K11a8

From Knot Atlas

Jump to: navigation, search

K11a7

K11a9

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 K11a8's page at Knotilus!

Visit K11a8's page at the original Knot Atlas!

[edit K11a8 Quick Notes]

[edit K11a8 Further Notes and Views]

[edit] Knot presentations

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

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	3
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a8/ThurstonBennequinNumber
Hyperbolic Volume	14.9162
A-Polynomial	See Data:K11a8/A-polynomial

[edit Notes for K11a8's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$[2,3]$
Rasmussen s-Invariant	0

[edit Notes for K11a8's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$-2 t 3 + 11 t 2 -27 t + 37-27 t -1 + 11 t -2 -2 t -3$
Conway polynomial	$-2 z 6 - z 4 - z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 117, 0 }
Jones polynomial	$q 4 -4 q 3 + 9 q 2 -13 q + 17-19 q -1 + 18 q -2 -15 q -3 + 11 q -4 -6 q -5 + 3 q -6 - q -7$
HOMFLY-PT polynomial (db, data sources)	$- z 2 a 6 - a 6 + 2 z 4 a 4 + 4 z 2 a 4 + 3 a 4 - z 6 a 2 -2 z 4 a 2 -3 z 2 a 2 -2 a 2 - z 6 -2 z 4 -2 z 2 + z 4 a -2 + z 2 a -2 + a -2$
Kauffman polynomial (db, data sources)	$a 4 z 10 + a 2 z 10 + 3 a 5 z 9 + 8 a 3 z 9 + 5 a z 9 + 3 a 6 z 8 + 9 a 4 z 8 + 16 a 2 z 8 + 10 z 8 + a 7 z 7 -6 a 5 z 7 -12 a 3 z 7 + 7 a z 7 + 12 z 7 a -1 -12 a 6 z 6 -39 a 4 z 6 -45 a 2 z 6 + 9 z 6 a -2 -9 z 6 -4 a 7 z 5 -5 a 5 z 5 -13 a 3 z 5 -32 a z 5 -16 z 5 a -1 + 4 z 5 a -3 + 16 a 6 z 4 + 46 a 4 z 4 + 35 a 2 z 4 -9 z 4 a -2 + z 4 a -4 -5 z 4 + 5 a 7 z 3 + 14 a 5 z 3 + 21 a 3 z 3 + 21 a z 3 + 8 z 3 a -1 - z 3 a -3 -8 a 6 z 2 -20 a 4 z 2 -13 a 2 z 2 + 4 z 2 a -2 + 3 z 2 -2 a 7 z -5 a 5 z -5 a 3 z -4 a z -2 z a -1 + a 6 + 3 a 4 + 2 a 2 - a -2$
The A2 invariant	Data:K11a8/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a8/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a8 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["K11a8"];

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

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

Out[4]= $-2 t 3 + 11 t 2 -27 t + 37-27 t -1 + 11 t -2 -2 t -3$

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

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

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]= { 117, 0 }

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

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

Out[8]= $q 4 -4 q 3 + 9 q 2 -13 q + 17-19 q -1 + 18 q -2 -15 q -3 + 11 q -4 -6 q -5 + 3 q -6 - q -7$

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

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

Out[9]= $- z 2 a 6 - a 6 + 2 z 4 a 4 + 4 z 2 a 4 + 3 a 4 - z 6 a 2 -2 z 4 a 2 -3 z 2 a 2 -2 a 2 - z 6 -2 z 4 -2 z 2 + z 4 a -2 + z 2 a -2 + a -2$

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

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

Out[10]= $a 4 z 10 + a 2 z 10 + 3 a 5 z 9 + 8 a 3 z 9 + 5 a z 9 + 3 a 6 z 8 + 9 a 4 z 8 + 16 a 2 z 8 + 10 z 8 + a 7 z 7 -6 a 5 z 7 -12 a 3 z 7 + 7 a z 7 + 12 z 7 a -1 -12 a 6 z 6 -39 a 4 z 6 -45 a 2 z 6 + 9 z 6 a -2 -9 z 6 -4 a 7 z 5 -5 a 5 z 5 -13 a 3 z 5 -32 a z 5 -16 z 5 a -1 + 4 z 5 a -3 + 16 a 6 z 4 + 46 a 4 z 4 + 35 a 2 z 4 -9 z 4 a -2 + z 4 a -4 -5 z 4 + 5 a 7 z 3 + 14 a 5 z 3 + 21 a 3 z 3 + 21 a z 3 + 8 z 3 a -1 - z 3 a -3 -8 a 6 z 2 -20 a 4 z 2 -13 a 2 z 2 + 4 z 2 a -2 + 3 z 2 -2 a 7 z -5 a 5 z -5 a 3 z -4 a z -2 z a -1 + a 6 + 3 a 4 + 2 a 2 - a -2$

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

Same Alexander/Conway Polynomial: {K11a38, K11a187, K11a249,}

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["K11a8"];

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]= { $-2 t 3 + 11 t 2 -27 t + 37-27 t -1 + 11 t -2 -2 t -3$ , $q 4 -4 q 3 + 9 q 2 -13 q + 17-19 q -1 + 18 q -2 -15 q -3 + 11 q -4 -6 q -5 + 3 q -6 - q -7$ }

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]= {K11a38, K11a187, K11a249,}

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:

(-1, -1)

[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 K11a8. 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

-2

-3

-1

-5

-7

-4

-9

-11

-3

-13

-15

-1

Integral Khovanov Homology

(db, data source)

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