K11a99

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K11a98

K11a100

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 K11a99's page at Knotilus!

Visit K11a99's page at the original Knot Atlas!

[edit K11a99 Quick Notes]

[edit K11a99 Further Notes and Views]

[edit] Knot presentations

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

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	${1,2}$
3-genus	4
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a99/ThurstonBennequinNumber
Hyperbolic Volume	16.6086
A-Polynomial	See Data:K11a99/A-polynomial

[edit Notes for K11a99'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 K11a99's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 4 + 6 t 3 -17 t 2 + 28 t -31 + 28 t -1 -17 t -2 + 6 t -3 - t -4$
Conway polynomial	$- z 8 -2 z 6 - z 4 -2 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 135, 2 }
Jones polynomial	$q 7 -4 q 6 + 9 q 5 -14 q 4 + 19 q 3 -22 q 2 + 21 q -18 + 14 q -1 -8 q -2 + 4 q -3 - q -4$
HOMFLY-PT polynomial (db, data sources)	$- z 8 a -2 -5 z 6 a -2 + z 6 a -4 + 2 z 6 - a 2 z 4 -10 z 4 a -2 + 3 z 4 a -4 + 7 z 4 -2 a 2 z 2 -10 z 2 a -2 + 3 z 2 a -4 + 7 z 2 -4 a -2 + 2 a -4 + 3$
Kauffman polynomial (db, data sources)	$2 z 10 a -2 + 2 z 10 + 5 a z 9 + 13 z 9 a -1 + 8 z 9 a -3 + 4 a 2 z 8 + 19 z 8 a -2 + 13 z 8 a -4 + 10 z 8 + a 3 z 7 -12 a z 7 -27 z 7 a -1 - z 7 a -3 + 13 z 7 a -5 -14 a 2 z 6 -62 z 6 a -2 -18 z 6 a -4 + 9 z 6 a -6 -49 z 6 -3 a 3 z 5 + a z 5 -25 z 5 a -3 -17 z 5 a -5 + 4 z 5 a -7 + 16 a 2 z 4 + 55 z 4 a -2 + 7 z 4 a -4 -8 z 4 a -6 + z 4 a -8 + 55 z 4 + 3 a 3 z 3 + 10 a z 3 + 15 z 3 a -1 + 18 z 3 a -3 + 9 z 3 a -5 - z 3 a -7 -6 a 2 z 2 -23 z 2 a -2 -5 z 2 a -4 + 3 z 2 a -6 -21 z 2 - a 3 z -3 a z -3 z a -1 -3 z a -3 -2 z a -5 + 4 a -2 + 2 a -4 + 3$
The A2 invariant	$- q 12 + q 10 + q 8 - q 6 + 4 q 4 -2 q 2 + 2 + q -2 -4 q -4 + 3 q -6 -5 q -8 + 3 q -10 - q -14 + 3 q -16 -2 q -18 + q -20$
The G2 invariant	$q 60 -3 q 58 + 9 q 56 -19 q 54 + 28 q 52 -33 q 50 + 17 q 48 + 26 q 46 -91 q 44 + 165 q 42 -204 q 40 + 167 q 38 -37 q 36 -174 q 34 + 394 q 32 -522 q 30 + 480 q 28 -242 q 26 -135 q 24 + 516 q 22 -741 q 20 + 713 q 18 -405 q 16 -48 q 14 + 467 q 12 -683 q 10 + 599 q 8 -269 q 6 -156 q 4 + 498 q 2 -586 + 391 q -2 + 18 q -4 -460 q -6 + 748 q -8 -756 q -10 + 454 q -12 + 43 q -14 -573 q -16 + 935 q -18 -987 q -20 + 703 q -22 -172 q -24 -401 q -26 + 798 q -28 -894 q -30 + 643 q -32 -188 q -34 -279 q -36 + 566 q -38 -560 q -40 + 292 q -42 + 105 q -44 -431 q -46 + 544 q -48 -402 q -50 + 68 q -52 + 293 q -54 -544 q -56 + 602 q -58 -444 q -60 + 168 q -62 + 140 q -64 -372 q -66 + 462 q -68 -420 q -70 + 276 q -72 -96 q -74 -65 q -76 + 179 q -78 -227 q -80 + 213 q -82 -152 q -84 + 78 q -86 -5 q -88 -46 q -90 + 68 q -92 -74 q -94 + 57 q -96 -33 q -98 + 13 q -100 + 4 q -102 -10 q -104 + 11 q -106 -10 q -108 + 6 q -110 -3 q -112 + q -114$

Further Quantum Invariants

K11a99 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["K11a99"];

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

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

Out[4]= $- t 4 + 6 t 3 -17 t 2 + 28 t -31 + 28 t -1 -17 t -2 + 6 t -3 - t -4$

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

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

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]= { 135, 2 }

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

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

Out[8]= $q 7 -4 q 6 + 9 q 5 -14 q 4 + 19 q 3 -22 q 2 + 21 q -18 + 14 q -1 -8 q -2 + 4 q -3 - q -4$

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

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

Out[9]= $- z 8 a -2 -5 z 6 a -2 + z 6 a -4 + 2 z 6 - a 2 z 4 -10 z 4 a -2 + 3 z 4 a -4 + 7 z 4 -2 a 2 z 2 -10 z 2 a -2 + 3 z 2 a -4 + 7 z 2 -4 a -2 + 2 a -4 + 3$

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

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

Out[10]= $2 z 10 a -2 + 2 z 10 + 5 a z 9 + 13 z 9 a -1 + 8 z 9 a -3 + 4 a 2 z 8 + 19 z 8 a -2 + 13 z 8 a -4 + 10 z 8 + a 3 z 7 -12 a z 7 -27 z 7 a -1 - z 7 a -3 + 13 z 7 a -5 -14 a 2 z 6 -62 z 6 a -2 -18 z 6 a -4 + 9 z 6 a -6 -49 z 6 -3 a 3 z 5 + a z 5 -25 z 5 a -3 -17 z 5 a -5 + 4 z 5 a -7 + 16 a 2 z 4 + 55 z 4 a -2 + 7 z 4 a -4 -8 z 4 a -6 + z 4 a -8 + 55 z 4 + 3 a 3 z 3 + 10 a z 3 + 15 z 3 a -1 + 18 z 3 a -3 + 9 z 3 a -5 - z 3 a -7 -6 a 2 z 2 -23 z 2 a -2 -5 z 2 a -4 + 3 z 2 a -6 -21 z 2 - a 3 z -3 a z -3 z a -1 -3 z a -3 -2 z a -5 + 4 a -2 + 2 a -4 + 3$

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

Same Alexander/Conway Polynomial: {K11a277,}

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["K11a99"];

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]= { $- t 4 + 6 t 3 -17 t 2 + 28 t -31 + 28 t -1 -17 t -2 + 6 t -3 - t -4$ , $q 7 -4 q 6 + 9 q 5 -14 q 4 + 19 q 3 -22 q 2 + 21 q -18 + 14 q -1 -8 q -2 + 4 q -3 - q -4$ }

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]= {K11a277,}

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, -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

-8

-16

32

$\frac{116}{3}$

$\frac{76}{3}$

128

$\frac{800}{3}$

$\frac{224}{3}$

80

$-\frac{256}{3}$

128

$-\frac{928}{3}$

$-\frac{608}{3}$

$\frac{1769}{15}$

$\frac{948}{5}$

$-\frac{13084}{45}$

$\frac{1111}{9}$

$-\frac{1591}{15}$

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 =

2 is the signature of K11a99. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

-3

-5

-3

-1

-4

-3

-5

-4

-7

-9

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$