K11a157

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K11a156

K11a158

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 K11a157's page at Knotilus!

Visit K11a157's page at the original Knot Atlas!

[edit K11a157 Quick Notes]

[edit K11a157 Further Notes and Views]

[edit] Knot presentations

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

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:K11a157/ThurstonBennequinNumber
Hyperbolic Volume	16.8128
A-Polynomial	See Data:K11a157/A-polynomial

[edit Notes for K11a157's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$2$
Rasmussen s-Invariant	2

[edit Notes for K11a157's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 4 + 6 t 3 -16 t 2 + 28 t -33 + 28 t -1 -16 t -2 + 6 t -3 - t -4$
Conway polynomial	$- z 8 -2 z 6 + 2 z 2 + 1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^4+t^2+1\right\}$
Determinant and Signature	{ 135, -2 }
Jones polynomial	$q 3 -4 q 2 + 9 q -14 + 19 q -1 -22 q -2 + 22 q -3 -18 q -4 + 14 q -5 -8 q -6 + 3 q -7 - q -8$
HOMFLY-PT polynomial (db, data sources)	$- a 2 z 8 + 2 a 4 z 6 -5 a 2 z 6 + z 6 - a 6 z 4 + 8 a 4 z 4 -10 a 2 z 4 + 3 z 4 -3 a 6 z 2 + 12 a 4 z 2 -10 a 2 z 2 + 3 z 2 -3 a 6 + 7 a 4 -5 a 2 + 2$
Kauffman polynomial (db, data sources)	$2 a 4 z 10 + 2 a 2 z 10 + 6 a 5 z 9 + 12 a 3 z 9 + 6 a z 9 + 8 a 6 z 8 + 13 a 4 z 8 + 12 a 2 z 8 + 7 z 8 + 6 a 7 z 7 -3 a 5 z 7 -20 a 3 z 7 -7 a z 7 + 4 z 7 a -1 + 3 a 8 z 6 -14 a 6 z 6 -39 a 4 z 6 -39 a 2 z 6 + z 6 a -2 -16 z 6 + a 9 z 5 -9 a 7 z 5 -8 a 5 z 5 + 4 a 3 z 5 -7 a z 5 -9 z 5 a -1 -4 a 8 z 4 + 16 a 6 z 4 + 46 a 4 z 4 + 38 a 2 z 4 -2 z 4 a -2 + 10 z 4 -2 a 9 z 3 + 6 a 7 z 3 + 13 a 5 z 3 + 8 a 3 z 3 + 8 a z 3 + 5 z 3 a -1 + a 8 z 2 -12 a 6 z 2 -28 a 4 z 2 -20 a 2 z 2 + z 2 a -2 -4 z 2 + a 9 z -2 a 7 z -5 a 5 z -3 a 3 z -2 a z - z a -1 + 3 a 6 + 7 a 4 + 5 a 2 + 2$
The A2 invariant	$- q 24 - q 20 -3 q 18 + 4 q 16 - q 14 + 4 q 12 + 3 q 10 -3 q 8 + 3 q 6 -6 q 4 + 3 q 2 - q -2 + 3 q -4 -2 q -6 + q -8$
The G2 invariant	$q 128 -2 q 126 + 5 q 124 -8 q 122 + 10 q 120 -10 q 118 + 4 q 116 + 10 q 114 -30 q 112 + 53 q 110 -73 q 108 + 74 q 106 -53 q 104 + 89 q 100 -184 q 98 + 263 q 96 -285 q 94 + 205 q 92 -43 q 90 -200 q 88 + 444 q 86 -594 q 84 + 578 q 82 -353 q 80 -33 q 78 + 442 q 76 -718 q 74 + 741 q 72 -503 q 70 + 62 q 68 + 382 q 66 -634 q 64 + 590 q 62 -242 q 60 -225 q 58 + 619 q 56 -714 q 54 + 457 q 52 + 41 q 50 -598 q 48 + 990 q 46 -1019 q 44 + 687 q 42 -80 q 40 -565 q 38 + 1029 q 36 -1139 q 34 + 855 q 32 -321 q 30 -291 q 28 + 734 q 26 -857 q 24 + 643 q 22 -176 q 20 -321 q 18 + 622 q 16 -611 q 14 + 275 q 12 + 201 q 10 -617 q 8 + 783 q 6 -619 q 4 + 209 q 2 + 290-671 q -2 + 802 q -4 -649 q -6 + 303 q -8 + 78 q -10 -371 q -12 + 490 q -14 -435 q -16 + 279 q -18 -80 q -20 -74 q -22 + 152 q -24 -163 q -26 + 122 q -28 -66 q -30 + 20 q -32 + 12 q -34 -24 q -36 + 22 q -38 -16 q -40 + 8 q -42 -3 q -44 + q -46$

Further Quantum Invariants

K11a157 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["K11a157"];

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

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

Out[4]= $- t 4 + 6 t 3 -16 t 2 + 28 t -33 + 28 t -1 -16 t -2 + 6 t -3 - t -4$

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

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

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]= $\left\{2,t^4+t^2+1\right\}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 135, -2 }

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

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

Out[8]= $q 3 -4 q 2 + 9 q -14 + 19 q -1 -22 q -2 + 22 q -3 -18 q -4 + 14 q -5 -8 q -6 + 3 q -7 - q -8$

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

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

Out[9]= $- a 2 z 8 + 2 a 4 z 6 -5 a 2 z 6 + z 6 - a 6 z 4 + 8 a 4 z 4 -10 a 2 z 4 + 3 z 4 -3 a 6 z 2 + 12 a 4 z 2 -10 a 2 z 2 + 3 z 2 -3 a 6 + 7 a 4 -5 a 2 + 2$

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

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

Out[10]= $2 a 4 z 10 + 2 a 2 z 10 + 6 a 5 z 9 + 12 a 3 z 9 + 6 a z 9 + 8 a 6 z 8 + 13 a 4 z 8 + 12 a 2 z 8 + 7 z 8 + 6 a 7 z 7 -3 a 5 z 7 -20 a 3 z 7 -7 a z 7 + 4 z 7 a -1 + 3 a 8 z 6 -14 a 6 z 6 -39 a 4 z 6 -39 a 2 z 6 + z 6 a -2 -16 z 6 + a 9 z 5 -9 a 7 z 5 -8 a 5 z 5 + 4 a 3 z 5 -7 a z 5 -9 z 5 a -1 -4 a 8 z 4 + 16 a 6 z 4 + 46 a 4 z 4 + 38 a 2 z 4 -2 z 4 a -2 + 10 z 4 -2 a 9 z 3 + 6 a 7 z 3 + 13 a 5 z 3 + 8 a 3 z 3 + 8 a z 3 + 5 z 3 a -1 + a 8 z 2 -12 a 6 z 2 -28 a 4 z 2 -20 a 2 z 2 + z 2 a -2 -4 z 2 + a 9 z -2 a 7 z -5 a 5 z -3 a 3 z -2 a z - z a -1 + 3 a 6 + 7 a 4 + 5 a 2 + 2$

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

Same Alexander/Conway Polynomial: {K11a264, K11a305,}

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["K11a157"];

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]= { $- t 4 + 6 t 3 -16 t 2 + 28 t -33 + 28 t -1 -16 t -2 + 6 t -3 - t -4$ , $q 3 -4 q 2 + 9 q -14 + 19 q -1 -22 q -2 + 22 q -3 -18 q -4 + 14 q -5 -8 q -6 + 3 q -7 - q -8$ }

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]= {K11a264, K11a305,}

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, -5)

[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 K11a157. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

-3

-5

-1

-3

-5

-7

-9

-4

-11

-13

-5

-15

-17

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{11}$
$r = 1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$