K11a162

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K11a161

K11a163

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 K11a162's page at Knotilus!

Visit K11a162's page at the original Knot Atlas!

[edit K11a162 Quick Notes]

[edit K11a162 Further Notes and Views]

[edit] Knot presentations

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

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:K11a162/ThurstonBennequinNumber
Hyperbolic Volume	18.245
A-Polynomial	See Data:K11a162/A-polynomial

[edit Notes for K11a162'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 K11a162's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 4 + 7 t 3 -20 t 2 + 35 t -41 + 35 t -1 -20 t -2 + 7 t -3 - t -4$
Conway polynomial	$- z 8 - z 6 + 2 z 4 + 2 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 167, -2 }
Jones polynomial	$q 3 -5 q 2 + 11 q -17 + 24 q -1 -27 q -2 + 27 q -3 -23 q -4 + 17 q -5 -10 q -6 + 4 q -7 - q -8$
HOMFLY-PT polynomial (db, data sources)	$- a 2 z 8 + 2 a 4 z 6 -4 a 2 z 6 + z 6 - a 6 z 4 + 6 a 4 z 4 -5 a 2 z 4 + 2 z 4 -2 a 6 z 2 + 5 a 4 z 2 - a 2 z 2 - a 6 + a 4 + a 2$
Kauffman polynomial (db, data sources)	$3 a 4 z 10 + 3 a 2 z 10 + 9 a 5 z 9 + 18 a 3 z 9 + 9 a z 9 + 12 a 6 z 8 + 20 a 4 z 8 + 18 a 2 z 8 + 10 z 8 + 9 a 7 z 7 -2 a 5 z 7 -27 a 3 z 7 -11 a z 7 + 5 z 7 a -1 + 4 a 8 z 6 -18 a 6 z 6 -53 a 4 z 6 -54 a 2 z 6 + z 6 a -2 -22 z 6 + a 9 z 5 -13 a 7 z 5 -18 a 5 z 5 -2 a 3 z 5 -7 a z 5 -9 z 5 a -1 -4 a 8 z 4 + 13 a 6 z 4 + 42 a 4 z 4 + 39 a 2 z 4 - z 4 a -2 + 13 z 4 - a 9 z 3 + 9 a 7 z 3 + 18 a 5 z 3 + 13 a 3 z 3 + 8 a z 3 + 3 z 3 a -1 + a 8 z 2 -6 a 6 z 2 -13 a 4 z 2 -7 a 2 z 2 - z 2 -3 a 7 z -5 a 5 z -3 a 3 z - a z + a 6 + a 4 - a 2$
The A2 invariant	$- q 24 + q 22 -3 q 18 + 4 q 16 -4 q 14 + 2 q 12 + 2 q 10 -3 q 8 + 6 q 6 -5 q 4 + 5 q 2 -2 q -2 + 3 q -4 -3 q -6 + q -8$
The G2 invariant	$q 128 -3 q 126 + 7 q 124 -13 q 122 + 17 q 120 -19 q 118 + 12 q 116 + 11 q 114 -46 q 112 + 94 q 110 -138 q 108 + 151 q 106 -118 q 104 + 17 q 102 + 152 q 100 -344 q 98 + 509 q 96 -563 q 94 + 427 q 92 -103 q 90 -372 q 88 + 858 q 86 -1159 q 84 + 1122 q 82 -679 q 80 -70 q 78 + 863 q 76 -1399 q 74 + 1437 q 72 -936 q 70 + 69 q 68 + 791 q 66 -1265 q 64 + 1134 q 62 -429 q 60 -505 q 58 + 1243 q 56 -1419 q 54 + 900 q 52 + 100 q 50 -1198 q 48 + 1938 q 46 -1983 q 44 + 1306 q 42 -122 q 40 -1126 q 38 + 1989 q 36 -2169 q 34 + 1607 q 32 -550 q 30 -614 q 28 + 1451 q 26 -1645 q 24 + 1181 q 22 -254 q 20 -695 q 18 + 1248 q 16 -1177 q 14 + 500 q 12 + 447 q 10 -1235 q 8 + 1534 q 6 -1190 q 4 + 367 q 2 + 595-1310 q -2 + 1523 q -4 -1208 q -6 + 535 q -8 + 190 q -10 -721 q -12 + 919 q -14 -794 q -16 + 480 q -18 -114 q -20 -162 q -22 + 286 q -24 -290 q -26 + 206 q -28 -102 q -30 + 23 q -32 + 28 q -34 -42 q -36 + 36 q -38 -24 q -40 + 11 q -42 -4 q -44 + q -46$

Further Quantum Invariants

K11a162 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["K11a162"];

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

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

Out[4]= $- t 4 + 7 t 3 -20 t 2 + 35 t -41 + 35 t -1 -20 t -2 + 7 t -3 - t -4$

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

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

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]= { 167, -2 }

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

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

Out[8]= $q 3 -5 q 2 + 11 q -17 + 24 q -1 -27 q -2 + 27 q -3 -23 q -4 + 17 q -5 -10 q -6 + 4 q -7 - q -8$

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

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

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

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

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

Out[10]= $3 a 4 z 10 + 3 a 2 z 10 + 9 a 5 z 9 + 18 a 3 z 9 + 9 a z 9 + 12 a 6 z 8 + 20 a 4 z 8 + 18 a 2 z 8 + 10 z 8 + 9 a 7 z 7 -2 a 5 z 7 -27 a 3 z 7 -11 a z 7 + 5 z 7 a -1 + 4 a 8 z 6 -18 a 6 z 6 -53 a 4 z 6 -54 a 2 z 6 + z 6 a -2 -22 z 6 + a 9 z 5 -13 a 7 z 5 -18 a 5 z 5 -2 a 3 z 5 -7 a z 5 -9 z 5 a -1 -4 a 8 z 4 + 13 a 6 z 4 + 42 a 4 z 4 + 39 a 2 z 4 - z 4 a -2 + 13 z 4 - a 9 z 3 + 9 a 7 z 3 + 18 a 5 z 3 + 13 a 3 z 3 + 8 a z 3 + 3 z 3 a -1 + a 8 z 2 -6 a 6 z 2 -13 a 4 z 2 -7 a 2 z 2 - z 2 -3 a 7 z -5 a 5 z -3 a 3 z - a z + a 6 + a 4 - a 2$

[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["K11a162"];

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]= { $- t 4 + 7 t 3 -20 t 2 + 35 t -41 + 35 t -1 -20 t -2 + 7 t -3 - t -4$ , $q 3 -5 q 2 + 11 q -17 + 24 q -1 -27 q -2 + 27 q -3 -23 q -4 + 17 q -5 -10 q -6 + 4 q -7 - q -8$ }

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]= {}

[edit] Vassiliev invariants

V₂ and V₃:

(2, -3)

[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 K11a162. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

-4

-6

-1

-3

-5

-7

-9

-6

-11

-13

-6

-15

-17

-1

Integral Khovanov Homology

(db, data source)

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