10 130

From Knot Atlas

Jump to: navigation, search

10_129

10_131

1 Knot presentations
2 Three dimensional invariants
3 Four dimensional invariants
4 Polynomial invariants
5 "Similar" Knots (within the Atlas)
6 Vassiliev invariants
7 Khovanov Homology
8 The Coloured Jones Polynomials
9 Computer Talk
10 Modifying This Page

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 130's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_130's page at Knotilus!

Visit 10 130's page at the original Knot Atlas!

[edit 10 130 Quick Notes]

[edit 10 130 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_5,14,6,15 X_15,20,16,1 X_9,16,10,17 X_19,10,20,11 X_11,18,12,19 X_17,12,18,13 X_13,6,14,7 X₂₈₃₇
Gauss code	1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -7, 8, -9, 3, -4, 5, -8, 7, -6, 4
Dowker-Thistlethwaite code	4 8 -14 2 -16 -18 -6 -20 -12 -10
Conway Notation	[311,3,2-]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 11, width is 4,

Braid index is 4

[{3, 10}, {2, 4}, {1, 3}, {13, 11}, {10, 12}, {11, 5}, {4, 8}, {7, 9}, {8, 6}, {5, 7}, {6, 13}, {12, 2}, {9, 1}]

[edit Notes on presentations of 10 130]

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.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["10 130"];

In[4]:= PD[K]

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

Out[4]= X₄₂₅₁ X₈₄₉₃ X_5,14,6,15 X_15,20,16,1 X_9,16,10,17 X_19,10,20,11 X_11,18,12,19 X_17,12,18,13 X_13,6,14,7 X₂₈₃₇

In[5]:= GaussCode[K]

Out[5]= 1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -7, 8, -9, 3, -4, 5, -8, 7, -6, 4

In[6]:= DTCode[K]

Out[6]= 4 8 -14 2 -16 -18 -6 -20 -12 -10

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:= ConwayNotation[K]

Out[8]= [311,3,2-]

In[9]:= br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]= $BR(4,{1,1,1,-2,-1,-1,-2,-2,-3,2,-3})$

In[10]:= {First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

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

Out[10]= { 4, 11, 4 }

In[11]:= Show[BraidPlot[br]]

Out[11]= -Graphics-

In[12]:= Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]= -Graphics-

In[13]:= ap = ArcPresentation[K]

Out[13]= ArcPresentation[{3, 10}, {2, 4}, {1, 3}, {13, 11}, {10, 12}, {11, 5}, {4, 8}, {7, 9}, {8, 6}, {5, 7}, {6, 13}, {12, 2}, {9, 1}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	2
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-8][-2]
Hyperbolic Volume	6.7782
A-Polynomial	See Data:10 130/A-polynomial

[edit Notes for 10 130's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	$1$
Topological 4 genus	$1$
Concordance genus	$2$
Rasmussen s-Invariant	0

[edit Notes for 10 130's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$2 t 2 -4 t + 5-4 t -1 + 2 t -2$
Conway polynomial	$2 z 4 + 4 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 17, 0 }
Jones polynomial	$- q + 2-2 q -1 + 3 q -2 -2 q -3 + 3 q -4 -2 q -5 + q -6 - q -7$
HOMFLY-PT polynomial (db, data sources)	$- z 2 a 6 -2 a 6 + z 4 a 4 + 3 z 2 a 4 + 2 a 4 + z 4 a 2 + 3 z 2 a 2 + 2 a 2 - z 2 -1$
Kauffman polynomial (db, data sources)	$a 6 z 8 + a 4 z 8 + a 7 z 7 + 3 a 5 z 7 + 2 a 3 z 7 -5 a 6 z 6 -3 a 4 z 6 + 2 a 2 z 6 -6 a 7 z 5 -15 a 5 z 5 -8 a 3 z 5 + a z 5 + 7 a 6 z 4 -7 a 2 z 4 + 11 a 7 z 3 + 21 a 5 z 3 + 8 a 3 z 3 -2 a z 3 -4 a 6 z 2 + 6 a 2 z 2 + 2 z 2 -6 a 7 z -9 a 5 z -3 a 3 z + a z + z a -1 + 2 a 6 + 2 a 4 -2 a 2 -1$
The A2 invariant	$- q 22 - q 20 - q 18 - q 16 + q 14 + q 12 + 2 q 10 + 2 q 8 + q 6 + q 4 - q -4$
The G2 invariant	$q 108 + 2 q 104 -2 q 102 + q 100 -2 q 96 + 3 q 94 -4 q 92 + 2 q 90 - q 88 -3 q 86 + 2 q 84 -4 q 82 - q 80 + q 78 -4 q 76 - q 74 -4 q 70 + 3 q 68 -3 q 66 + q 62 -2 q 60 + 4 q 58 -2 q 56 + 4 q 54 + 3 q 50 + 2 q 48 + 4 q 44 - q 42 + 3 q 40 + 2 q 38 - q 36 + 2 q 34 + 3 q 32 -3 q 30 + 4 q 28 - q 26 - q 24 + 4 q 22 -4 q 20 + 3 q 18 - q 16 + q 14 + q 12 - q 10 + 1-2 q -2 + q -4 - q -6 - q -12 - q -16 - q -20 + q -24$

Further Quantum Invariants

Further quantum knot invariants for 10_130.

A1 Invariants.

Weight	Invariant
1	$- q 15 - q 11 + q 9 + q 7 + q 5 + q 3 + q -1 - q -3$
2	$q 44 - q 40 + q 38 + q 36 -2 q 34 - q 32 - q 28 - q 26 + q 22 - q 20 + 2 q 16 + 3 q 10 + q 8 + q 2 - q -2$
3	$- q 87 + q 83 + q 81 - q 79 -2 q 77 + 3 q 73 + 2 q 71 - q 69 -2 q 67 + 2 q 63 + 2 q 61 -3 q 57 -2 q 55 + q 53 + 3 q 51 -2 q 49 -4 q 47 - q 45 + 3 q 43 -3 q 39 -2 q 37 + 2 q 35 + q 33 -2 q 31 - q 29 + 4 q 27 + 2 q 25 -2 q 23 - q 21 + 2 q 19 + 3 q 17 + q 15 - q 13 -2 q 11 + 2 q 9 + 5 q 7 + 2 q 5 -6 q 3 -3 q + 4 q -1 + 4 q -3 -3 q -5 -2 q -7 + q -9 + q -11 - q -13 - q -15 + q -17$

A2 Invariants.

Weight	Invariant
1,0	$- q 22 - q 20 - q 18 - q 16 + q 14 + q 12 + 2 q 10 + 2 q 8 + q 6 + q 4 - q -4$
1,1	$q 60 + 4 q 56 -4 q 54 + 6 q 52 -8 q 50 + 6 q 48 -8 q 46 + q 44 -4 q 40 + 6 q 38 -7 q 36 + 8 q 34 -10 q 32 + 6 q 30 -11 q 28 + 4 q 26 -6 q 24 + 4 q 22 + 3 q 20 + 2 q 18 + 8 q 16 + 4 q 14 + 5 q 12 + 2 q 10 + 2 q 8 -2 q 6 + 2 q 4 -4 q 2 + 2-4 q -2 + q -4 -2 q -6 + q -12$
2,0	$q 58 + q 56 + q 54 + q 50 + q 48 -3 q 44 -3 q 42 -3 q 40 -2 q 38 -2 q 36 -3 q 34 - q 32 - q 30 + q 28 + 3 q 24 + 4 q 22 + 5 q 20 + 3 q 18 + 3 q 16 + 2 q 14 + q 12 + q 10 - q 8 - q 6 -1-2 q -2 + q -6$

A3 Invariants.

Weight	Invariant
0,1,0	$q 46 + 2 q 42 + q 40 - q 36 -3 q 34 -5 q 32 -6 q 30 -4 q 28 -2 q 26 + 2 q 24 + 3 q 22 + 6 q 20 + 5 q 18 + 4 q 16 + 3 q 14 + 2 q 12 + q 10 + q 8 - q 2 -1- q -2 - q -4 - q -6 + q -10$
1,0,0	$- q 29 - q 27 -2 q 25 - q 23 - q 21 + q 19 + q 17 + 2 q 15 + 2 q 13 + 2 q 11 + 2 q 9 + q 7 + q 5 - q -1 - q -5$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q 60 + q 58 + 2 q 56 + 3 q 54 + 3 q 52 + 2 q 50 + q 48 -3 q 46 -7 q 44 -10 q 42 -11 q 40 -12 q 38 -10 q 36 -3 q 34 + 3 q 32 + 6 q 30 + 10 q 28 + 14 q 26 + 11 q 24 + 9 q 22 + 6 q 20 + 4 q 18 - q 10 -2 q 8 - q 4 -3 q 2 -2- q -2 - q -4 - q -6 + q -10 + q -12$
1,0,0,0	$- q 36 - q 34 -2 q 32 -2 q 30 - q 28 - q 26 + q 24 + q 22 + 2 q 20 + 2 q 18 + 2 q 16 + 2 q 14 + 2 q 12 + 2 q 10 + q 8 + q 6 -1- q -2 - q -6$

B2 Invariants.

Weight	Invariant
0,1	$- q 46 -2 q 42 + q 40 -2 q 38 + q 36 - q 34 + q 32 + 2 q 26 -2 q 24 + 3 q 22 -2 q 20 + 3 q 18 -2 q 16 + 3 q 14 + q 10 + q 8 + 2 q 4 - q 2 + 1- q -2 + q -4 - q -6 - q -10$
1,0	$q 76 + 2 q 68 + q 66 - q 64 - q 62 + q 60 -2 q 56 -3 q 54 -2 q 52 - q 50 -2 q 48 -3 q 46 -3 q 44 + q 40 + q 38 + 2 q 34 + 3 q 32 + 3 q 30 + q 28 + 2 q 26 + 3 q 24 + 3 q 22 + q 16 + 2 q 14 - q 10 - q 8 + q 6 - q 2 -1- q -8 - q -10 + q -16$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q 62 + 2 q 58 + 3 q 54 - q 52 -3 q 48 -3 q 46 -6 q 44 -6 q 42 -5 q 40 -4 q 38 - q 34 + 5 q 32 + 3 q 30 + 8 q 28 + 3 q 26 + 6 q 24 + q 22 + 4 q 20 + q 18 + 2 q 16 + q 14 + q 12 + 2 q 10 + q 6 -2 q 4 -2- q -2 -2 q -4 - q -8 + q -14$

G2 Invariants.

Weight

Invariant

1,0

q 108 + 2 q 104 -2 q 102 + q 100 -2 q 96 + 3 q 94 -4 q 92 + 2 q 90 - q 88 -3 q 86 + 2 q 84 -4 q 82 - q 80 + q 78 -4 q 76 - q 74 -4 q 70 + 3 q 68 -3 q 66 + q 62 -2 q 60 + 4 q 58 -2 q 56 + 4 q 54 + 3 q 50 + 2 q 48 + 4 q 44 - q 42 + 3 q 40 + 2 q 38 - q 36 + 2 q 34 + 3 q 32 -3 q 30 + 4 q 28 - q 26 - q 24 + 4 q 22 -4 q 20 + 3 q 18 - q 16 + q 14 + q 12 - q 10 + 1-2 q -2 + q -4 - q -6 - q -12 - q -16 - q -20 + q -24

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["10 130"];

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

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

Out[4]= $2 t 2 -4 t + 5-4 t -1 + 2 t -2$

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

Out[5]= $2 z 4 + 4 z 2 + 1$

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]= { 17, 0 }

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

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

Out[8]= $- q + 2-2 q -1 + 3 q -2 -2 q -3 + 3 q -4 -2 q -5 + q -6 - q -7$

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

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

Out[9]= $- z 2 a 6 -2 a 6 + z 4 a 4 + 3 z 2 a 4 + 2 a 4 + z 4 a 2 + 3 z 2 a 2 + 2 a 2 - z 2 -1$

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

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

Out[10]= $a 6 z 8 + a 4 z 8 + a 7 z 7 + 3 a 5 z 7 + 2 a 3 z 7 -5 a 6 z 6 -3 a 4 z 6 + 2 a 2 z 6 -6 a 7 z 5 -15 a 5 z 5 -8 a 3 z 5 + a z 5 + 7 a 6 z 4 -7 a 2 z 4 + 11 a 7 z 3 + 21 a 5 z 3 + 8 a 3 z 3 -2 a z 3 -4 a 6 z 2 + 6 a 2 z 2 + 2 z 2 -6 a 7 z -9 a 5 z -3 a 3 z + a z + z a -1 + 2 a 6 + 2 a 4 -2 a 2 -1$

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

Same Alexander/Conway Polynomial: {7_5,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11n61,}

Computer Talk

(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["10 130"];

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]= { $2 t 2 -4 t + 5-4 t -1 + 2 t -2$ , $- q + 2-2 q -1 + 3 q -2 -2 q -3 + 3 q -4 -2 q -5 + q -6 - q -7$ }

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]= {7_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]= {K11n61,}

[edit] Vassiliev invariants

V₂ and V₃:

(4, -6)

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

16

-48

128

$\frac{776}{3}$

$\frac{136}{3}$

-768

-1344

-192

-272

$\frac{2048}{3}$

1152

$\frac{12416}{3}$

$\frac{2176}{3}$

$\frac{109382}{15}$

$-\frac{2056}{5}$

$\frac{160808}{45}$

$\frac{1162}{9}$

$\frac{7862}{15}$

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 =

0 is the signature of 10 130. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

-3

-5

-7

-9

-11

-1

-13

-15

-1

Integral Khovanov Homology

(db, data source)

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

[edit] The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the

n + 1

-dimensional representation of

s l (2)

)

$n$	$J n$
2	$-1 + q -1 + q -2 -2 q -3 + q -4 + 2 q -5 -2 q -7 + 2 q -8 + 2 q -9 -4 q -10 + q -11 + 4 q -12 -5 q -13 + 4 q -15 -4 q -16 - q -17 + 3 q -18 - q -19 - q -20 + q -21$
3	$q 7 -2 q 6 + 2 q 4 + q 3 -5 q 2 - q + 9 + q -1 -12 q -2 -4 q -3 + 17 q -4 + 4 q -5 -15 q -6 -8 q -7 + 18 q -8 + 6 q -9 -13 q -10 -9 q -11 + 15 q -12 + 5 q -13 -9 q -14 -7 q -15 + 10 q -16 + 4 q -17 -6 q -18 -6 q -19 + 6 q -20 + 3 q -21 -3 q -22 -3 q -23 + 2 q -24 - q -26 + 2 q -27 -3 q -29 -2 q -30 + 5 q -31 + 2 q -32 -3 q -33 -4 q -34 + 3 q -35 + 3 q -36 -3 q -38 + q -40 + q -41 - q -42$
4	$- q 12 + 2 q 11 -2 q 9 + 2 q 8 -2 q 7 + 4 q 6 -5 q 5 -5 q 4 + 10 q 3 + q 2 + 7 q -17-15 q -1 + 19 q -2 + 13 q -3 + 16 q -4 -28 q -5 -31 q -6 + 18 q -7 + 22 q -8 + 30 q -9 -27 q -10 -41 q -11 + 11 q -12 + 19 q -13 + 35 q -14 -17 q -15 -38 q -16 + 4 q -17 + 11 q -18 + 32 q -19 -8 q -20 -33 q -21 + q -22 + 5 q -23 + 29 q -24 -29 q -26 -5 q -27 + 27 q -29 + 10 q -30 -24 q -31 -11 q -32 -7 q -33 + 21 q -34 + 20 q -35 -15 q -36 -12 q -37 -13 q -38 + 9 q -39 + 22 q -40 -5 q -41 -5 q -42 -11 q -43 -3 q -44 + 14 q -45 -2 q -46 + 3 q -47 -2 q -48 -5 q -49 + 5 q -50 -7 q -51 + 3 q -52 + 4 q -53 + 5 q -55 -9 q -56 -2 q -57 + q -58 + 2 q -59 + 7 q -60 -4 q -61 -2 q -62 -2 q -63 - q -64 + 4 q -65 - q -68 - q -69 + q -70$

[edit] Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.