K11a62

From Knot Atlas

Jump to: navigation, search

K11a61

K11a63

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 K11a62's page at Knotilus!

Visit K11a62's page at the original Knot Atlas!

[edit K11a62 Quick Notes]

[edit K11a62 Further Notes and Views]

[edit] Knot presentations

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

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	3
3-genus	4
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a62/ThurstonBennequinNumber
Hyperbolic Volume	10.6386
A-Polynomial	See Data:K11a62/A-polynomial

[edit Notes for K11a62's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$4$
Rasmussen s-Invariant	6

[edit Notes for K11a62's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 4 + 5 t 3 -8 t 2 + 9 t -9 + 9 t -1 -8 t -2 + 5 t -3 - t -4$
Conway polynomial	$- z 8 -3 z 6 + 2 z 4 + 6 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 55, -6 }
Jones polynomial	$q -1 -2 q -2 + 4 q -3 -5 q -4 + 7 q -5 -8 q -6 + 8 q -7 -7 q -8 + 6 q -9 -4 q -10 + 2 q -11 - q -12$
HOMFLY-PT polynomial (db, data sources)	$- z 4 a 10 -4 z 2 a 10 -3 a 10 + 2 z 6 a 8 + 10 z 4 a 8 + 14 z 2 a 8 + 6 a 8 - z 8 a 6 -6 z 6 a 6 -12 z 4 a 6 -11 z 2 a 6 -5 a 6 + z 6 a 4 + 5 z 4 a 4 + 7 z 2 a 4 + 3 a 4$
Kauffman polynomial (db, data sources)	$z 3 a 15 - z a 15 + 2 z 4 a 14 - z 2 a 14 + 3 z 5 a 13 -2 z 3 a 13 + z a 13 + 4 z 6 a 12 -6 z 4 a 12 + 4 z 2 a 12 + 4 z 7 a 11 -7 z 5 a 11 + 2 z 3 a 11 + 4 z 8 a 10 -11 z 6 a 10 + 10 z 4 a 10 -9 z 2 a 10 + 3 a 10 + 3 z 9 a 9 -9 z 7 a 9 + 6 z 5 a 9 -3 z 3 a 9 + z 10 a 8 + 2 z 8 a 8 -24 z 6 a 8 + 41 z 4 a 8 -27 z 2 a 8 + 6 a 8 + 5 z 9 a 7 -24 z 7 a 7 + 34 z 5 a 7 -17 z 3 a 7 + 3 z a 7 + z 10 a 6 - z 8 a 6 -15 z 6 a 6 + 35 z 4 a 6 -23 z 2 a 6 + 5 a 6 + 2 z 9 a 5 -11 z 7 a 5 + 18 z 5 a 5 -9 z 3 a 5 + z a 5 + z 8 a 4 -6 z 6 a 4 + 12 z 4 a 4 -10 z 2 a 4 + 3 a 4$
The A2 invariant	$- q 36 - q 34 - q 30 + q 28 + 2 q 24 + q 22 - q 20 + q 18 -2 q 16 + q 14 + q 10 + q 8 + q 4$
The G2 invariant	$q 196 - q 194 + 2 q 192 -2 q 190 + q 188 -2 q 184 + 4 q 182 -5 q 180 + 6 q 178 -6 q 176 + 3 q 174 + 2 q 172 -6 q 170 + 11 q 168 -12 q 166 + 10 q 164 -8 q 162 - q 160 + 7 q 158 -14 q 156 + 15 q 154 -12 q 152 + 5 q 150 -6 q 146 + 8 q 144 -8 q 142 + 6 q 140 -6 q 138 + 4 q 136 -5 q 134 + 3 q 132 + 2 q 130 -8 q 128 + 14 q 126 -13 q 124 + 7 q 122 + q 120 -10 q 118 + 11 q 116 -4 q 114 -5 q 112 + 14 q 110 -14 q 108 + 7 q 106 + 11 q 104 -24 q 102 + 31 q 100 -26 q 98 + 12 q 96 + 7 q 94 -21 q 92 + 33 q 90 -30 q 88 + 23 q 86 -9 q 84 -7 q 82 + 19 q 80 -24 q 78 + 18 q 76 -9 q 74 -6 q 72 + 16 q 70 -18 q 68 + 7 q 66 + 10 q 64 -25 q 62 + 29 q 60 -22 q 58 + q 56 + 20 q 54 -34 q 52 + 38 q 50 -26 q 48 + 9 q 46 + 11 q 44 -22 q 42 + 25 q 40 -17 q 38 + 10 q 36 -4 q 32 + 6 q 30 -5 q 28 + 4 q 26 - q 24 + q 22$

Further Quantum Invariants

K11a62 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["K11a62"];

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

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

Out[4]= $- t 4 + 5 t 3 -8 t 2 + 9 t -9 + 9 t -1 -8 t -2 + 5 t -3 - t -4$

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

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

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]= { 55, -6 }

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

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

Out[8]= $q -1 -2 q -2 + 4 q -3 -5 q -4 + 7 q -5 -8 q -6 + 8 q -7 -7 q -8 + 6 q -9 -4 q -10 + 2 q -11 - q -12$

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

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

Out[9]= $- z 4 a 10 -4 z 2 a 10 -3 a 10 + 2 z 6 a 8 + 10 z 4 a 8 + 14 z 2 a 8 + 6 a 8 - z 8 a 6 -6 z 6 a 6 -12 z 4 a 6 -11 z 2 a 6 -5 a 6 + z 6 a 4 + 5 z 4 a 4 + 7 z 2 a 4 + 3 a 4$

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

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

Out[10]= $z 3 a 15 - z a 15 + 2 z 4 a 14 - z 2 a 14 + 3 z 5 a 13 -2 z 3 a 13 + z a 13 + 4 z 6 a 12 -6 z 4 a 12 + 4 z 2 a 12 + 4 z 7 a 11 -7 z 5 a 11 + 2 z 3 a 11 + 4 z 8 a 10 -11 z 6 a 10 + 10 z 4 a 10 -9 z 2 a 10 + 3 a 10 + 3 z 9 a 9 -9 z 7 a 9 + 6 z 5 a 9 -3 z 3 a 9 + z 10 a 8 + 2 z 8 a 8 -24 z 6 a 8 + 41 z 4 a 8 -27 z 2 a 8 + 6 a 8 + 5 z 9 a 7 -24 z 7 a 7 + 34 z 5 a 7 -17 z 3 a 7 + 3 z a 7 + z 10 a 6 - z 8 a 6 -15 z 6 a 6 + 35 z 4 a 6 -23 z 2 a 6 + 5 a 6 + 2 z 9 a 5 -11 z 7 a 5 + 18 z 5 a 5 -9 z 3 a 5 + z a 5 + z 8 a 4 -6 z 6 a 4 + 12 z 4 a 4 -10 z 2 a 4 + 3 a 4$

[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["K11a62"];

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]= { $- t 4 + 5 t 3 -8 t 2 + 9 t -9 + 9 t -1 -8 t -2 + 5 t -3 - t -4$ , $q -1 -2 q -2 + 4 q -3 -5 q -4 + 7 q -5 -8 q -6 + 8 q -7 -7 q -8 + 6 q -9 -4 q -10 + 2 q -11 - q -12$ }

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:

(6, -17)

[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 =

-6 is the signature of K11a62. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

-3

-1

-5

-7

-1

-9

-11

-1

-13

-15

-17

-1

-19

-21

-2

-23

-25

-1

Integral Khovanov Homology

(db, data source)

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