T(9,5)

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T(35,2)

Image:T(37,2).jpg

T(37,2)

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(9,5)'s page at Knotilus!

Visit T(9,5)'s page at the original Knot Atlas!

Edit T(9,5) Quick Notes

Edit T(9,5) Further Notes and Views

[edit] Knot presentations

Planar diagram presentation	X_51,37,52,36 X_66,38,67,37 X_9,39,10,38 X_24,40,25,39 X_67,53,68,52 X_10,54,11,53 X_25,55,26,54 X_40,56,41,55 X_11,69,12,68 X_26,70,27,69 X_41,71,42,70 X_56,72,57,71 X_27,13,28,12 X_42,14,43,13 X_57,15,58,14 X_72,16,1,15 X_43,29,44,28 X_58,30,59,29 X_1,31,2,30 X_16,32,17,31 X_59,45,60,44 X_2,46,3,45 X_17,47,18,46 X_32,48,33,47 X_3,61,4,60 X_18,62,19,61 X_33,63,34,62 X_48,64,49,63 X_19,5,20,4 X_34,6,35,5 X_49,7,50,6 X_64,8,65,7 X_35,21,36,20 X_50,22,51,21 X_65,23,66,22 X_8,24,9,23
Gauss code	-19, -22, -25, 29, 30, 31, 32, -36, -3, -6, -9, 13, 14, 15, 16, -20, -23, -26, -29, 33, 34, 35, 36, -4, -7, -10, -13, 17, 18, 19, 20, -24, -27, -30, -33, 1, 2, 3, 4, -8, -11, -14, -17, 21, 22, 23, 24, -28, -31, -34, -1, 5, 6, 7, 8, -12, -15, -18, -21, 25, 26, 27, 28, -32, -35, -2, -5, 9, 10, 11, 12, -16
Dowker-Thistlethwaite code	30 60 -34 -64 38 68 -42 -72 46 4 -50 -8 54 12 -58 -16 62 20 -66 -24 70 28 -2 -32 6 36 -10 -40 14 44 -18 -48 22 52 -26 -56

Braid presentation

[edit] Polynomial invariants

Alexander polynomial	$t 16 - t 15 + t 11 - t 10 + t 7 - t 5 + t 2 -1 + t -2 - t -5 + t -7 - t -10 + t -11 - t -15 + t -16$
Conway polynomial	$z 32 + 31 z 30 + 434 z 28 + 3627 z 26 + 20150 z 24 + 78431 z 22 + 219625 z 20 + 447240 z 18 + 661810 z 16 + 703851 z 14 + 526423 z 12 + 267421 z 10 + 87560 z 8 + 17094 z 6 + 1772 z 4 + 80 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 1, 24 }
Jones polynomial	$- q 28 - q 26 + q 20 + q 18 + q 16$
HOMFLY-PT polynomial (db, data sources)	Data:T(9,5)/HOMFLYPT Polynomial
Kauffman polynomial (db, data sources)	Data:T(9,5)/Kauffman Polynomial
The A2 invariant	Data:T(9,5)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(9,5)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(9,5) 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(9,5)"];

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

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

Out[4]= $t 16 - t 15 + t 11 - t 10 + t 7 - t 5 + t 2 -1 + t -2 - t -5 + t -7 - t -10 + t -11 - t -15 + t -16$

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

Out[5]= $z 32 + 31 z 30 + 434 z 28 + 3627 z 26 + 20150 z 24 + 78431 z 22 + 219625 z 20 + 447240 z 18 + 661810 z 16 + 703851 z 14 + 526423 z 12 + 267421 z 10 + 87560 z 8 + 17094 z 6 + 1772 z 4 + 80 z 2 + 1$

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]= { 1, 24 }

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

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

Out[8]= $- q 28 - q 26 + q 20 + q 18 + q 16$

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

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

Out[9]= Data:T(9,5)/HOMFLYPT Polynomial

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

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

Out[10]= Data:T(9,5)/Kauffman Polynomial

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

Same Alexander/Conway Polynomial: {}

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

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(9,5)"];

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]= { $t 16 - t 15 + t 11 - t 10 + t 7 - t 5 + t 2 -1 + t -2 - t -5 + t -7 - t -10 + t -11 - t -15 + t -16$ , $- q 28 - q 26 + q 20 + q 18 + q 16$ }

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]= {}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

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

Out[6]= {}

[edit] Vassiliev invariants

V₂ and V₃:

(80, 600)

[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 =

24 is the signature of T(9,5). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 19$	$i = 21$	$i = 23$	$i = 25$	$i = 27$	$i = 29$	$i = 31$	$i = 33$
$r = 0$							${\mathbb Z}$	${\mathbb Z}$
$r = 1$
$r = 2$							${\mathbb Z}$
$r = 3$							${\mathbb Z}_2$	${\mathbb Z}$
$r = 4$						${\mathbb Z}$	${\mathbb Z}$
$r = 5$							${\mathbb Z}$	${\mathbb Z}$
$r = 6$					${\mathbb Z}$	${\mathbb Z}$
$r = 7$					${\mathbb Z}_2$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 8$				${\mathbb Z}$	${\mathbb Z}^{2}$
$r = 9$					${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = 10$				${\mathbb Z}^{2}$	${\mathbb Z}_2$	${\mathbb Z}_2$
$r = 11$				${\mathbb Z}_2^{2}\oplus{\mathbb Z}_5$	${\mathbb Z}^{3}$
$r = 12$			${\mathbb Z}^{2}$	${\mathbb Z}^{2}$	${\mathbb Z}_2\oplus{\mathbb Z}_5$	${\mathbb Z}$
$r = 13$			${\mathbb Z}_2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = 14$		${\mathbb Z}$	${\mathbb Z}^{2}$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 15$		${\mathbb Z}_2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = 16$		${\mathbb Z}^{2}$	${\mathbb Z}_2$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 17$		${\mathbb Z}_2\oplus{\mathbb Z}_4\oplus{\mathbb Z}_5$	${\mathbb Z}^{3}$
$r = 18$	${\mathbb Z}$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}_2^{2}\oplus{\mathbb Z}_5$	${\mathbb Z}$
$r = 19$	${\mathbb Z}_2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 20$	${\mathbb Z}$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 21$	${\mathbb Z}_4$	${\mathbb Z}$
$r = 22$	${\mathbb Z}_2$	${\mathbb Z}_2$