K11a53

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K11a52

K11a54

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 K11a53's page at Knotilus!

Visit K11a53's page at the original Knot Atlas!

[edit K11a53 Quick Notes]

[edit K11a53 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₃₉₄ X_14,6,15,5 X_16,7,17,8 X_2,9,3,10 X_18,11,19,12 X_20,13,21,14 X_22,16,1,15 X_10,17,11,18 X_12,19,13,20 X_6,21,7,22
Gauss code	1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -10, 7, -3, 8, -4, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code	4 8 14 16 2 18 20 22 10 12 6

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Chiral
Unknotting number	${2,3}$
3-genus	4
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a53/ThurstonBennequinNumber
Hyperbolic Volume	14.416
A-Polynomial	See Data:K11a53/A-polynomial

[edit Notes for K11a53'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 K11a53's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$t 4 -6 t 3 + 14 t 2 -18 t + 19-18 t -1 + 14 t -2 -6 t -3 + t -4$
Conway polynomial	$z 8 + 2 z 6 -2 z 4 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 97, -4 }
Jones polynomial	$- q + 4-6 q -1 + 10 q -2 -13 q -3 + 15 q -4 -15 q -5 + 13 q -6 -10 q -7 + 6 q -8 -3 q -9 + q -10$
HOMFLY-PT polynomial (db, data sources)	$z 4 a 8 + 3 z 2 a 8 + a 8 -2 z 6 a 6 -8 z 4 a 6 -8 z 2 a 6 -2 a 6 + z 8 a 4 + 5 z 6 a 4 + 8 z 4 a 4 + 5 z 2 a 4 - z 6 a 2 -3 z 4 a 2 + 2 a 2$
Kauffman polynomial (db, data sources)	$z 4 a 12 - z 2 a 12 + 3 z 5 a 11 -3 z 3 a 11 + z a 11 + 5 z 6 a 10 -5 z 4 a 10 + 2 z 2 a 10 + 6 z 7 a 9 -6 z 5 a 9 + z 3 a 9 + z a 9 + 6 z 8 a 8 -8 z 6 a 8 + 4 z 4 a 8 -3 z 2 a 8 + a 8 + 5 z 9 a 7 -9 z 7 a 7 + 7 z 5 a 7 -6 z 3 a 7 + z a 7 + 2 z 10 a 6 + 4 z 8 a 6 -26 z 6 a 6 + 32 z 4 a 6 -16 z 2 a 6 + 2 a 6 + 10 z 9 a 5 -35 z 7 a 5 + 38 z 5 a 5 -17 z 3 a 5 + 3 z a 5 + 2 z 10 a 4 + 2 z 8 a 4 -29 z 6 a 4 + 38 z 4 a 4 -12 z 2 a 4 + 5 z 9 a 3 -19 z 7 a 3 + 19 z 5 a 3 -6 z 3 a 3 + 2 z a 3 + 4 z 8 a 2 -16 z 6 a 2 + 16 z 4 a 2 -2 z 2 a 2 -2 a 2 + z 7 a -3 z 5 a + z 3 a$
The A2 invariant	$q 30 - q 26 + q 24 -2 q 22 + 2 q 20 - q 18 - q 16 + q 14 -4 q 12 + 3 q 10 - q 8 + 2 q 6 + 2 q 4 + 2- q -2$
The G2 invariant	$q 162 -2 q 160 + 4 q 158 -6 q 156 + 5 q 154 -4 q 152 -2 q 150 + 11 q 148 -20 q 146 + 28 q 144 -30 q 142 + 22 q 140 -4 q 138 -21 q 136 + 50 q 134 -68 q 132 + 71 q 130 -56 q 128 + 20 q 126 + 22 q 124 -60 q 122 + 91 q 120 -96 q 118 + 87 q 116 -61 q 114 + 18 q 112 + 33 q 110 -87 q 108 + 125 q 106 -133 q 104 + 99 q 102 -29 q 100 -57 q 98 + 125 q 96 -140 q 94 + 94 q 92 -100 q 88 + 143 q 86 -110 q 84 + 9 q 82 + 119 q 80 -205 q 78 + 213 q 76 -129 q 74 -20 q 72 + 170 q 70 -272 q 68 + 278 q 66 -195 q 64 + 47 q 62 + 107 q 60 -213 q 58 + 249 q 56 -202 q 54 + 87 q 52 + 43 q 50 -151 q 48 + 181 q 46 -129 q 44 + 16 q 42 + 113 q 40 -183 q 38 + 167 q 36 -68 q 34 -73 q 32 + 199 q 30 -249 q 28 + 203 q 26 -79 q 24 -66 q 22 + 181 q 20 -216 q 18 + 181 q 16 -90 q 14 -4 q 12 + 72 q 10 -101 q 8 + 91 q 6 -55 q 4 + 21 q 2 + 7-18 q -2 + 17 q -4 -14 q -6 + 7 q -8 -3 q -10 + q -12$

Further Quantum Invariants

K11a53 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["K11a53"];

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

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

Out[4]= $t 4 -6 t 3 + 14 t 2 -18 t + 19-18 t -1 + 14 t -2 -6 t -3 + t -4$

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

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

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]= { 97, -4 }

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

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

Out[8]= $- q + 4-6 q -1 + 10 q -2 -13 q -3 + 15 q -4 -15 q -5 + 13 q -6 -10 q -7 + 6 q -8 -3 q -9 + q -10$

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

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

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

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

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

Out[10]= $z 4 a 12 - z 2 a 12 + 3 z 5 a 11 -3 z 3 a 11 + z a 11 + 5 z 6 a 10 -5 z 4 a 10 + 2 z 2 a 10 + 6 z 7 a 9 -6 z 5 a 9 + z 3 a 9 + z a 9 + 6 z 8 a 8 -8 z 6 a 8 + 4 z 4 a 8 -3 z 2 a 8 + a 8 + 5 z 9 a 7 -9 z 7 a 7 + 7 z 5 a 7 -6 z 3 a 7 + z a 7 + 2 z 10 a 6 + 4 z 8 a 6 -26 z 6 a 6 + 32 z 4 a 6 -16 z 2 a 6 + 2 a 6 + 10 z 9 a 5 -35 z 7 a 5 + 38 z 5 a 5 -17 z 3 a 5 + 3 z a 5 + 2 z 10 a 4 + 2 z 8 a 4 -29 z 6 a 4 + 38 z 4 a 4 -12 z 2 a 4 + 5 z 9 a 3 -19 z 7 a 3 + 19 z 5 a 3 -6 z 3 a 3 + 2 z a 3 + 4 z 8 a 2 -16 z 6 a 2 + 16 z 4 a 2 -2 z 2 a 2 -2 a 2 + z 7 a -3 z 5 a + z 3 a$

[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["K11a53"];

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]= { $t 4 -6 t 3 + 14 t 2 -18 t + 19-18 t -1 + 14 t -2 -6 t -3 + t -4$ , $- q + 4-6 q -1 + 10 q -2 -13 q -3 + 15 q -4 -15 q -5 + 13 q -6 -10 q -7 + 6 q -8 -3 q -9 + q -10$ }

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:

(0, 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 K11a53. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-8

-7

-6

-5

-4

-3

-2

-1

-2

-3

-5

-3

-7

-9

-11

-2

-13

-15

-4

-17

-19

-2

-21

Integral Khovanov Homology

(db, data source)

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