K11a156

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K11a155

K11a157

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 K11a156's page at Knotilus!

Visit K11a156's page at the original Knot Atlas!

[edit K11a156 Quick Notes]

[edit K11a156 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X_10,3,11,4 X_18,5,19,6 X_12,8,13,7 X_14,10,15,9 X_2,11,3,12 X_8,14,9,13 X_20,15,21,16 X_22,17,1,18 X_6,19,7,20 X_16,21,17,22
Gauss code	1, -6, 2, -1, 3, -10, 4, -7, 5, -2, 6, -4, 7, -5, 8, -11, 9, -3, 10, -8, 11, -9
Dowker-Thistlethwaite code	4 10 18 12 14 2 8 20 22 6 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:K11a156/ThurstonBennequinNumber
Hyperbolic Volume	14.0856
A-Polynomial	See Data:K11a156/A-polynomial

[edit Notes for K11a156'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 K11a156's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 4 + 5 t 3 -11 t 2 + 18 t -21 + 18 t -1 -11 t -2 + 5 t -3 - t -4$
Conway polynomial	$- z 8 -3 z 6 - z 4 + 3 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 91, -2 }
Jones polynomial	$q 3 -3 q 2 + 6 q -10 + 13 q -1 -14 q -2 + 15 q -3 -12 q -4 + 9 q -5 -5 q -6 + 2 q -7 - q -8$
HOMFLY-PT polynomial (db, data sources)	$- a 2 z 8 + 2 a 4 z 6 -6 a 2 z 6 + z 6 - a 6 z 4 + 10 a 4 z 4 -14 a 2 z 4 + 4 z 4 -4 a 6 z 2 + 17 a 4 z 2 -15 a 2 z 2 + 5 z 2 -4 a 6 + 9 a 4 -6 a 2 + 2$
Kauffman polynomial (db, data sources)	$a 4 z 10 + a 2 z 10 + 3 a 5 z 9 + 6 a 3 z 9 + 3 a z 9 + 4 a 6 z 8 + 6 a 4 z 8 + 6 a 2 z 8 + 4 z 8 + 3 a 7 z 7 -4 a 5 z 7 -13 a 3 z 7 -3 a z 7 + 3 z 7 a -1 + 2 a 8 z 6 -10 a 6 z 6 -25 a 4 z 6 -24 a 2 z 6 + z 6 a -2 -10 z 6 + a 9 z 5 -5 a 7 z 5 + 7 a 3 z 5 -8 a z 5 -9 z 5 a -1 -4 a 8 z 4 + 16 a 6 z 4 + 43 a 4 z 4 + 32 a 2 z 4 -3 z 4 a -2 + 6 z 4 -3 a 9 z 3 + 2 a 7 z 3 + 10 a 5 z 3 + 8 a 3 z 3 + 10 a z 3 + 7 z 3 a -1 + a 8 z 2 -13 a 6 z 2 -30 a 4 z 2 -22 a 2 z 2 + 2 z 2 a -2 -4 z 2 + 2 a 9 z - a 7 z -6 a 5 z -6 a 3 z -5 a z -2 z a -1 + 4 a 6 + 9 a 4 + 6 a 2 + 2$
The A2 invariant	$- q 24 - q 22 - q 20 -2 q 18 + 3 q 16 + 3 q 12 + 3 q 10 - q 8 + 3 q 6 -4 q 4 + q 2 -1- q -2 + 2 q -4 - q -6 + q -8$
The G2 invariant	$q 128 - q 126 + 3 q 124 -4 q 122 + 4 q 120 -3 q 118 - q 116 + 7 q 114 -13 q 112 + 19 q 110 -22 q 108 + 18 q 106 -9 q 104 -9 q 102 + 32 q 100 -54 q 98 + 66 q 96 -69 q 94 + 44 q 92 -5 q 90 -54 q 88 + 112 q 86 -150 q 84 + 147 q 82 -100 q 80 + 4 q 78 + 100 q 76 -183 q 74 + 210 q 72 -158 q 70 + 50 q 68 + 73 q 66 -160 q 64 + 172 q 62 -100 q 60 -16 q 58 + 133 q 56 -179 q 54 + 140 q 52 -14 q 50 -128 q 48 + 248 q 46 -272 q 44 + 200 q 42 -47 q 40 -126 q 38 + 268 q 36 -318 q 34 + 269 q 32 -130 q 30 -44 q 28 + 185 q 26 -253 q 24 + 216 q 22 -105 q 20 -41 q 18 + 145 q 16 -177 q 14 + 114 q 12 + 8 q 10 -135 q 8 + 203 q 6 -185 q 4 + 81 q 2 + 55-173 q -2 + 231 q -4 -201 q -6 + 115 q -8 -100 q -12 + 152 q -14 -149 q -16 + 107 q -18 -44 q -20 -11 q -22 + 47 q -24 -60 q -26 + 53 q -28 -33 q -30 + 15 q -32 + q -34 -10 q -36 + 10 q -38 -9 q -40 + 5 q -42 -2 q -44 + q -46$

Further Quantum Invariants

K11a156 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["K11a156"];

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

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

Out[4]= $- t 4 + 5 t 3 -11 t 2 + 18 t -21 + 18 t -1 -11 t -2 + 5 t -3 - t -4$

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

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

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]= { 91, -2 }

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

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

Out[8]= $q 3 -3 q 2 + 6 q -10 + 13 q -1 -14 q -2 + 15 q -3 -12 q -4 + 9 q -5 -5 q -6 + 2 q -7 - q -8$

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

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

Out[9]= $- a 2 z 8 + 2 a 4 z 6 -6 a 2 z 6 + z 6 - a 6 z 4 + 10 a 4 z 4 -14 a 2 z 4 + 4 z 4 -4 a 6 z 2 + 17 a 4 z 2 -15 a 2 z 2 + 5 z 2 -4 a 6 + 9 a 4 -6 a 2 + 2$

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

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

Out[10]= $a 4 z 10 + a 2 z 10 + 3 a 5 z 9 + 6 a 3 z 9 + 3 a z 9 + 4 a 6 z 8 + 6 a 4 z 8 + 6 a 2 z 8 + 4 z 8 + 3 a 7 z 7 -4 a 5 z 7 -13 a 3 z 7 -3 a z 7 + 3 z 7 a -1 + 2 a 8 z 6 -10 a 6 z 6 -25 a 4 z 6 -24 a 2 z 6 + z 6 a -2 -10 z 6 + a 9 z 5 -5 a 7 z 5 + 7 a 3 z 5 -8 a z 5 -9 z 5 a -1 -4 a 8 z 4 + 16 a 6 z 4 + 43 a 4 z 4 + 32 a 2 z 4 -3 z 4 a -2 + 6 z 4 -3 a 9 z 3 + 2 a 7 z 3 + 10 a 5 z 3 + 8 a 3 z 3 + 10 a z 3 + 7 z 3 a -1 + a 8 z 2 -13 a 6 z 2 -30 a 4 z 2 -22 a 2 z 2 + 2 z 2 a -2 -4 z 2 + 2 a 9 z - a 7 z -6 a 5 z -6 a 3 z -5 a z -2 z a -1 + 4 a 6 + 9 a 4 + 6 a 2 + 2$

[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["K11a156"];

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]= { $- t 4 + 5 t 3 -11 t 2 + 18 t -21 + 18 t -1 -11 t -2 + 5 t -3 - t -4$ , $q 3 -3 q 2 + 6 q -10 + 13 q -1 -14 q -2 + 15 q -3 -12 q -4 + 9 q -5 -5 q -6 + 2 q -7 - q -8$ }

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]= {}

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:

(3, -7)

[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 K11a156. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

-2

-4

-1

-3

-1

-5

-7

-9

-3

-11

-13

-3

-15

-17

-1

Integral Khovanov Homology

(db, data source)

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