T(7,2)

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Visit T(7,2)'s page at the original Knot Atlas!

Knot presentations

Planar diagram presentation	X_5,13,6,12 X_13,7,14,6 X_7,1,8,14 X₁₉₂₈ X_9,3,10,2 X_3,11,4,10 X_11,5,12,4
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	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^3-t^2+t-1+ t^{-1} - t^{-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 z^6+5 z^4+6 z^2+1}
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 7, 6 }
Jones polynomial	$-q^{10}+q^{9}-q^{8}+q^{7}-q^{6}+q^{5}+q^{3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-6}+6z^{4}a^{-6}-z^{4}a^{-8}+10z^{2}a^{-6}-4z^{2}a^{-8}+4a^{-6}-3a^{-8}$
Kauffman polynomial (db, data sources)	$z^{6}a^{-6}+z^{6}a^{-8}+z^{5}a^{-7}+z^{5}a^{-9}-6z^{4}a^{-6}-5z^{4}a^{-8}+z^{4}a^{-10}-4z^{3}a^{-7}-3z^{3}a^{-9}+z^{3}a^{-11}+10z^{2}a^{-6}+7z^{2}a^{-8}-2z^{2}a^{-10}+z^{2}a^{-12}+3za^{-7}+za^{-9}-za^{-11}+za^{-13}-4a^{-6}-3a^{-8}$
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.

In[3]:= K = Knot["T(7,2)"];

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 t^3-t^2+t-1+ t^{-1} - t^{-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 z^6+5 z^4+6 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]= $\{1\}$

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

Out[7]= { 7, 6 }

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

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

Out[8]= $-q^{10}+q^{9}-q^{8}+q^{7}-q^{6}+q^{5}+q^{3}$

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

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

Out[9]= $z^{6}a^{-6}+6z^{4}a^{-6}-z^{4}a^{-8}+10z^{2}a^{-6}-4z^{2}a^{-8}+4a^{-6}-3a^{-8}$

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

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

Out[10]= $z^{6}a^{-6}+z^{6}a^{-8}+z^{5}a^{-7}+z^{5}a^{-9}-6z^{4}a^{-6}-5z^{4}a^{-8}+z^{4}a^{-10}-4z^{3}a^{-7}-3z^{3}a^{-9}+z^{3}a^{-11}+10z^{2}a^{-6}+7z^{2}a^{-8}-2z^{2}a^{-10}+z^{2}a^{-12}+3za^{-7}+za^{-9}-za^{-11}+za^{-13}-4a^{-6}-3a^{-8}$

Vassiliev invariants

V₂ and V₃:

(6, 14)

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

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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (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=} 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

0

15

0

13

1

0

11

0

9

1

7

1

5

1

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[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 2 3 -1 + t - t + - + t - t + t t
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 9 2 13 3 13 4 17 5 17 6 21 7 q + q + q t + q t + q t + q t + q t + q t

This category should contain all the individual knots pages, like 7_5, K11n67, L8a2 and T(5,3)

T(7,2)

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Knot presentations

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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