K11a88

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K11a87

K11a89

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 K11a88's page at Knotilus!

Visit K11a88's page at the original Knot Atlas!

[edit K11a88 Quick Notes]

[edit K11a88 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_18,9,19,10 X_2,11,3,12 X_20,14,21,13 X_22,16,1,15 X_8,17,9,18 X_6,19,7,20 X_14,22,15,21
Gauss code	1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, -7, 11, -8
Dowker-Thistlethwaite code	4 10 12 16 18 2 20 22 8 6 14

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	4
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a88/ThurstonBennequinNumber
Hyperbolic Volume	14.1444
A-Polynomial	See Data:K11a88/A-polynomial

[edit Notes for K11a88's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$4$
Rasmussen s-Invariant	0

[edit Notes for K11a88's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$t 4 -5 t 3 + 12 t 2 -20 t + 25-20 t -1 + 12 t -2 -5 t -3 + t -4$
Conway polynomial	$z 8 + 3 z 6 + 2 z 4 - z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 101, 0 }
Jones polynomial	$- q 5 + 3 q 4 -6 q 3 + 11 q 2 -14 q + 16-16 q -1 + 14 q -2 -10 q -3 + 6 q -4 -3 q -5 + q -6$
HOMFLY-PT polynomial (db, data sources)	$z 8 -2 a 2 z 6 - z 6 a -2 + 6 z 6 + a 4 z 4 -9 a 2 z 4 -4 z 4 a -2 + 14 z 4 + 3 a 4 z 2 -13 a 2 z 2 -5 z 2 a -2 + 14 z 2 + 2 a 4 -5 a 2 - a -2 + 5$
Kauffman polynomial (db, data sources)	$a 2 z 10 + z 10 + 3 a 3 z 9 + 7 a z 9 + 4 z 9 a -1 + 4 a 4 z 8 + 7 a 2 z 8 + 6 z 8 a -2 + 9 z 8 + 3 a 5 z 7 -3 a 3 z 7 -16 a z 7 -5 z 7 a -1 + 5 z 7 a -3 + a 6 z 6 -10 a 4 z 6 -25 a 2 z 6 -14 z 6 a -2 + 3 z 6 a -4 -31 z 6 -9 a 5 z 5 -4 a 3 z 5 + 18 a z 5 + 2 z 5 a -1 -10 z 5 a -3 + z 5 a -5 -3 a 6 z 4 + 7 a 4 z 4 + 33 a 2 z 4 + 16 z 4 a -2 -6 z 4 a -4 + 45 z 4 + 7 a 5 z 3 + 2 a 3 z 3 -10 a z 3 + 3 z 3 a -1 + 6 z 3 a -3 -2 z 3 a -5 + 2 a 6 z 2 -4 a 4 z 2 -23 a 2 z 2 -7 z 2 a -2 + 2 z 2 a -4 -26 z 2 - a 5 z + a z - z a -1 - z a -3 + 2 a 4 + 5 a 2 + a -2 + 5$
The A2 invariant	$q 18 + q 12 -3 q 10 + 2 q 8 - q 6 - q 4 + 2 q 2 -3 + 4 q -2 - q -4 + 2 q -6 + 2 q -8 -2 q -10 + q -12 - q -14$
The G2 invariant	$q 94 -2 q 92 + 5 q 90 -9 q 88 + 10 q 86 -10 q 84 + 2 q 82 + 15 q 80 -34 q 78 + 55 q 76 -63 q 74 + 49 q 72 -15 q 70 -43 q 68 + 110 q 66 -157 q 64 + 169 q 62 -122 q 60 + 26 q 58 + 100 q 56 -212 q 54 + 273 q 52 -249 q 50 + 137 q 48 + 23 q 46 -180 q 44 + 267 q 42 -248 q 40 + 141 q 38 + 20 q 36 -159 q 34 + 209 q 32 -161 q 30 + 8 q 28 + 159 q 26 -270 q 24 + 270 q 22 -147 q 20 -56 q 18 + 264 q 16 -395 q 14 + 397 q 12 -269 q 10 + 41 q 8 + 194 q 6 -357 q 4 + 399 q 2 -297 + 117 q -2 + 85 q -4 -224 q -6 + 254 q -8 -171 q -10 + 19 q -12 + 132 q -14 -205 q -16 + 178 q -18 -50 q -20 -109 q -22 + 236 q -24 -274 q -26 + 216 q -28 -84 q -30 -82 q -32 + 206 q -34 -259 q -36 + 232 q -38 -136 q -40 + 24 q -42 + 70 q -44 -129 q -46 + 139 q -48 -114 q -50 + 67 q -52 -16 q -54 -22 q -56 + 41 q -58 -45 q -60 + 37 q -62 -23 q -64 + 11 q -66 + q -68 -7 q -70 + 7 q -72 -7 q -74 + 4 q -76 -2 q -78 + q -80$

Further Quantum Invariants

K11a88 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["K11a88"];

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

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

Out[4]= $t 4 -5 t 3 + 12 t 2 -20 t + 25-20 t -1 + 12 t -2 -5 t -3 + t -4$

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

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

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]= { 101, 0 }

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

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

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

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

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

Out[9]= $z 8 -2 a 2 z 6 - z 6 a -2 + 6 z 6 + a 4 z 4 -9 a 2 z 4 -4 z 4 a -2 + 14 z 4 + 3 a 4 z 2 -13 a 2 z 2 -5 z 2 a -2 + 14 z 2 + 2 a 4 -5 a 2 - a -2 + 5$

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

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

Out[10]= $a 2 z 10 + z 10 + 3 a 3 z 9 + 7 a z 9 + 4 z 9 a -1 + 4 a 4 z 8 + 7 a 2 z 8 + 6 z 8 a -2 + 9 z 8 + 3 a 5 z 7 -3 a 3 z 7 -16 a z 7 -5 z 7 a -1 + 5 z 7 a -3 + a 6 z 6 -10 a 4 z 6 -25 a 2 z 6 -14 z 6 a -2 + 3 z 6 a -4 -31 z 6 -9 a 5 z 5 -4 a 3 z 5 + 18 a z 5 + 2 z 5 a -1 -10 z 5 a -3 + z 5 a -5 -3 a 6 z 4 + 7 a 4 z 4 + 33 a 2 z 4 + 16 z 4 a -2 -6 z 4 a -4 + 45 z 4 + 7 a 5 z 3 + 2 a 3 z 3 -10 a z 3 + 3 z 3 a -1 + 6 z 3 a -3 -2 z 3 a -5 + 2 a 6 z 2 -4 a 4 z 2 -23 a 2 z 2 -7 z 2 a -2 + 2 z 2 a -4 -26 z 2 - a 5 z + a z - z a -1 - z a -3 + 2 a 4 + 5 a 2 + a -2 + 5$

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11a84,}

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["K11a88"];

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]= { $t 4 -5 t 3 + 12 t 2 -20 t + 25-20 t -1 + 12 t -2 -5 t -3 + t -4$ , $- q 5 + 3 q 4 -6 q 3 + 11 q 2 -14 q + 16-16 q -1 + 14 q -2 -10 q -3 + 6 q -4 -3 q -5 + q -6$ }

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]= {K11a84,}

[edit] Vassiliev invariants

V₂ and V₃:

(-1, 2)

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

-4

16

8

$-\frac{62}{3}$

$-\frac{34}{3}$

-64

$-\frac{320}{3}$

48

$-\frac{32}{3}$

128

$\frac{248}{3}$

$\frac{136}{3}$

$\frac{11729}{30}$

$-\frac{1778}{15}$

$\frac{18658}{45}$

$-\frac{1265}{18}$

$\frac{2609}{30}$

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 K11a88. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

-3

-1

-3

-2

-5

-7

-4

-9

-11

-2

-13

Integral Khovanov Homology

(db, data source)

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