T(7,6)

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T(17,3)

T(35,2)

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(7,6)'s page at Knotilus!

Visit T(7,6)'s page at the original Knot Atlas!

Edit T(7,6) Quick Notes

Edit T(7,6) Further Notes and Views

[edit] Knot presentations

Planar diagram presentation	X_53,65,54,64 X_42,66,43,65 X_31,67,32,66 X_20,68,21,67 X_9,69,10,68 X_43,55,44,54 X_32,56,33,55 X_21,57,22,56 X_10,58,11,57 X_69,59,70,58 X_33,45,34,44 X_22,46,23,45 X_11,47,12,46 X_70,48,1,47 X_59,49,60,48 X_23,35,24,34 X_12,36,13,35 X_1,37,2,36 X_60,38,61,37 X_49,39,50,38 X_13,25,14,24 X_2,26,3,25 X_61,27,62,26 X_50,28,51,27 X_39,29,40,28 X_3,15,4,14 X_62,16,63,15 X_51,17,52,16 X_40,18,41,17 X_29,19,30,18 X_63,5,64,4 X_52,6,53,5 X_41,7,42,6 X_30,8,31,7 X_19,9,20,8
Gauss code	-18, -22, -26, 31, 32, 33, 34, 35, -5, -9, -13, -17, -21, 26, 27, 28, 29, 30, -35, -4, -8, -12, -16, 21, 22, 23, 24, 25, -30, -34, -3, -7, -11, 16, 17, 18, 19, 20, -25, -29, -33, -2, -6, 11, 12, 13, 14, 15, -20, -24, -28, -32, -1, 6, 7, 8, 9, 10, -15, -19, -23, -27, -31, 1, 2, 3, 4, 5, -10, -14
Dowker-Thistlethwaite code	36 14 -52 -30 68 46 24 -62 -40 8 56 34 -2 -50 18 66 44 -12 -60 28 6 54 -22 -70 38 16 64 -32 -10 48 26 4 -42 -20 58

Braid presentation

[edit] Polynomial invariants

Alexander polynomial	$t 15 - t 14 + t 9 - t 7 + t 3 -1 + t -3 - t -7 + t -9 - t -14 + t -15$
Conway polynomial	$z 30 + 29 z 28 + 377 z 26 + 2900 z 24 + 14674 z 22 + 51359 z 20 + 127282 z 18 + 224826 z 16 + 281144 z 14 + 244074 z 12 + 142208 z 10 + 52844 z 8 + 11649 z 6 + 1365 z 4 + 70 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 7, 18 }
Jones polynomial	$- q 26 - q 24 - q 22 + q 21 + q 19 + q 17 + q 15$
HOMFLY-PT polynomial (db, data sources)	Data:T(7,6)/HOMFLYPT Polynomial
Kauffman polynomial (db, data sources)	Data:T(7,6)/Kauffman Polynomial
The A2 invariant	Data:T(7,6)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(7,6)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(7,6) 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(7,6)"];

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

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

Out[4]= $t 15 - t 14 + t 9 - t 7 + t 3 -1 + t -3 - t -7 + t -9 - t -14 + t -15$

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

Out[5]= $z 30 + 29 z 28 + 377 z 26 + 2900 z 24 + 14674 z 22 + 51359 z 20 + 127282 z 18 + 224826 z 16 + 281144 z 14 + 244074 z 12 + 142208 z 10 + 52844 z 8 + 11649 z 6 + 1365 z 4 + 70 z 2 + 1$

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, 18 }

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

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

Out[8]= $- q 26 - q 24 - q 22 + q 21 + q 19 + q 17 + q 15$

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

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

Out[9]= Data:T(7,6)/HOMFLYPT Polynomial

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

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

Out[10]= Data:T(7,6)/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(7,6)"];

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]= { $t 15 - t 14 + t 9 - t 7 + t 3 -1 + t -3 - t -7 + t -9 - t -14 + t -15$ , $- q 26 - q 24 - q 22 + q 21 + q 19 + q 17 + q 15$ }

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:

(70, 490)

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.

[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 =

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

\		r
	\
j		\

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 17$	$i = 19$	$i = 21$	$i = 23$	$i = 25$	$i = 27$	$i = 29$	$i = 31$
$r = 0$							${\mathbb Z}$	${\mathbb Z}$
$r = 1$
$r = 2$							${\mathbb Z}$
$r = 3$							${\mathbb Z}_2$	${\mathbb Z}$
$r = 4$						${\mathbb Z}$	${\mathbb Z}$
$r = 5$							${\mathbb Z}$	${\mathbb Z}$
$r = 6$					${\mathbb Z}$	${\mathbb Z}$
$r = 7$					${\mathbb Z}_2$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 8$				${\mathbb Z}$	${\mathbb Z}^{2}$
$r = 9$					${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = 10$			${\mathbb Z}$	${\mathbb Z}^{2}$	${\mathbb Z}_2$	${\mathbb Z}_2$
$r = 11$			${\mathbb Z}_2$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = 12$			${\mathbb Z}^{2}$	${\mathbb Z}$	${\mathbb Z}_2\oplus{\mathbb Z}_5$	${\mathbb Z}$
$r = 13$			${\mathbb Z}_2^{2}$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 14$		${\mathbb Z}$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}_2^{2}$	${\mathbb Z}$
$r = 15$		${\mathbb Z}_2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = 16$	${\mathbb Z}$	${\mathbb Z}$	${\mathbb Z}_2$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 17$	${\mathbb Z}_2$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 18$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}_2^{2}$	${\mathbb Z}_4$
$r = 19$	${\mathbb Z}_2\oplus{\mathbb Z}_3$	${\mathbb Z}$
$r = 20$	${\mathbb Z}_2$	${\mathbb Z}_2\oplus{\mathbb Z}_3$	${\mathbb Z}_3$
$r = 21$	${\mathbb Z}_2$	${\mathbb Z}_2$