K11a66

From Knot Atlas

Jump to: navigation, search

K11a65

K11a67

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 K11a66's page at Knotilus!

Visit K11a66's page at the original Knot Atlas!

[edit K11a66 Quick Notes]

[edit K11a66 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₃₉₄ X_16,5,17,6 X_12,8,13,7 X_2,9,3,10 X_18,12,19,11 X_20,13,21,14 X_22,15,1,16 X_10,18,11,17 X_6,19,7,20 X_14,21,15,22
Gauss code	1, -5, 2, -1, 3, -10, 4, -2, 5, -9, 6, -4, 7, -11, 8, -3, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code	4 8 16 12 2 18 20 22 10 6 14

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Chiral
Unknotting number	${1,2}$
3-genus	4
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a66/ThurstonBennequinNumber
Hyperbolic Volume	15.8011
A-Polynomial	See Data:K11a66/A-polynomial

[edit Notes for K11a66'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 K11a66's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 4 + 6 t 3 -15 t 2 + 24 t -27 + 24 t -1 -15 t -2 + 6 t -3 - t -4$
Conway polynomial	$- z 8 -2 z 6 + z 4 + 2 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 119, -2 }
Jones polynomial	$q 3 -4 q 2 + 8 q -12 + 17 q -1 -19 q -2 + 19 q -3 -16 q -4 + 12 q -5 -7 q -6 + 3 q -7 - q -8$
HOMFLY-PT polynomial (db, data sources)	$- a 2 z 8 + 2 a 4 z 6 -5 a 2 z 6 + z 6 - a 6 z 4 + 8 a 4 z 4 -9 a 2 z 4 + 3 z 4 -3 a 6 z 2 + 10 a 4 z 2 -7 a 2 z 2 + 2 z 2 -2 a 6 + 4 a 4 -2 a 2 + 1$
Kauffman polynomial (db, data sources)	$2 a 4 z 10 + 2 a 2 z 10 + 5 a 5 z 9 + 11 a 3 z 9 + 6 a z 9 + 6 a 6 z 8 + 7 a 4 z 8 + 8 a 2 z 8 + 7 z 8 + 5 a 7 z 7 -6 a 5 z 7 -28 a 3 z 7 -13 a z 7 + 4 z 7 a -1 + 3 a 8 z 6 -9 a 6 z 6 -26 a 4 z 6 -35 a 2 z 6 + z 6 a -2 -20 z 6 + a 9 z 5 -7 a 7 z 5 + 5 a 5 z 5 + 29 a 3 z 5 + 6 a z 5 -10 z 5 a -1 -5 a 8 z 4 + 8 a 6 z 4 + 36 a 4 z 4 + 41 a 2 z 4 -2 z 4 a -2 + 16 z 4 -2 a 9 z 3 + 2 a 7 z 3 -3 a 5 z 3 -12 a 3 z 3 - a z 3 + 4 z 3 a -1 + 2 a 8 z 2 -7 a 6 z 2 -21 a 4 z 2 -17 a 2 z 2 -5 z 2 + a 9 z + 2 a 3 z + a z + 2 a 6 + 4 a 4 + 2 a 2 + 1$
The A2 invariant	$- q 24 -2 q 18 + 3 q 16 -2 q 14 + 2 q 12 + 2 q 10 -2 q 8 + 4 q 6 -4 q 4 + 3 q 2 - q -2 + 2 q -4 -2 q -6 + q -8$
The G2 invariant	$q 128 -2 q 126 + 5 q 124 -8 q 122 + 9 q 120 -8 q 118 + q 116 + 13 q 114 -30 q 112 + 48 q 110 -58 q 108 + 49 q 106 -22 q 104 -26 q 102 + 86 q 100 -138 q 98 + 166 q 96 -158 q 94 + 93 q 92 + 8 q 90 -140 q 88 + 268 q 86 -338 q 84 + 321 q 82 -195 q 80 -17 q 78 + 243 q 76 -408 q 74 + 439 q 72 -312 q 70 + 66 q 68 + 200 q 66 -367 q 64 + 359 q 62 -166 q 60 -109 q 58 + 341 q 56 -418 q 54 + 280 q 52 + 13 q 50 -346 q 48 + 586 q 46 -611 q 44 + 417 q 42 -53 q 40 -338 q 38 + 616 q 36 -692 q 34 + 539 q 32 -221 q 30 -147 q 28 + 437 q 26 -537 q 24 + 432 q 22 -158 q 20 -154 q 18 + 360 q 16 -388 q 14 + 209 q 12 + 79 q 10 -342 q 8 + 475 q 6 -401 q 4 + 155 q 2 + 161-417 q -2 + 514 q -4 -431 q -6 + 212 q -8 + 43 q -10 -244 q -12 + 335 q -14 -307 q -16 + 202 q -18 -63 q -20 -51 q -22 + 109 q -24 -122 q -26 + 95 q -28 -53 q -30 + 19 q -32 + 9 q -34 -19 q -36 + 18 q -38 -14 q -40 + 7 q -42 -3 q -44 + q -46$

Further Quantum Invariants

K11a66 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["K11a66"];

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

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

Out[4]= $- t 4 + 6 t 3 -15 t 2 + 24 t -27 + 24 t -1 -15 t -2 + 6 t -3 - t -4$

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]= { 119, -2 }

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

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

Out[8]= $q 3 -4 q 2 + 8 q -12 + 17 q -1 -19 q -2 + 19 q -3 -16 q -4 + 12 q -5 -7 q -6 + 3 q -7 - q -8$

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

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

Out[9]= $- a 2 z 8 + 2 a 4 z 6 -5 a 2 z 6 + z 6 - a 6 z 4 + 8 a 4 z 4 -9 a 2 z 4 + 3 z 4 -3 a 6 z 2 + 10 a 4 z 2 -7 a 2 z 2 + 2 z 2 -2 a 6 + 4 a 4 -2 a 2 + 1$

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

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

Out[10]= $2 a 4 z 10 + 2 a 2 z 10 + 5 a 5 z 9 + 11 a 3 z 9 + 6 a z 9 + 6 a 6 z 8 + 7 a 4 z 8 + 8 a 2 z 8 + 7 z 8 + 5 a 7 z 7 -6 a 5 z 7 -28 a 3 z 7 -13 a z 7 + 4 z 7 a -1 + 3 a 8 z 6 -9 a 6 z 6 -26 a 4 z 6 -35 a 2 z 6 + z 6 a -2 -20 z 6 + a 9 z 5 -7 a 7 z 5 + 5 a 5 z 5 + 29 a 3 z 5 + 6 a z 5 -10 z 5 a -1 -5 a 8 z 4 + 8 a 6 z 4 + 36 a 4 z 4 + 41 a 2 z 4 -2 z 4 a -2 + 16 z 4 -2 a 9 z 3 + 2 a 7 z 3 -3 a 5 z 3 -12 a 3 z 3 - a z 3 + 4 z 3 a -1 + 2 a 8 z 2 -7 a 6 z 2 -21 a 4 z 2 -17 a 2 z 2 -5 z 2 + a 9 z + 2 a 3 z + a z + 2 a 6 + 4 a 4 + 2 a 2 + 1$

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

Same Alexander/Conway Polynomial: {K11a163,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11a121,}

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["K11a66"];

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]= { $- t 4 + 6 t 3 -15 t 2 + 24 t -27 + 24 t -1 -15 t -2 + 6 t -3 - t -4$ , $q 3 -4 q 2 + 8 q -12 + 17 q -1 -19 q -2 + 19 q -3 -16 q -4 + 12 q -5 -7 q -6 + 3 q -7 - q -8$ }

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]= {K11a163,}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]= {K11a121,}

[edit] Vassiliev invariants

V₂ and V₃:

(2, -4)

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

-32

32

$\frac{316}{3}$

$\frac{20}{3}$

-256

$-\frac{1376}{3}$

$-\frac{32}{3}$

-96

$\frac{256}{3}$

512

$\frac{2528}{3}$

$\frac{160}{3}$

$\frac{30991}{15}$

$-\frac{764}{15}$

$\frac{30604}{45}$

$\frac{1025}{9}$

$\frac{991}{15}$

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 K11a66. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

-3

-4

-1

-3

-2

-5

-7

-9

-4

-11

-13

-4

-15

-17

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -3$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$