K11a92

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K11a91

K11a93

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 K11a92's page at Knotilus!

Visit K11a92's page at the original Knot Atlas!

[edit K11a92 Quick Notes]

[edit K11a92 Further Notes and Views]

[edit] Knot presentations

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

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	${1,2}$
3-genus	4
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a92/ThurstonBennequinNumber
Hyperbolic Volume	14.2422
A-Polynomial	See Data:K11a92/A-polynomial

[edit Notes for K11a92'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 K11a92's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 4 + 5 t 3 -12 t 2 + 21 t -25 + 21 t -1 -12 t -2 + 5 t -3 - t -4$
Conway polynomial	$- z 8 -3 z 6 -2 z 4 + 2 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 103, -2 }
Jones polynomial	$q 3 -3 q 2 + 6 q -10 + 14 q -1 -16 q -2 + 17 q -3 -14 q -4 + 11 q -5 -7 q -6 + 3 q -7 - q -8$
HOMFLY-PT polynomial (db, data sources)	$- a 2 z 8 + 2 a 4 z 6 -6 a 2 z 6 + z 6 - a 6 z 4 + 9 a 4 z 4 -14 a 2 z 4 + 4 z 4 -3 a 6 z 2 + 14 a 4 z 2 -14 a 2 z 2 + 5 z 2 -3 a 6 + 7 a 4 -5 a 2 + 2$
Kauffman polynomial (db, data sources)	$a 4 z 10 + a 2 z 10 + 4 a 5 z 9 + 7 a 3 z 9 + 3 a z 9 + 6 a 6 z 8 + 10 a 4 z 8 + 8 a 2 z 8 + 4 z 8 + 5 a 7 z 7 -3 a 5 z 7 -13 a 3 z 7 -2 a z 7 + 3 z 7 a -1 + 3 a 8 z 6 -12 a 6 z 6 -34 a 4 z 6 -30 a 2 z 6 + z 6 a -2 -10 z 6 + a 9 z 5 -8 a 7 z 5 -5 a 5 z 5 + 3 a 3 z 5 -10 a z 5 -9 z 5 a -1 -5 a 8 z 4 + 13 a 6 z 4 + 47 a 4 z 4 + 39 a 2 z 4 -3 z 4 a -2 + 7 z 4 -2 a 9 z 3 + 3 a 7 z 3 + 12 a 5 z 3 + 12 a 3 z 3 + 12 a z 3 + 7 z 3 a -1 + a 8 z 2 -10 a 6 z 2 -27 a 4 z 2 -23 a 2 z 2 + 2 z 2 a -2 -5 z 2 + a 9 z -2 a 7 z -6 a 5 z -6 a 3 z -5 a z -2 z a -1 + 3 a 6 + 7 a 4 + 5 a 2 + 2$
The A2 invariant	$- q 24 - q 20 -2 q 18 + 3 q 16 - q 14 + 3 q 12 + 2 q 10 - q 8 + 3 q 6 -4 q 4 + 2 q 2 -1- q -2 + 2 q -4 - q -6 + q -8$
The G2 invariant	$q 128 -2 q 126 + 5 q 124 -8 q 122 + 9 q 120 -7 q 118 + 13 q 114 -28 q 112 + 43 q 110 -53 q 108 + 43 q 106 -19 q 104 -24 q 102 + 81 q 100 -127 q 98 + 153 q 96 -137 q 94 + 64 q 92 + 41 q 90 -162 q 88 + 249 q 86 -269 q 84 + 202 q 82 -65 q 80 -105 q 78 + 239 q 76 -286 q 74 + 225 q 72 -84 q 70 -83 q 68 + 194 q 66 -206 q 64 + 112 q 62 + 54 q 60 -202 q 58 + 279 q 56 -224 q 54 + 60 q 52 + 159 q 50 -339 q 48 + 423 q 46 -355 q 44 + 170 q 42 + 81 q 40 -299 q 38 + 419 q 36 -394 q 34 + 241 q 32 -23 q 30 -180 q 28 + 282 q 26 -261 q 24 + 132 q 22 + 42 q 20 -183 q 18 + 218 q 16 -148 q 14 -7 q 12 + 172 q 10 -275 q 8 + 271 q 6 -166 q 4 -2 q 2 + 163-265 q -2 + 280 q -4 -203 q -6 + 83 q -8 + 43 q -10 -134 q -12 + 170 q -14 -152 q -16 + 102 q -18 -37 q -20 -18 q -22 + 51 q -24 -61 q -26 + 52 q -28 -32 q -30 + 15 q -32 + q -34 -10 q -36 + 10 q -38 -9 q -40 + 5 q -42 -2 q -44 + q -46$

Further Quantum Invariants

K11a92 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["K11a92"];

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

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

Out[4]= $- t 4 + 5 t 3 -12 t 2 + 21 t -25 + 21 t -1 -12 t -2 + 5 t -3 - t -4$

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

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

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]= { 103, -2 }

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

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

Out[8]= $q 3 -3 q 2 + 6 q -10 + 14 q -1 -16 q -2 + 17 q -3 -14 q -4 + 11 q -5 -7 q -6 + 3 q -7 - q -8$

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

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

Out[9]= $- a 2 z 8 + 2 a 4 z 6 -6 a 2 z 6 + z 6 - a 6 z 4 + 9 a 4 z 4 -14 a 2 z 4 + 4 z 4 -3 a 6 z 2 + 14 a 4 z 2 -14 a 2 z 2 + 5 z 2 -3 a 6 + 7 a 4 -5 a 2 + 2$

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

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

Out[10]= $a 4 z 10 + a 2 z 10 + 4 a 5 z 9 + 7 a 3 z 9 + 3 a z 9 + 6 a 6 z 8 + 10 a 4 z 8 + 8 a 2 z 8 + 4 z 8 + 5 a 7 z 7 -3 a 5 z 7 -13 a 3 z 7 -2 a z 7 + 3 z 7 a -1 + 3 a 8 z 6 -12 a 6 z 6 -34 a 4 z 6 -30 a 2 z 6 + z 6 a -2 -10 z 6 + a 9 z 5 -8 a 7 z 5 -5 a 5 z 5 + 3 a 3 z 5 -10 a z 5 -9 z 5 a -1 -5 a 8 z 4 + 13 a 6 z 4 + 47 a 4 z 4 + 39 a 2 z 4 -3 z 4 a -2 + 7 z 4 -2 a 9 z 3 + 3 a 7 z 3 + 12 a 5 z 3 + 12 a 3 z 3 + 12 a z 3 + 7 z 3 a -1 + a 8 z 2 -10 a 6 z 2 -27 a 4 z 2 -23 a 2 z 2 + 2 z 2 a -2 -5 z 2 + a 9 z -2 a 7 z -6 a 5 z -6 a 3 z -5 a z -2 z a -1 + 3 a 6 + 7 a 4 + 5 a 2 + 2$

[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["K11a92"];

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]= { $- t 4 + 5 t 3 -12 t 2 + 21 t -25 + 21 t -1 -12 t -2 + 5 t -3 - t -4$ , $q 3 -3 q 2 + 6 q -10 + 14 q -1 -16 q -2 + 17 q -3 -14 q -4 + 11 q -5 -7 q -6 + 3 q -7 - q -8$ }

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]= {}

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, -5)

[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 K11a92. 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

-2

-5

-7

-9

-3

-11

-13

-4

-15

-17

-1

Integral Khovanov Homology

(db, data source)

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