K11a274

From Knot Atlas

Jump to: navigation, search

K11a273

K11a275

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 K11a274's page at Knotilus!

Visit K11a274's page at the original Knot Atlas!

[edit K11a274 Quick Notes]

[edit K11a274 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₆₂₇₁ X_10,4,11,3 X_14,5,15,6 X_18,7,19,8 X_2,10,3,9 X_20,11,21,12 X_22,14,1,13 X_4,15,5,16 X_12,17,13,18 X_8,19,9,20 X_16,22,17,21
Gauss code	1, -5, 2, -8, 3, -1, 4, -10, 5, -2, 6, -9, 7, -3, 8, -11, 9, -4, 10, -6, 11, -7
Dowker-Thistlethwaite code	6 10 14 18 2 20 22 4 12 8 16

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Chiral
Unknotting number	${1,2}$
3-genus	4
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a274/ThurstonBennequinNumber
Hyperbolic Volume	18.6505
A-Polynomial	See Data:K11a274/A-polynomial

[edit Notes for K11a274'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 K11a274's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$t 4 -6 t 3 + 18 t 2 -35 t + 45-35 t -1 + 18 t -2 -6 t -3 + t -4$
Conway polynomial	$z 8 + 2 z 6 + 2 z 4 - z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 165, 0 }
Jones polynomial	$- q 5 + 4 q 4 -10 q 3 + 18 q 2 -23 q + 27-27 q -1 + 23 q -2 -17 q -3 + 10 q -4 -4 q -5 + q -6$
HOMFLY-PT polynomial (db, data sources)	$z 8 -2 a 2 z 6 - z 6 a -2 + 5 z 6 + a 4 z 4 -7 a 2 z 4 -3 z 4 a -2 + 11 z 4 + 2 a 4 z 2 -10 a 2 z 2 -4 z 2 a -2 + 11 z 2 + 2 a 4 -5 a 2 - a -2 + 5$
Kauffman polynomial (db, data sources)	$3 a 2 z 10 + 3 z 10 + 8 a 3 z 9 + 18 a z 9 + 10 z 9 a -1 + 8 a 4 z 8 + 16 a 2 z 8 + 13 z 8 a -2 + 21 z 8 + 4 a 5 z 7 -10 a 3 z 7 -30 a z 7 -7 z 7 a -1 + 9 z 7 a -3 + a 6 z 6 -17 a 4 z 6 -51 a 2 z 6 -22 z 6 a -2 + 4 z 6 a -4 -59 z 6 -8 a 5 z 5 -3 a 3 z 5 + 10 a z 5 -8 z 5 a -1 -12 z 5 a -3 + z 5 a -5 -2 a 6 z 4 + 13 a 4 z 4 + 48 a 2 z 4 + 19 z 4 a -2 -4 z 4 a -4 + 56 z 4 + 5 a 5 z 3 + 4 a 3 z 3 - a z 3 + 8 z 3 a -1 + 7 z 3 a -3 - z 3 a -5 + a 6 z 2 -6 a 4 z 2 -24 a 2 z 2 -9 z 2 a -2 + z 2 a -4 -27 z 2 - a 5 z + a z - z a -1 - z a -3 + 2 a 4 + 5 a 2 + a -2 + 5$
The A2 invariant	$q 18 - q 16 + q 14 + 3 q 12 -5 q 10 + 3 q 8 -3 q 6 -2 q 4 + 4 q 2 -4 + 7 q -2 -3 q -4 + 2 q -6 + 3 q -8 -4 q -10 + 2 q -12 - q -14$
The G2 invariant	Data:K11a274/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a274 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["K11a274"];

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

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

Out[4]= $t 4 -6 t 3 + 18 t 2 -35 t + 45-35 t -1 + 18 t -2 -6 t -3 + t -4$

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

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

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]= { 165, 0 }

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

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

Out[8]= $- q 5 + 4 q 4 -10 q 3 + 18 q 2 -23 q + 27-27 q -1 + 23 q -2 -17 q -3 + 10 q -4 -4 q -5 + q -6$

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

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

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

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

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

Out[10]= $3 a 2 z 10 + 3 z 10 + 8 a 3 z 9 + 18 a z 9 + 10 z 9 a -1 + 8 a 4 z 8 + 16 a 2 z 8 + 13 z 8 a -2 + 21 z 8 + 4 a 5 z 7 -10 a 3 z 7 -30 a z 7 -7 z 7 a -1 + 9 z 7 a -3 + a 6 z 6 -17 a 4 z 6 -51 a 2 z 6 -22 z 6 a -2 + 4 z 6 a -4 -59 z 6 -8 a 5 z 5 -3 a 3 z 5 + 10 a z 5 -8 z 5 a -1 -12 z 5 a -3 + z 5 a -5 -2 a 6 z 4 + 13 a 4 z 4 + 48 a 2 z 4 + 19 z 4 a -2 -4 z 4 a -4 + 56 z 4 + 5 a 5 z 3 + 4 a 3 z 3 - a z 3 + 8 z 3 a -1 + 7 z 3 a -3 - z 3 a -5 + a 6 z 2 -6 a 4 z 2 -24 a 2 z 2 -9 z 2 a -2 + z 2 a -4 -27 z 2 - a 5 z + a z - z a -1 - z a -3 + 2 a 4 + 5 a 2 + a -2 + 5$

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

Same Alexander/Conway Polynomial: {}

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

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["K11a274"];

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]= { $t 4 -6 t 3 + 18 t 2 -35 t + 45-35 t -1 + 18 t -2 -6 t -3 + t -4$ , $- q 5 + 4 q 4 -10 q 3 + 18 q 2 -23 q + 27-27 q -1 + 23 q -2 -17 q -3 + 10 q -4 -4 q -5 + q -6$ }

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]= {}

[edit] Vassiliev invariants

V₂ and V₃:

(-1, 2)

[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 K11a274. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

-6

-5

-1

-3

-4

-5

-7

-9

-11

-3

-13

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -2$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -1$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 0$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{15}$
$r = 1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$