K11a174

From Knot Atlas

Jump to: navigation, search

K11a173

K11a175

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 K11a174's page at Knotilus!

Visit K11a174's page at the original Knot Atlas!

[edit K11a174 Quick Notes]

[edit K11a174 Further Notes and Views]

[edit] Knot presentations

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

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:K11a174/ThurstonBennequinNumber
Hyperbolic Volume	12.1343
A-Polynomial	See Data:K11a174/A-polynomial

[edit Notes for K11a174'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 K11a174's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 4 + 5 t 3 -11 t 2 + 15 t -15 + 15 t -1 -11 t -2 + 5 t -3 - t -4$
Conway polynomial	$- z 8 -3 z 6 - z 4 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 79, -2 }
Jones polynomial	$- q 4 + 3 q 3 -5 q 2 + 8 q -10 + 12 q -1 -12 q -2 + 11 q -3 -8 q -4 + 5 q -5 -3 q -6 + q -7$
HOMFLY-PT polynomial (db, data sources)	$- a 2 z 8 + a 4 z 6 -6 a 2 z 6 + 2 z 6 + 4 a 4 z 4 -13 a 2 z 4 - z 4 a -2 + 9 z 4 + 4 a 4 z 2 -12 a 2 z 2 -3 z 2 a -2 + 11 z 2 + a 4 -3 a 2 - a -2 + 4$
Kauffman polynomial (db, data sources)	$a 2 z 10 + z 10 + 3 a 3 z 9 + 6 a z 9 + 3 z 9 a -1 + 4 a 4 z 8 + 5 a 2 z 8 + 3 z 8 a -2 + 4 z 8 + 4 a 5 z 7 -4 a 3 z 7 -19 a z 7 -10 z 7 a -1 + z 7 a -3 + 4 a 6 z 6 -5 a 4 z 6 -24 a 2 z 6 -13 z 6 a -2 -28 z 6 + 3 a 7 z 5 -2 a 5 z 5 + a 3 z 5 + 15 a z 5 + 5 z 5 a -1 -4 z 5 a -3 + a 8 z 4 -4 a 6 z 4 + 3 a 4 z 4 + 31 a 2 z 4 + 16 z 4 a -2 + 39 z 4 -4 a 7 z 3 -3 a 5 z 3 - a 3 z 3 -2 a z 3 + 4 z 3 a -1 + 4 z 3 a -3 - a 8 z 2 + a 6 z 2 -2 a 4 z 2 -17 a 2 z 2 -7 z 2 a -2 -20 z 2 + a 7 z + 2 a 5 z + a 3 z - a z -2 z a -1 - z a -3 + a 4 + 3 a 2 + a -2 + 4$
The A2 invariant	$q 20 - q 18 + q 16 - q 14 - q 12 + q 10 -2 q 8 + 3 q 6 - q 4 + q 2 + 1- q -2 + 2 q -4 + q -8 - q -12$
The G2 invariant	$q 114 -2 q 112 + 4 q 110 -6 q 108 + 4 q 106 -2 q 104 -4 q 102 + 12 q 100 -17 q 98 + 22 q 96 -21 q 94 + 10 q 92 + 5 q 90 -22 q 88 + 36 q 86 -43 q 84 + 41 q 82 -27 q 80 + 7 q 78 + 20 q 76 -40 q 74 + 56 q 72 -55 q 70 + 45 q 68 -29 q 66 - q 64 + 29 q 62 -52 q 60 + 63 q 58 -55 q 56 + 33 q 54 -4 q 52 -30 q 50 + 47 q 48 -45 q 46 + 21 q 44 + 11 q 42 -42 q 40 + 48 q 38 -24 q 36 -19 q 34 + 67 q 32 -95 q 30 + 89 q 28 -45 q 26 -24 q 24 + 92 q 22 -133 q 20 + 133 q 18 -86 q 16 + 15 q 14 + 58 q 12 -105 q 10 + 117 q 8 -88 q 6 + 32 q 4 + 25 q 2 -67 + 72 q -2 -39 q -4 -7 q -6 + 56 q -8 -76 q -10 + 62 q -12 -15 q -14 -46 q -16 + 98 q -18 -115 q -20 + 92 q -22 -34 q -24 -32 q -26 + 86 q -28 -105 q -30 + 93 q -32 -53 q -34 + 4 q -36 + 32 q -38 -53 q -40 + 49 q -42 -34 q -44 + 16 q -46 + q -48 -10 q -50 + 10 q -52 -9 q -54 + 5 q -56 -2 q -58 + q -60$

Further Quantum Invariants

K11a174 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["K11a174"];

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

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

Out[4]= $- t 4 + 5 t 3 -11 t 2 + 15 t -15 + 15 t -1 -11 t -2 + 5 t -3 - t -4$

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

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

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]= { 79, -2 }

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

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

Out[8]= $- q 4 + 3 q 3 -5 q 2 + 8 q -10 + 12 q -1 -12 q -2 + 11 q -3 -8 q -4 + 5 q -5 -3 q -6 + q -7$

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

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

Out[9]= $- a 2 z 8 + a 4 z 6 -6 a 2 z 6 + 2 z 6 + 4 a 4 z 4 -13 a 2 z 4 - z 4 a -2 + 9 z 4 + 4 a 4 z 2 -12 a 2 z 2 -3 z 2 a -2 + 11 z 2 + a 4 -3 a 2 - a -2 + 4$

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

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

Out[10]= $a 2 z 10 + z 10 + 3 a 3 z 9 + 6 a z 9 + 3 z 9 a -1 + 4 a 4 z 8 + 5 a 2 z 8 + 3 z 8 a -2 + 4 z 8 + 4 a 5 z 7 -4 a 3 z 7 -19 a z 7 -10 z 7 a -1 + z 7 a -3 + 4 a 6 z 6 -5 a 4 z 6 -24 a 2 z 6 -13 z 6 a -2 -28 z 6 + 3 a 7 z 5 -2 a 5 z 5 + a 3 z 5 + 15 a z 5 + 5 z 5 a -1 -4 z 5 a -3 + a 8 z 4 -4 a 6 z 4 + 3 a 4 z 4 + 31 a 2 z 4 + 16 z 4 a -2 + 39 z 4 -4 a 7 z 3 -3 a 5 z 3 - a 3 z 3 -2 a z 3 + 4 z 3 a -1 + 4 z 3 a -3 - a 8 z 2 + a 6 z 2 -2 a 4 z 2 -17 a 2 z 2 -7 z 2 a -2 -20 z 2 + a 7 z + 2 a 5 z + a 3 z - a z -2 z a -1 - z a -3 + a 4 + 3 a 2 + a -2 + 4$

[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["K11a174"];

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]= { $- t 4 + 5 t 3 -11 t 2 + 15 t -15 + 15 t -1 -11 t -2 + 5 t -3 - t -4$ , $- q 4 + 3 q 3 -5 q 2 + 8 q -10 + 12 q -1 -12 q -2 + 11 q -3 -8 q -4 + 5 q -5 -3 q -6 + q -7$ }

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]= {}

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, 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 K11a174. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

-2

-1

-3

-5

-1

-7

-9

-3

-11

-13

-2

-15

Integral Khovanov Homology

(db, data source)

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