T(9,2)

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T(4,3)

T(5,3)

1 Knot presentations
2 Polynomial invariants
3 "Similar" Knots (within the Atlas)
4 Vassiliev invariants
5 Khovanov Homology
6 Computer Talk
7 Modifying This Page

See other torus knots

Visit T(9,2)'s page at Knotilus!

Visit T(9,2)'s page at the original Knot Atlas!

Edit T(9,2) Quick Notes

[edit] Knot presentations

Planar diagram presentation	X_7,17,8,16 X_17,9,18,8 X_9,1,10,18 X_1,11,2,10 X_11,3,12,2 X_3,13,4,12 X_13,5,14,4 X_5,15,6,14 X_15,7,16,6
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

[edit] Polynomial invariants

Alexander polynomial	$t 4 - t 3 + t 2 - t + 1- t -1 + t -2 - t -3 + t -4$
Conway polynomial	$z 8 + 7 z 6 + 15 z 4 + 10 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 9, 8 }
Jones polynomial	$- q 13 + q 12 - q 11 + q 10 - q 9 + q 8 - q 7 + q 6 + q 4$
HOMFLY-PT polynomial (db, data sources)	$z 8 a -8 + 8 z 6 a -8 - z 6 a -10 + 21 z 4 a -8 -6 z 4 a -10 + 20 z 2 a -8 -10 z 2 a -10 + 5 a -8 -4 a -10$
Kauffman polynomial (db, data sources)	$z 8 a -8 + z 8 a -10 + z 7 a -9 + z 7 a -11 -8 z 6 a -8 -7 z 6 a -10 + z 6 a -12 -6 z 5 a -9 -5 z 5 a -11 + z 5 a -13 + 21 z 4 a -8 + 16 z 4 a -10 -4 z 4 a -12 + z 4 a -14 + 10 z 3 a -9 + 6 z 3 a -11 -3 z 3 a -13 + z 3 a -15 -20 z 2 a -8 -14 z 2 a -10 + 3 z 2 a -12 -2 z 2 a -14 + z 2 a -16 -4 z a -9 - z a -11 + z a -13 - z a -15 + z a -17 + 5 a -8 + 4 a -10$
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.

In[3]:= K = Knot["T(9,2)"];

In[4]:= Alexander[K][t]

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

Out[4]= $t 4 - t 3 + t 2 - t + 1- t -1 + t -2 - t -3 + t -4$

In[5]:= Conway[K][z]

Out[5]= $z 8 + 7 z 6 + 15 z 4 + 10 z 2 + 1$

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]= { 9, 8 }

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

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

Out[8]= $- q 13 + q 12 - q 11 + q 10 - q 9 + q 8 - q 7 + q 6 + q 4$

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

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

Out[9]= $z 8 a -8 + 8 z 6 a -8 - z 6 a -10 + 21 z 4 a -8 -6 z 4 a -10 + 20 z 2 a -8 -10 z 2 a -10 + 5 a -8 -4 a -10$

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

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

Out[10]= $z 8 a -8 + z 8 a -10 + z 7 a -9 + z 7 a -11 -8 z 6 a -8 -7 z 6 a -10 + z 6 a -12 -6 z 5 a -9 -5 z 5 a -11 + z 5 a -13 + 21 z 4 a -8 + 16 z 4 a -10 -4 z 4 a -12 + z 4 a -14 + 10 z 3 a -9 + 6 z 3 a -11 -3 z 3 a -13 + z 3 a -15 -20 z 2 a -8 -14 z 2 a -10 + 3 z 2 a -12 -2 z 2 a -14 + z 2 a -16 -4 z a -9 - z a -11 + z a -13 - z a -15 + z a -17 + 5 a -8 + 4 a -10$

[edit] "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_1,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {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.

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]= { $t 4 - t 3 + t 2 - t + 1- t -1 + t -2 - t -3 + t -4$ , $- q 13 + q 12 - q 11 + q 10 - q 9 + q 8 - q 7 + q 6 + q 4$ }

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,}

[edit] Vassiliev invariants

V₂ and V₃:

(10, 30)

[edit] Khovanov Homology

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

8 is the signature of T(9,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 7$	$i = 9$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$
$r = 1$
$r = 2$	${\mathbb Z}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 4$	${\mathbb Z}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 6$	${\mathbb Z}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 8$	${\mathbb Z}$
$r = 9$	${\mathbb Z}_2$	${\mathbb Z}$