K11a3

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K11a2

K11a4

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 K11a3's page at Knotilus!

Visit K11a3's page at the original Knot Atlas!

[edit K11a3 Quick Notes]

[edit K11a3 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₃₉₄ X_10,6,11,5 X_14,8,15,7 X_2,9,3,10 X_18,11,19,12 X_6,14,7,13 X_20,15,21,16 X_22,17,1,18 X_12,19,13,20 X_16,21,17,22
Gauss code	1, -5, 2, -1, 3, -7, 4, -2, 5, -3, 6, -10, 7, -4, 8, -11, 9, -6, 10, -8, 11, -9
Dowker-Thistlethwaite code	4 8 10 14 2 18 6 20 22 12 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:K11a3/ThurstonBennequinNumber
Hyperbolic Volume	15.55
A-Polynomial	See Data:K11a3/A-polynomial

[edit Notes for K11a3'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 K11a3's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 4 + 5 t 3 -13 t 2 + 24 t -29 + 24 t -1 -13 t -2 + 5 t -3 - t -4$
Conway polynomial	$- z 8 -3 z 6 -3 z 4 + z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 115, -2 }
Jones polynomial	$q 3 -3 q 2 + 7 q -12 + 16 q -1 -18 q -2 + 19 q -3 -16 q -4 + 12 q -5 -7 q -6 + 3 q -7 - q -8$
HOMFLY-PT polynomial (db, data sources)	$- a 2 z 8 + 2 a 4 z 6 -6 a 2 z 6 + z 6 - a 6 z 4 + 9 a 4 z 4 -15 a 2 z 4 + 4 z 4 -3 a 6 z 2 + 15 a 4 z 2 -17 a 2 z 2 + 6 z 2 -3 a 6 + 8 a 4 -7 a 2 + 3$
Kauffman polynomial (db, data sources)	$a 4 z 10 + a 2 z 10 + 4 a 5 z 9 + 8 a 3 z 9 + 4 a z 9 + 6 a 6 z 8 + 13 a 4 z 8 + 12 a 2 z 8 + 5 z 8 + 5 a 7 z 7 + a 5 z 7 -11 a 3 z 7 -4 a z 7 + 3 z 7 a -1 + 3 a 8 z 6 -9 a 6 z 6 -38 a 4 z 6 -40 a 2 z 6 + z 6 a -2 -13 z 6 + a 9 z 5 -7 a 7 z 5 -11 a 5 z 5 -2 a 3 z 5 -7 a z 5 -8 z 5 a -1 -5 a 8 z 4 + 9 a 6 z 4 + 48 a 4 z 4 + 49 a 2 z 4 -3 z 4 a -2 + 12 z 4 -2 a 9 z 3 + 3 a 7 z 3 + 12 a 5 z 3 + 10 a 3 z 3 + 9 a z 3 + 6 z 3 a -1 + 2 a 8 z 2 -8 a 6 z 2 -30 a 4 z 2 -30 a 2 z 2 + 2 z 2 a -2 -8 z 2 + a 9 z - a 7 z -4 a 5 z -4 a 3 z -4 a z -2 z a -1 + 3 a 6 + 8 a 4 + 7 a 2 + 3$
The A2 invariant	$- q 24 - q 20 -2 q 18 + 4 q 16 - q 14 + 3 q 12 + 2 q 10 -2 q 8 + 3 q 6 -5 q 4 + 2 q 2 -1- q -2 + 3 q -4 - q -6 + q -8$
The G2 invariant	$q 128 -2 q 126 + 5 q 124 -8 q 122 + 9 q 120 -8 q 118 + q 116 + 13 q 114 -29 q 112 + 47 q 110 -59 q 108 + 53 q 106 -30 q 104 -20 q 102 + 87 q 100 -150 q 98 + 191 q 96 -186 q 94 + 112 q 92 + 17 q 90 -181 q 88 + 324 q 86 -389 q 84 + 334 q 82 -167 q 80 -81 q 78 + 315 q 76 -446 q 74 + 423 q 72 -237 q 70 -30 q 68 + 264 q 66 -367 q 64 + 285 q 62 -53 q 60 -216 q 58 + 409 q 56 -407 q 54 + 208 q 52 + 130 q 50 -455 q 48 + 647 q 46 -610 q 44 + 353 q 42 + 36 q 40 -415 q 38 + 658 q 36 -671 q 34 + 468 q 32 -122 q 30 -234 q 28 + 454 q 26 -482 q 24 + 308 q 22 -27 q 20 -243 q 18 + 372 q 16 -315 q 14 + 91 q 12 + 196 q 10 -420 q 8 + 474 q 6 -340 q 4 + 65 q 2 + 227-431 q -2 + 485 q -4 -368 q -6 + 159 q -8 + 69 q -10 -235 q -12 + 294 q -14 -251 q -16 + 153 q -18 -39 q -20 -45 q -22 + 88 q -24 -92 q -26 + 69 q -28 -36 q -30 + 11 q -32 + 6 q -34 -13 q -36 + 11 q -38 -9 q -40 + 5 q -42 -2 q -44 + q -46$

Further Quantum Invariants

K11a3 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["K11a3"];

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

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

Out[4]= $- t 4 + 5 t 3 -13 t 2 + 24 t -29 + 24 t -1 -13 t -2 + 5 t -3 - t -4$

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

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

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]= { 115, -2 }

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

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

Out[8]= $q 3 -3 q 2 + 7 q -12 + 16 q -1 -18 q -2 + 19 q -3 -16 q -4 + 12 q -5 -7 q -6 + 3 q -7 - q -8$

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

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

Out[9]= $- a 2 z 8 + 2 a 4 z 6 -6 a 2 z 6 + z 6 - a 6 z 4 + 9 a 4 z 4 -15 a 2 z 4 + 4 z 4 -3 a 6 z 2 + 15 a 4 z 2 -17 a 2 z 2 + 6 z 2 -3 a 6 + 8 a 4 -7 a 2 + 3$

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

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

Out[10]= $a 4 z 10 + a 2 z 10 + 4 a 5 z 9 + 8 a 3 z 9 + 4 a z 9 + 6 a 6 z 8 + 13 a 4 z 8 + 12 a 2 z 8 + 5 z 8 + 5 a 7 z 7 + a 5 z 7 -11 a 3 z 7 -4 a z 7 + 3 z 7 a -1 + 3 a 8 z 6 -9 a 6 z 6 -38 a 4 z 6 -40 a 2 z 6 + z 6 a -2 -13 z 6 + a 9 z 5 -7 a 7 z 5 -11 a 5 z 5 -2 a 3 z 5 -7 a z 5 -8 z 5 a -1 -5 a 8 z 4 + 9 a 6 z 4 + 48 a 4 z 4 + 49 a 2 z 4 -3 z 4 a -2 + 12 z 4 -2 a 9 z 3 + 3 a 7 z 3 + 12 a 5 z 3 + 10 a 3 z 3 + 9 a z 3 + 6 z 3 a -1 + 2 a 8 z 2 -8 a 6 z 2 -30 a 4 z 2 -30 a 2 z 2 + 2 z 2 a -2 -8 z 2 + a 9 z - a 7 z -4 a 5 z -4 a 3 z -4 a z -2 z a -1 + 3 a 6 + 8 a 4 + 7 a 2 + 3$

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11a51, K11a331,}

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["K11a3"];

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]= { $- t 4 + 5 t 3 -13 t 2 + 24 t -29 + 24 t -1 -13 t -2 + 5 t -3 - t -4$ , $q 3 -3 q 2 + 7 q -12 + 16 q -1 -18 q -2 + 19 q -3 -16 q -4 + 12 q -5 -7 q -6 + 3 q -7 - q -8$ }

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]= {}

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

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

Out[6]= {K11a51, K11a331,}

[edit] Vassiliev invariants

V₂ and V₃:

(1, -4)

[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 K11a3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

-2

-5

-1

-3

-2

-5

-7

-9

-4

-11

-13

-4

-15

-17

-1

Integral Khovanov Homology

(db, data source)

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