnot[7, 2]]}
Out[9]=	{7, 6}
In[10]:=	J=Jones[TorusKnot[7, 2]][q]
Out[10]=	3 5 6 7 8 9 10 q + q - q + q - q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[7, 1]}
In[12]:=	A2Invariant[TorusKnot[7, 2]][q]
Out[12]=	10 12 14 16 18 26 28 30 q + q + 2 q + q + q - q - q - q
In[13]:=	Kauffman[TorusKnot[7, 2]][a, z]
Out[13]=	2 2 2 2 3 -3 4 z z z 3 z z 2 z 7 z 10 z z -- - -- + --- - --- + -- + --- + --- - ---- + ---- + ----- + --- - 8 6 13 11 9 7 12 10 8 6 11 a a a a a a a a a a a 3 3 4 4 4 5 5 6 6 3 z 4 z z 5 z 6 z z z z z ---- - ---- + --- - ---- - ---- + -- + -- + -- + -- 9 7 10 8 6 9 7 8 6 a a a a a a a a a
In[14]:=	{Vassiliev[2][TorusKnot[7, 2]], Vassiliev[3][TorusKnot[7, 2]]}
Out[14]=	{0, 14}
In[15]:=	Kh[TorusKnot[7, 2]][q, t]
Out[15]=	5 7 2 9 3 13 4 13 q + q + Alternating q + Alternating q + Alternating q + 5 17 6 17 7 21 Alternating q + Alternating q + Alternating q