K11a305

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K11a304

K11a306

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 K11a305's page at Knotilus!

Visit K11a305's page at the original Knot Atlas!

[edit K11a305 Quick Notes]

[edit K11a305 Further Notes and Views]

[edit] Knot presentations

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

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	${1,2}$
3-genus	4
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a305/ThurstonBennequinNumber
Hyperbolic Volume	17.2512
A-Polynomial	See Data:K11a305/A-polynomial

[edit Notes for K11a305's three dimensional invariants]

[edit] Four dimensional invariants

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

[edit Notes for K11a305's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 4 + 6 t 3 -16 t 2 + 28 t -33 + 28 t -1 -16 t -2 + 6 t -3 - t -4$
Conway polynomial	$- z 8 -2 z 6 + 2 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 135, -2 }
Jones polynomial	$- q 4 + 4 q 3 -9 q 2 + 14 q -18 + 22 q -1 -21 q -2 + 19 q -3 -14 q -4 + 8 q -5 -4 q -6 + q -7$
HOMFLY-PT polynomial (db, data sources)	$- a 2 z 8 + a 4 z 6 -5 a 2 z 6 + 2 z 6 + 3 a 4 z 4 -9 a 2 z 4 - z 4 a -2 + 7 z 4 + 2 a 4 z 2 -5 a 2 z 2 -2 z 2 a -2 + 7 z 2 - a 4 + a 2 - a -2 + 2$
Kauffman polynomial (db, data sources)	$3 a 2 z 10 + 3 z 10 + 10 a 3 z 9 + 16 a z 9 + 6 z 9 a -1 + 14 a 4 z 8 + 15 a 2 z 8 + 4 z 8 a -2 + 5 z 8 + 12 a 5 z 7 -14 a 3 z 7 -44 a z 7 -17 z 7 a -1 + z 7 a -3 + 8 a 6 z 6 -27 a 4 z 6 -64 a 2 z 6 -13 z 6 a -2 -42 z 6 + 4 a 7 z 5 -15 a 5 z 5 -5 a 3 z 5 + 26 a z 5 + 9 z 5 a -1 -3 z 5 a -3 + a 8 z 4 -6 a 6 z 4 + 20 a 4 z 4 + 64 a 2 z 4 + 13 z 4 a -2 + 50 z 4 -2 a 7 z 3 + 4 a 5 z 3 + 9 a 3 z 3 + 3 a z 3 + 3 z 3 a -1 + 3 z 3 a -3 -6 a 4 z 2 -19 a 2 z 2 -6 z 2 a -2 -19 z 2 + a 5 z -2 a z -2 z a -1 - z a -3 - a 4 - a 2 + a -2 + 2$
The A2 invariant	$q 20 -2 q 18 + 2 q 16 -3 q 14 -2 q 12 + 3 q 10 -3 q 8 + 6 q 6 - q 4 + 2 q 2 + 2-3 q -2 + 3 q -4 -2 q -6 + q -10 - q -12$
The G2 invariant	Data:K11a305/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a305 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["K11a305"];

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

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

Out[4]= $- t 4 + 6 t 3 -16 t 2 + 28 t -33 + 28 t -1 -16 t -2 + 6 t -3 - t -4$

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

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

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

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

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

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

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

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

Out[10]= $3 a 2 z 10 + 3 z 10 + 10 a 3 z 9 + 16 a z 9 + 6 z 9 a -1 + 14 a 4 z 8 + 15 a 2 z 8 + 4 z 8 a -2 + 5 z 8 + 12 a 5 z 7 -14 a 3 z 7 -44 a z 7 -17 z 7 a -1 + z 7 a -3 + 8 a 6 z 6 -27 a 4 z 6 -64 a 2 z 6 -13 z 6 a -2 -42 z 6 + 4 a 7 z 5 -15 a 5 z 5 -5 a 3 z 5 + 26 a z 5 + 9 z 5 a -1 -3 z 5 a -3 + a 8 z 4 -6 a 6 z 4 + 20 a 4 z 4 + 64 a 2 z 4 + 13 z 4 a -2 + 50 z 4 -2 a 7 z 3 + 4 a 5 z 3 + 9 a 3 z 3 + 3 a z 3 + 3 z 3 a -1 + 3 z 3 a -3 -6 a 4 z 2 -19 a 2 z 2 -6 z 2 a -2 -19 z 2 + a 5 z -2 a z -2 z a -1 - z a -3 - a 4 - a 2 + a -2 + 2$

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

Same Alexander/Conway Polynomial: {K11a157, K11a264,}

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["K11a305"];

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]= { $- t 4 + 6 t 3 -16 t 2 + 28 t -33 + 28 t -1 -16 t -2 + 6 t -3 - t -4$ , $- q 4 + 4 q 3 -9 q 2 + 14 q -18 + 22 q -1 -21 q -2 + 19 q -3 -14 q -4 + 8 q -5 -4 q -6 + q -7$ }

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]= {K11a157, K11a264,}

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, -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 =

-2 is the signature of K11a305. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

-5

-4

-1

-3

-5

-2

-7

-9

-6

-11

-13

-3

-15

Integral Khovanov Homology

(db, data source)

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