T(7,5)

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T(27,2)

T(14,3)

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(7,5)'s page at Knotilus!

Visit T(7,5)'s page at the original Knot Atlas!

Edit T(7,5) Quick Notes

Edit T(7,5) Further Notes and Views

[edit] Knot presentations

Planar diagram presentation	X_25,3,26,2 X_48,4,49,3 X_15,5,16,4 X_38,6,39,5 X_49,27,50,26 X_16,28,17,27 X_39,29,40,28 X_6,30,7,29 X_17,51,18,50 X_40,52,41,51 X_7,53,8,52 X_30,54,31,53 X_41,19,42,18 X_8,20,9,19 X_31,21,32,20 X_54,22,55,21 X_9,43,10,42 X_32,44,33,43 X_55,45,56,44 X_22,46,23,45 X_33,11,34,10 X_56,12,1,11 X_23,13,24,12 X_46,14,47,13 X_1,35,2,34 X_24,36,25,35 X_47,37,48,36 X_14,38,15,37
Gauss code	-25, 1, 2, 3, 4, -8, -11, -14, -17, 21, 22, 23, 24, -28, -3, -6, -9, 13, 14, 15, 16, -20, -23, -26, -1, 5, 6, 7, 8, -12, -15, -18, -21, 25, 26, 27, 28, -4, -7, -10, -13, 17, 18, 19, 20, -24, -27, -2, -5, 9, 10, 11, 12, -16, -19, -22
Dowker-Thistlethwaite code	34 -48 -38 52 42 -56 -46 4 50 -8 -54 12 2 -16 -6 20 10 -24 -14 28 18 -32 -22 36 26 -40 -30 44

Braid presentation

[edit] Polynomial invariants

Alexander polynomial	$t 12 - t 11 + t 7 - t 6 + t 5 - t 4 + t 2 - t + 1- t -1 + t -2 - t -4 + t -5 - t -6 + t -7 - t -11 + t -12$
Conway polynomial	$z 24 + 23 z 22 + 230 z 20 + 1311 z 18 + 4692 z 16 + 10949 z 14 + 16757 z 12 + 16511 z 10 + 10032 z 8 + 3498 z 6 + 628 z 4 + 48 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 1, 16 }
Jones polynomial	$- q 22 - q 20 + q 16 + q 14 + q 12$
HOMFLY-PT polynomial (db, data sources)	Data:T(7,5)/HOMFLYPT Polynomial
Kauffman polynomial (db, data sources)	Data:T(7,5)/Kauffman Polynomial
The A2 invariant	Data:T(7,5)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(7,5)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(7,5) 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(7,5)"];

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

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

Out[4]= $t 12 - t 11 + t 7 - t 6 + t 5 - t 4 + t 2 - t + 1- t -1 + t -2 - t -4 + t -5 - t -6 + t -7 - t -11 + t -12$

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

Out[5]= $z 24 + 23 z 22 + 230 z 20 + 1311 z 18 + 4692 z 16 + 10949 z 14 + 16757 z 12 + 16511 z 10 + 10032 z 8 + 3498 z 6 + 628 z 4 + 48 z 2 + 1$

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, 16 }

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

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

Out[8]= $- q 22 - q 20 + q 16 + q 14 + q 12$

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

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

Out[9]= Data:T(7,5)/HOMFLYPT Polynomial

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

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

Out[10]= Data:T(7,5)/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(7,5)"];

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]= { $t 12 - t 11 + t 7 - t 6 + t 5 - t 4 + t 2 - t + 1- t -1 + t -2 - t -4 + t -5 - t -6 + t -7 - t -11 + t -12$ , $- q 22 - q 20 + q 16 + q 14 + q 12$ }

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:

(48, 280)

[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 =

16 is the signature of T(7,5). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-1

Integral Khovanov Homology

(db, data source)

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