T(8,5)

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T(31,2)

T(16,3)

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(8,5)'s page at Knotilus!

Visit T(8,5)'s page at the original Knot Atlas!

Edit T(8,5) Quick Notes

Edit T(8,5) Further Notes and Views

[edit] Knot presentations

Planar diagram presentation	X_54,16,55,15 X_29,17,30,16 X_4,18,5,17 X_43,19,44,18 X_30,56,31,55 X_5,57,6,56 X_44,58,45,57 X_19,59,20,58 X_6,32,7,31 X_45,33,46,32 X_20,34,21,33 X_59,35,60,34 X_46,8,47,7 X_21,9,22,8 X_60,10,61,9 X_35,11,36,10 X_22,48,23,47 X_61,49,62,48 X_36,50,37,49 X_11,51,12,50 X_62,24,63,23 X_37,25,38,24 X_12,26,13,25 X_51,27,52,26 X_38,64,39,63 X_13,1,14,64 X_52,2,53,1 X_27,3,28,2 X_14,40,15,39 X_53,41,54,40 X_28,42,29,41 X_3,43,4,42
Gauss code	27, 28, -32, -3, -6, -9, 13, 14, 15, 16, -20, -23, -26, -29, 1, 2, 3, 4, -8, -11, -14, -17, 21, 22, 23, 24, -28, -31, -2, -5, 9, 10, 11, 12, -16, -19, -22, -25, 29, 30, 31, 32, -4, -7, -10, -13, 17, 18, 19, 20, -24, -27, -30, -1, 5, 6, 7, 8, -12, -15, -18, -21, 25, 26
Dowker-Thistlethwaite code	52 -42 -56 46 60 -50 -64 54 4 -58 -8 62 12 -2 -16 6 20 -10 -24 14 28 -18 -32 22 36 -26 -40 30 44 -34 -48 38

Braid presentation

[edit] Polynomial invariants

Alexander polynomial	$t 14 - t 13 + t 9 - t 8 + t 6 - t 5 + t 4 - t 3 + t -1 + t -1 - t -3 + t -4 - t -5 + t -6 - t -8 + t -9 - t -13 + t -14$
Conway polynomial	$z 28 + 27 z 26 + 324 z 24 + 2277 z 22 + 10395 z 20 + 32320 z 18 + 69785 z 16 + 104771 z 14 + 107849 z 12 + 73876 z 10 + 32055 z 8 + 8169 z 6 + 1092 z 4 + 63 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 5, 20 }
Jones polynomial	$- q 25 - q 23 + q 18 + q 16 + q 14$
HOMFLY-PT polynomial (db, data sources)	Data:T(8,5)/HOMFLYPT Polynomial
Kauffman polynomial (db, data sources)	Data:T(8,5)/Kauffman Polynomial
The A2 invariant	Data:T(8,5)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(8,5)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(8,5) 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(8,5)"];

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

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

Out[4]= $t 14 - t 13 + t 9 - t 8 + t 6 - t 5 + t 4 - t 3 + t -1 + t -1 - t -3 + t -4 - t -5 + t -6 - t -8 + t -9 - t -13 + t -14$

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

Out[5]= $z 28 + 27 z 26 + 324 z 24 + 2277 z 22 + 10395 z 20 + 32320 z 18 + 69785 z 16 + 104771 z 14 + 107849 z 12 + 73876 z 10 + 32055 z 8 + 8169 z 6 + 1092 z 4 + 63 z 2 + 1$

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]= { 5, 20 }

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

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

Out[8]= $- q 25 - q 23 + q 18 + q 16 + q 14$

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

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

Out[9]= Data:T(8,5)/HOMFLYPT Polynomial

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

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

Out[10]= Data:T(8,5)/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(8,5)"];

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]= { $t 14 - t 13 + t 9 - t 8 + t 6 - t 5 + t 4 - t 3 + t -1 + t -1 - t -3 + t -4 - t -5 + t -6 - t -8 + t -9 - t -13 + t -14$ , $- q 25 - q 23 + q 18 + q 16 + q 14$ }

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:

(63, 420)

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 =

20 is the signature of T(8,5). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-1

Integral Khovanov Homology

(db, data source)

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