K11a131

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K11a130

K11a132

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 Computer Talk
9 Modifying This Page

(Knotscape image)

See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a131's page at Knotilus!

Visit K11a131's page at the original Knot Atlas!

[edit K11a131 Quick Notes]

[edit K11a131 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X_10,3,11,4 X_14,6,15,5 X_22,8,1,7 X_18,9,19,10 X_2,11,3,12 X_20,13,21,14 X_6,16,7,15 X_8,17,9,18 X_12,19,13,20 X_16,21,17,22
Gauss code	1, -6, 2, -1, 3, -8, 4, -9, 5, -2, 6, -10, 7, -3, 8, -11, 9, -5, 10, -7, 11, -4
Dowker-Thistlethwaite code	4 10 14 22 18 2 20 6 8 12 16

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Chiral
Unknotting number	${1,2}$
3-genus	4
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a131/ThurstonBennequinNumber
Hyperbolic Volume	16.9146
A-Polynomial	See Data:K11a131/A-polynomial

[edit Notes for K11a131's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$4$
Rasmussen s-Invariant	2

[edit Notes for K11a131's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 4 + 6 t 3 -16 t 2 + 27 t -31 + 27 t -1 -16 t -2 + 6 t -3 - t -4$
Conway polynomial	$- z 8 -2 z 6 + z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 131, -2 }
Jones polynomial	$q 3 -4 q 2 + 9 q -14 + 19 q -1 -21 q -2 + 21 q -3 -18 q -4 + 13 q -5 -7 q -6 + 3 q -7 - q -8$
HOMFLY-PT polynomial (db, data sources)	$- a 2 z 8 + 2 a 4 z 6 -5 a 2 z 6 + z 6 - a 6 z 4 + 8 a 4 z 4 -10 a 2 z 4 + 3 z 4 -3 a 6 z 2 + 11 a 4 z 2 -10 a 2 z 2 + 3 z 2 -2 a 6 + 5 a 4 -4 a 2 + 2$
Kauffman polynomial (db, data sources)	$2 a 4 z 10 + 2 a 2 z 10 + 6 a 5 z 9 + 12 a 3 z 9 + 6 a z 9 + 7 a 6 z 8 + 12 a 4 z 8 + 12 a 2 z 8 + 7 z 8 + 5 a 7 z 7 -7 a 5 z 7 -23 a 3 z 7 -7 a z 7 + 4 z 7 a -1 + 3 a 8 z 6 -12 a 6 z 6 -40 a 4 z 6 -42 a 2 z 6 + z 6 a -2 -16 z 6 + a 9 z 5 -6 a 7 z 5 + 3 a 5 z 5 + 11 a 3 z 5 -8 a z 5 -9 z 5 a -1 -5 a 8 z 4 + 13 a 6 z 4 + 50 a 4 z 4 + 44 a 2 z 4 -2 z 4 a -2 + 10 z 4 -2 a 9 z 3 + a 7 z 3 + 3 a 5 z 3 + 3 a 3 z 3 + 8 a z 3 + 5 z 3 a -1 + 2 a 8 z 2 -8 a 6 z 2 -26 a 4 z 2 -22 a 2 z 2 + z 2 a -2 -5 z 2 + a 9 z -2 a 5 z -2 a 3 z -2 a z - z a -1 + 2 a 6 + 5 a 4 + 4 a 2 + 2$
The A2 invariant	$- q 24 -2 q 18 + 4 q 16 -2 q 14 + 2 q 12 + 2 q 10 -3 q 8 + 4 q 6 -5 q 4 + 3 q 2 - q -2 + 3 q -4 -2 q -6 + q -8$
The G2 invariant	$q 128 -2 q 126 + 5 q 124 -8 q 122 + 9 q 120 -8 q 118 + q 116 + 12 q 114 -27 q 112 + 44 q 110 -56 q 108 + 54 q 106 -36 q 104 -6 q 102 + 67 q 100 -134 q 98 + 189 q 96 -215 q 94 + 175 q 92 -65 q 90 -116 q 88 + 328 q 86 -483 q 84 + 513 q 82 -369 q 80 + 57 q 78 + 313 q 76 -618 q 74 + 723 q 72 -554 q 70 + 171 q 68 + 276 q 66 -586 q 64 + 623 q 62 -357 q 60 -81 q 58 + 486 q 56 -662 q 54 + 512 q 52 -88 q 50 -432 q 48 + 841 q 46 -941 q 44 + 692 q 42 -174 q 40 -429 q 38 + 894 q 36 -1061 q 34 + 873 q 32 -404 q 30 -173 q 28 + 650 q 26 -849 q 24 + 707 q 22 -295 q 20 -203 q 18 + 556 q 16 -628 q 14 + 376 q 12 + 72 q 10 -503 q 8 + 729 q 6 -637 q 4 + 269 q 2 + 211-607 q -2 + 766 q -4 -646 q -6 + 328 q -8 + 51 q -10 -350 q -12 + 482 q -14 -435 q -16 + 280 q -18 -84 q -20 -71 q -22 + 150 q -24 -163 q -26 + 123 q -28 -66 q -30 + 20 q -32 + 12 q -34 -24 q -36 + 22 q -38 -16 q -40 + 8 q -42 -3 q -44 + q -46$

Further Quantum Invariants

K11a131 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["K11a131"];

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

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

Out[4]= $- t 4 + 6 t 3 -16 t 2 + 27 t -31 + 27 t -1 -16 t -2 + 6 t -3 - t -4$

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

Out[5]= $- z 8 -2 z 6 + z 2 + 1$

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]= { 131, -2 }

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

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

Out[8]= $q 3 -4 q 2 + 9 q -14 + 19 q -1 -21 q -2 + 21 q -3 -18 q -4 + 13 q -5 -7 q -6 + 3 q -7 - q -8$

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

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

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

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

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

Out[10]= $2 a 4 z 10 + 2 a 2 z 10 + 6 a 5 z 9 + 12 a 3 z 9 + 6 a z 9 + 7 a 6 z 8 + 12 a 4 z 8 + 12 a 2 z 8 + 7 z 8 + 5 a 7 z 7 -7 a 5 z 7 -23 a 3 z 7 -7 a z 7 + 4 z 7 a -1 + 3 a 8 z 6 -12 a 6 z 6 -40 a 4 z 6 -42 a 2 z 6 + z 6 a -2 -16 z 6 + a 9 z 5 -6 a 7 z 5 + 3 a 5 z 5 + 11 a 3 z 5 -8 a z 5 -9 z 5 a -1 -5 a 8 z 4 + 13 a 6 z 4 + 50 a 4 z 4 + 44 a 2 z 4 -2 z 4 a -2 + 10 z 4 -2 a 9 z 3 + a 7 z 3 + 3 a 5 z 3 + 3 a 3 z 3 + 8 a z 3 + 5 z 3 a -1 + 2 a 8 z 2 -8 a 6 z 2 -26 a 4 z 2 -22 a 2 z 2 + z 2 a -2 -5 z 2 + a 9 z -2 a 5 z -2 a 3 z -2 a z - z a -1 + 2 a 6 + 5 a 4 + 4 a 2 + 2$

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

Same Alexander/Conway Polynomial: {K11a252, K11a254,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11a218,}

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["K11a131"];

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]= { $- t 4 + 6 t 3 -16 t 2 + 27 t -31 + 27 t -1 -16 t -2 + 6 t -3 - t -4$ , $q 3 -4 q 2 + 9 q -14 + 19 q -1 -21 q -2 + 21 q -3 -18 q -4 + 13 q -5 -7 q -6 + 3 q -7 - q -8$ }

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]= {K11a252, K11a254,}

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

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

Out[6]= {K11a218,}

[edit] Vassiliev invariants

V₂ and V₃:

(1, -3)

[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 =

-2 is the signature of K11a131. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

-3

-5

-1

-3

-2

-5

-7

-9

-5

-11

-13

-4

-15

-17

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{11}$
$r = 1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$