K11a184

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K11a183

K11a185

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 K11a184's page at Knotilus!

Visit K11a184's page at the original Knot Atlas!

[edit K11a184 Quick Notes]

[edit K11a184 Further Notes and Views]

[edit] Knot presentations

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

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	4
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a184/ThurstonBennequinNumber
Hyperbolic Volume	12.4499
A-Polynomial	See Data:K11a184/A-polynomial

[edit Notes for K11a184'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 K11a184's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 4 + 5 t 3 -11 t 2 + 17 t -19 + 17 t -1 -11 t -2 + 5 t -3 - t -4$
Conway polynomial	$- z 8 -3 z 6 - z 4 + 2 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 87, -2 }
Jones polynomial	$q 3 -3 q 2 + 5 q -8 + 12 q -1 -13 q -2 + 14 q -3 -12 q -4 + 9 q -5 -6 q -6 + 3 q -7 - q -8$
HOMFLY-PT polynomial (db, data sources)	$- a 2 z 8 + 2 a 4 z 6 -6 a 2 z 6 + z 6 - a 6 z 4 + 9 a 4 z 4 -13 a 2 z 4 + 4 z 4 -3 a 6 z 2 + 12 a 4 z 2 -11 a 2 z 2 + 4 z 2 -2 a 6 + 4 a 4 -2 a 2 + 1$
Kauffman polynomial (db, data sources)	$a 4 z 10 + a 2 z 10 + 3 a 5 z 9 + 6 a 3 z 9 + 3 a z 9 + 4 a 6 z 8 + 5 a 4 z 8 + 5 a 2 z 8 + 4 z 8 + 4 a 7 z 7 -4 a 5 z 7 -17 a 3 z 7 -6 a z 7 + 3 z 7 a -1 + 3 a 8 z 6 -6 a 6 z 6 -21 a 4 z 6 -26 a 2 z 6 + z 6 a -2 -13 z 6 + a 9 z 5 -6 a 7 z 5 + 3 a 5 z 5 + 18 a 3 z 5 -2 a z 5 -10 z 5 a -1 -6 a 8 z 4 + 4 a 6 z 4 + 34 a 4 z 4 + 39 a 2 z 4 -3 z 4 a -2 + 12 z 4 -2 a 9 z 3 -2 a 3 z 3 + 7 a z 3 + 7 z 3 a -1 + 2 a 8 z 2 -5 a 6 z 2 -20 a 4 z 2 -20 a 2 z 2 + z 2 a -2 -6 z 2 + a 9 z - a 5 z - a 3 z -2 a z - z a -1 + 2 a 6 + 4 a 4 + 2 a 2 + 1$
The A2 invariant	$- q 24 - q 18 + 2 q 16 -2 q 14 + q 12 + q 10 + 4 q 6 -2 q 4 + 2 q 2 -1- q -2 + q -4 - q -6 + q -8$
The G2 invariant	$q 128 -2 q 126 + 5 q 124 -8 q 122 + 8 q 120 -5 q 118 -3 q 116 + 16 q 114 -27 q 112 + 36 q 110 -37 q 108 + 21 q 106 + 2 q 104 -35 q 102 + 65 q 100 -82 q 98 + 80 q 96 -56 q 94 + 11 q 92 + 43 q 90 -94 q 88 + 126 q 86 -123 q 84 + 82 q 82 -19 q 80 -56 q 78 + 110 q 76 -126 q 74 + 101 q 72 -35 q 70 -37 q 68 + 85 q 66 -88 q 64 + 40 q 62 + 34 q 60 -101 q 58 + 126 q 56 -95 q 54 + 15 q 52 + 89 q 50 -172 q 48 + 207 q 46 -167 q 44 + 71 q 42 + 48 q 40 -152 q 38 + 207 q 36 -190 q 34 + 121 q 32 -15 q 30 -80 q 28 + 140 q 26 -134 q 24 + 74 q 22 + 9 q 20 -79 q 18 + 102 q 16 -75 q 14 + 4 q 12 + 79 q 10 -129 q 8 + 135 q 6 -89 q 4 - q 2 + 85-143 q -2 + 151 q -4 -113 q -6 + 48 q -8 + 25 q -10 -78 q -12 + 103 q -14 -94 q -16 + 64 q -18 -24 q -20 -12 q -22 + 33 q -24 -41 q -26 + 36 q -28 -23 q -30 + 12 q -32 + q -34 -7 q -36 + 7 q -38 -7 q -40 + 4 q -42 -2 q -44 + q -46$

Further Quantum Invariants

K11a184 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["K11a184"];

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

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

Out[4]= $- t 4 + 5 t 3 -11 t 2 + 17 t -19 + 17 t -1 -11 t -2 + 5 t -3 - t -4$

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

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

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]= { 87, -2 }

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

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

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

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

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

Out[9]= $- a 2 z 8 + 2 a 4 z 6 -6 a 2 z 6 + z 6 - a 6 z 4 + 9 a 4 z 4 -13 a 2 z 4 + 4 z 4 -3 a 6 z 2 + 12 a 4 z 2 -11 a 2 z 2 + 4 z 2 -2 a 6 + 4 a 4 -2 a 2 + 1$

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

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

Out[10]= $a 4 z 10 + a 2 z 10 + 3 a 5 z 9 + 6 a 3 z 9 + 3 a z 9 + 4 a 6 z 8 + 5 a 4 z 8 + 5 a 2 z 8 + 4 z 8 + 4 a 7 z 7 -4 a 5 z 7 -17 a 3 z 7 -6 a z 7 + 3 z 7 a -1 + 3 a 8 z 6 -6 a 6 z 6 -21 a 4 z 6 -26 a 2 z 6 + z 6 a -2 -13 z 6 + a 9 z 5 -6 a 7 z 5 + 3 a 5 z 5 + 18 a 3 z 5 -2 a z 5 -10 z 5 a -1 -6 a 8 z 4 + 4 a 6 z 4 + 34 a 4 z 4 + 39 a 2 z 4 -3 z 4 a -2 + 12 z 4 -2 a 9 z 3 -2 a 3 z 3 + 7 a z 3 + 7 z 3 a -1 + 2 a 8 z 2 -5 a 6 z 2 -20 a 4 z 2 -20 a 2 z 2 + z 2 a -2 -6 z 2 + a 9 z - a 5 z - a 3 z -2 a z - z a -1 + 2 a 6 + 4 a 4 + 2 a 2 + 1$

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

Same Alexander/Conway Polynomial: {10_112,}

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["K11a184"];

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]= { $- t 4 + 5 t 3 -11 t 2 + 17 t -19 + 17 t -1 -11 t -2 + 5 t -3 - t -4$ , $q 3 -3 q 2 + 5 q -8 + 12 q -1 -13 q -2 + 14 q -3 -12 q -4 + 9 q -5 -6 q -6 + 3 q -7 - q -8$ }

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]= {10_112,}

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:

(2, -4)

[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 K11a184. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

-2

-3

-1

-3

-1

-5

-7

-9

-3

-11

-13

-3

-15

-17

-1

Integral Khovanov Homology

(db, data source)

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