K11a182

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K11a181

K11a183

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 K11a182's page at Knotilus!

Visit K11a182's page at the original Knot Atlas!

[edit K11a182 Quick Notes]

[edit K11a182 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X_12,3,13,4 X_14,6,15,5 X_16,7,17,8 X_18,9,19,10 X_20,11,21,12 X_2,13,3,14 X_22,16,1,15 X_8,17,9,18 X_10,19,11,20 X_6,21,7,22
Gauss code	1, -7, 2, -1, 3, -11, 4, -9, 5, -10, 6, -2, 7, -3, 8, -4, 9, -5, 10, -6, 11, -8
Dowker-Thistlethwaite code	4 12 14 16 18 20 2 22 8 10 6

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	4
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a182/ThurstonBennequinNumber
Hyperbolic Volume	11.2693
A-Polynomial	See Data:K11a182/A-polynomial

[edit Notes for K11a182's three dimensional invariants]

[edit] Four dimensional invariants

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

[edit Notes for K11a182's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$t 4 -5 t 3 + 11 t 2 -13 t + 13-13 t -1 + 11 t -2 -5 t -3 + t -4$
Conway polynomial	$z 8 + 3 z 6 + z 4 + 2 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 73, -4 }
Jones polynomial	$- q + 3-4 q -1 + 7 q -2 -9 q -3 + 11 q -4 -11 q -5 + 10 q -6 -8 q -7 + 5 q -8 -3 q -9 + q -10$
HOMFLY-PT polynomial (db, data sources)	$z 4 a 8 + 3 z 2 a 8 + a 8 -2 z 6 a 6 -9 z 4 a 6 -11 z 2 a 6 -4 a 6 + z 8 a 4 + 6 z 6 a 4 + 13 z 4 a 4 + 13 z 2 a 4 + 4 a 4 - z 6 a 2 -4 z 4 a 2 -3 z 2 a 2$
Kauffman polynomial (db, data sources)	$z 4 a 12 - z 2 a 12 + 3 z 5 a 11 -4 z 3 a 11 + z a 11 + 4 z 6 a 10 -4 z 4 a 10 + 4 z 7 a 9 -3 z 5 a 9 -2 z 3 a 9 + 2 z a 9 + 4 z 8 a 8 -7 z 6 a 8 + 7 z 4 a 8 -4 z 2 a 8 + a 8 + 3 z 9 a 7 -6 z 7 a 7 + 5 z 5 a 7 -2 z 3 a 7 + z 10 a 6 + 4 z 8 a 6 -24 z 6 a 6 + 36 z 4 a 6 -21 z 2 a 6 + 4 a 6 + 6 z 9 a 5 -23 z 7 a 5 + 27 z 5 a 5 -11 z 3 a 5 + z 10 a 4 + 3 z 8 a 4 -27 z 6 a 4 + 42 z 4 a 4 -23 z 2 a 4 + 4 a 4 + 3 z 9 a 3 -12 z 7 a 3 + 12 z 5 a 3 -4 z 3 a 3 + z a 3 + 3 z 8 a 2 -14 z 6 a 2 + 18 z 4 a 2 -7 z 2 a 2 + z 7 a -4 z 5 a + 3 z 3 a$
The A2 invariant	$q 30 - q 26 -2 q 22 + q 20 - q 18 + q 14 -2 q 12 + 3 q 10 + 2 q 6 + q 4 + 1- q -2$
The G2 invariant	$q 162 -2 q 160 + 4 q 158 -6 q 156 + 4 q 154 -2 q 152 -4 q 150 + 12 q 148 -18 q 146 + 22 q 144 -19 q 142 + 8 q 140 + 7 q 138 -23 q 136 + 37 q 134 -40 q 132 + 34 q 130 -20 q 128 -2 q 126 + 25 q 124 -39 q 122 + 48 q 120 -41 q 118 + 29 q 116 -11 q 114 -11 q 112 + 28 q 110 -40 q 108 + 40 q 106 -30 q 104 + 11 q 102 + 8 q 100 -24 q 98 + 27 q 96 -18 q 94 -2 q 92 + 21 q 90 -36 q 88 + 25 q 86 + q 84 -35 q 82 + 64 q 80 -71 q 78 + 51 q 76 -12 q 74 -39 q 72 + 78 q 70 -98 q 68 + 84 q 66 -43 q 64 -8 q 62 + 55 q 60 -76 q 58 + 75 q 56 -48 q 54 + 8 q 52 + 27 q 50 -49 q 48 + 44 q 46 -13 q 44 -19 q 42 + 51 q 40 -54 q 38 + 32 q 36 + 8 q 34 -49 q 32 + 79 q 30 -81 q 28 + 56 q 26 -9 q 24 -36 q 22 + 70 q 20 -76 q 18 + 61 q 16 -29 q 14 -3 q 12 + 26 q 10 -37 q 8 + 35 q 6 -23 q 4 + 11 q 2 + 1-7 q -2 + 6 q -4 -7 q -6 + 4 q -8 -2 q -10 + q -12$

Further Quantum Invariants

K11a182 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["K11a182"];

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

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

Out[4]= $t 4 -5 t 3 + 11 t 2 -13 t + 13-13 t -1 + 11 t -2 -5 t -3 + t -4$

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

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

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]= { 73, -4 }

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

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

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

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

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

Out[9]= $z 4 a 8 + 3 z 2 a 8 + a 8 -2 z 6 a 6 -9 z 4 a 6 -11 z 2 a 6 -4 a 6 + z 8 a 4 + 6 z 6 a 4 + 13 z 4 a 4 + 13 z 2 a 4 + 4 a 4 - z 6 a 2 -4 z 4 a 2 -3 z 2 a 2$

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

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

Out[10]= $z 4 a 12 - z 2 a 12 + 3 z 5 a 11 -4 z 3 a 11 + z a 11 + 4 z 6 a 10 -4 z 4 a 10 + 4 z 7 a 9 -3 z 5 a 9 -2 z 3 a 9 + 2 z a 9 + 4 z 8 a 8 -7 z 6 a 8 + 7 z 4 a 8 -4 z 2 a 8 + a 8 + 3 z 9 a 7 -6 z 7 a 7 + 5 z 5 a 7 -2 z 3 a 7 + z 10 a 6 + 4 z 8 a 6 -24 z 6 a 6 + 36 z 4 a 6 -21 z 2 a 6 + 4 a 6 + 6 z 9 a 5 -23 z 7 a 5 + 27 z 5 a 5 -11 z 3 a 5 + z 10 a 4 + 3 z 8 a 4 -27 z 6 a 4 + 42 z 4 a 4 -23 z 2 a 4 + 4 a 4 + 3 z 9 a 3 -12 z 7 a 3 + 12 z 5 a 3 -4 z 3 a 3 + z a 3 + 3 z 8 a 2 -14 z 6 a 2 + 18 z 4 a 2 -7 z 2 a 2 + z 7 a -4 z 5 a + 3 z 3 a$

[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["K11a182"];

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]= { $t 4 -5 t 3 + 11 t 2 -13 t + 13-13 t -1 + 11 t -2 -5 t -3 + t -4$ , $- q + 3-4 q -1 + 7 q -2 -9 q -3 + 11 q -4 -11 q -5 + 10 q -6 -8 q -7 + 5 q -8 -3 q -9 + q -10$ }

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:

(2, -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 =

-4 is the signature of K11a182. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-8

-7

-6

-5

-4

-3

-2

-1

-3

-5

-2

-7

-9

-11

-1

-13

-15

-3

-17

-19

-2

-21

Integral Khovanov Homology

(db, data source)

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