T(9,2)

T(4,3)

T(5,3)

	See other torus knots Visit T(9,2) at Knotilus!
	Edit T(9,2) Quick Notes See also 9_1.

Edit T(9,2) Further Notes and Views

Knot presentations

Planar diagram presentation	X7,17,8,16 X17,9,18,8 X9,1,10,18 X1,11,2,10 X11,3,12,2 X3,13,4,12 X13,5,14,4 X5,15,6,14 X15,7,16,6
Gauss code	-4, 5, -6, 7, -8, 9, -1, 2, -3, 4, -5, 6, -7, 8, -9, 1, -2, 3
Dowker-Thistlethwaite code	10 12 14 16 18 2 4 6 8

Braid presentation

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-t^3+t^2-t+1- t^{-1} + t^{-2} - t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+7 z^6+15 z^4+10 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 9, 8 }
Jones polynomial	[math]\displaystyle{ -q^{13}+q^{12}-q^{11}+q^{10}-q^9+q^8-q^7+q^6+q^4 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-8} +8 z^6 a^{-8} -z^6 a^{-10} +21 z^4 a^{-8} -6 z^4 a^{-10} +20 z^2 a^{-8} -10 z^2 a^{-10} +5 a^{-8} -4 a^{-10} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-9} +z^7 a^{-11} -8 z^6 a^{-8} -7 z^6 a^{-10} +z^6 a^{-12} -6 z^5 a^{-9} -5 z^5 a^{-11} +z^5 a^{-13} +21 z^4 a^{-8} +16 z^4 a^{-10} -4 z^4 a^{-12} +z^4 a^{-14} +10 z^3 a^{-9} +6 z^3 a^{-11} -3 z^3 a^{-13} +z^3 a^{-15} -20 z^2 a^{-8} -14 z^2 a^{-10} +3 z^2 a^{-12} -2 z^2 a^{-14} +z^2 a^{-16} -4 z a^{-9} -z a^{-11} +z a^{-13} -z a^{-15} +z a^{-17} +5 a^{-8} +4 a^{-10} }[/math]
The A2 invariant	Data:T(9,2)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(9,2)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(9,2) 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(9,2)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-t^3+t^2-t+1- t^{-1} + t^{-2} - t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+7 z^6+15 z^4+10 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]=

{ 9, 8 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^{13}+q^{12}-q^{11}+q^{10}-q^9+q^8-q^7+q^6+q^4 }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^8 a^{-8} +8 z^6 a^{-8} -z^6 a^{-10} +21 z^4 a^{-8} -6 z^4 a^{-10} +20 z^2 a^{-8} -10 z^2 a^{-10} +5 a^{-8} -4 a^{-10} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-9} +z^7 a^{-11} -8 z^6 a^{-8} -7 z^6 a^{-10} +z^6 a^{-12} -6 z^5 a^{-9} -5 z^5 a^{-11} +z^5 a^{-13} +21 z^4 a^{-8} +16 z^4 a^{-10} -4 z^4 a^{-12} +z^4 a^{-14} +10 z^3 a^{-9} +6 z^3 a^{-11} -3 z^3 a^{-13} +z^3 a^{-15} -20 z^2 a^{-8} -14 z^2 a^{-10} +3 z^2 a^{-12} -2 z^2 a^{-14} +z^2 a^{-16} -4 z a^{-9} -z a^{-11} +z a^{-13} -z a^{-15} +z a^{-17} +5 a^{-8} +4 a^{-10} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_1,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {9_1,}

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`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["T(9,2)"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

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

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

Out[4]=

{ [math]\displaystyle{ t^4-t^3+t^2-t+1- t^{-1} + t^{-2} - t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q^{13}+q^{12}-q^{11}+q^{10}-q^9+q^8-q^7+q^6+q^4 }[/math] }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{9_1,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{9_1,}

Vassiliev invariants

V2 and V3:

(10, 30)

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(9,2)/V 2,1 Data:T(9,2)/V 3,1 Data:T(9,2)/V 4,1 Data:T(9,2)/V 4,2 Data:T(9,2)/V 4,3 Data:T(9,2)/V 5,1 Data:T(9,2)/V 5,2 Data:T(9,2)/V 5,3 Data:T(9,2)/V 5,4 Data:T(9,2)/V 6,1 Data:T(9,2)/V 6,2 Data:T(9,2)/V 6,3 Data:T(9,2)/V 6,4 Data:T(9,2)/V 6,5 Data:T(9,2)/V 6,6 Data:T(9,2)/V 6,7 Data:T(9,2)/V 6,8 Data:T(9,2)/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]8 is the signature of T(9,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 χ 27                   1 -1 25                     0 23               1 1   0 21                     0 19           1 1       0 17                     0 15       1 1           0 13                     0 11     1               1 9 1                   1 7 1                   1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=7 }[/math] [math]\displaystyle{ i=9 }[/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] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=9 }[/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.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Torus Knot Page master template (intermediate). See/edit the Torus Knot_Splice_Base (expert). Back to the top.

T(4,3) T(5,3)