K11a315

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K11a314

K11a316

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 K11a315's page at Knotilus!

Visit K11a315's page at the original Knot Atlas!

[edit K11a315 Quick Notes]

[edit K11a315 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₆₂₇₁ X_12,4,13,3 X_16,5,17,6 X_18,8,19,7 X_22,10,1,9 X_4,12,5,11 X_20,13,21,14 X_10,15,11,16 X_2,17,3,18 X_8,20,9,19 X_14,21,15,22
Gauss code	1, -9, 2, -6, 3, -1, 4, -10, 5, -8, 6, -2, 7, -11, 8, -3, 9, -4, 10, -7, 11, -5
Dowker-Thistlethwaite code	6 12 16 18 22 4 20 10 2 8 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:K11a315/ThurstonBennequinNumber
Hyperbolic Volume	18.2725
A-Polynomial	See Data:K11a315/A-polynomial

[edit Notes for K11a315's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$4$
Rasmussen s-Invariant	0

[edit Notes for K11a315's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$t 4 -6 t 3 + 18 t 2 -33 t + 41-33 t -1 + 18 t -2 -6 t -3 + t -4$
Conway polynomial	$z 8 + 2 z 6 + 2 z 4 + z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 157, 0 }
Jones polynomial	$q 6 -4 q 5 + 9 q 4 -16 q 3 + 22 q 2 -25 q + 26-22 q -1 + 17 q -2 -10 q -3 + 4 q -4 - q -5$
HOMFLY-PT polynomial (db, data sources)	$z 8 - a 2 z 6 -2 z 6 a -2 + 5 z 6 -3 a 2 z 4 -7 z 4 a -2 + z 4 a -4 + 11 z 4 -4 a 2 z 2 -9 z 2 a -2 + 2 z 2 a -4 + 12 z 2 -2 a 2 -4 a -2 + a -4 + 6$
Kauffman polynomial (db, data sources)	$3 z 10 a -2 + 3 z 10 + 10 a z 9 + 17 z 9 a -1 + 7 z 9 a -3 + 13 a 2 z 8 + 12 z 8 a -2 + 7 z 8 a -4 + 18 z 8 + 9 a 3 z 7 -11 a z 7 -32 z 7 a -1 -8 z 7 a -3 + 4 z 7 a -5 + 4 a 4 z 6 -25 a 2 z 6 -41 z 6 a -2 -14 z 6 a -4 + z 6 a -6 -55 z 6 + a 5 z 5 -13 a 3 z 5 -2 a z 5 + 15 z 5 a -1 -6 z 5 a -3 -9 z 5 a -5 -4 a 4 z 4 + 23 a 2 z 4 + 39 z 4 a -2 + 8 z 4 a -4 -2 z 4 a -6 + 56 z 4 - a 5 z 3 + 7 a 3 z 3 + 9 a z 3 + 3 z 3 a -1 + 8 z 3 a -3 + 6 z 3 a -5 -11 a 2 z 2 -19 z 2 a -2 -3 z 2 a -4 + z 2 a -6 -26 z 2 -2 a 3 z -4 a z -4 z a -1 -3 z a -3 - z a -5 + 2 a 2 + 4 a -2 + a -4 + 6$
The A2 invariant	$- q 14 + 2 q 12 -4 q 10 + 2 q 8 + q 6 -3 q 4 + 7 q 2 -3 + 5 q -2 - q -4 -2 q -6 + 3 q -8 -5 q -10 + 2 q -12 - q -16 + q -18$
The G2 invariant	Data:K11a315/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a315 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["K11a315"];

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

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

Out[4]= $t 4 -6 t 3 + 18 t 2 -33 t + 41-33 t -1 + 18 t -2 -6 t -3 + t -4$

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

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

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]= { 157, 0 }

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

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

Out[8]= $q 6 -4 q 5 + 9 q 4 -16 q 3 + 22 q 2 -25 q + 26-22 q -1 + 17 q -2 -10 q -3 + 4 q -4 - q -5$

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

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

Out[9]= $z 8 - a 2 z 6 -2 z 6 a -2 + 5 z 6 -3 a 2 z 4 -7 z 4 a -2 + z 4 a -4 + 11 z 4 -4 a 2 z 2 -9 z 2 a -2 + 2 z 2 a -4 + 12 z 2 -2 a 2 -4 a -2 + a -4 + 6$

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

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

Out[10]= $3 z 10 a -2 + 3 z 10 + 10 a z 9 + 17 z 9 a -1 + 7 z 9 a -3 + 13 a 2 z 8 + 12 z 8 a -2 + 7 z 8 a -4 + 18 z 8 + 9 a 3 z 7 -11 a z 7 -32 z 7 a -1 -8 z 7 a -3 + 4 z 7 a -5 + 4 a 4 z 6 -25 a 2 z 6 -41 z 6 a -2 -14 z 6 a -4 + z 6 a -6 -55 z 6 + a 5 z 5 -13 a 3 z 5 -2 a z 5 + 15 z 5 a -1 -6 z 5 a -3 -9 z 5 a -5 -4 a 4 z 4 + 23 a 2 z 4 + 39 z 4 a -2 + 8 z 4 a -4 -2 z 4 a -6 + 56 z 4 - a 5 z 3 + 7 a 3 z 3 + 9 a z 3 + 3 z 3 a -1 + 8 z 3 a -3 + 6 z 3 a -5 -11 a 2 z 2 -19 z 2 a -2 -3 z 2 a -4 + z 2 a -6 -26 z 2 -2 a 3 z -4 a z -4 z a -1 -3 z a -3 - z a -5 + 2 a 2 + 4 a -2 + a -4 + 6$

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

Same Alexander/Conway Polynomial: {K11a24, K11a26,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11a24, K11a26,}

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["K11a315"];

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]= { $t 4 -6 t 3 + 18 t 2 -33 t + 41-33 t -1 + 18 t -2 -6 t -3 + t -4$ , $q 6 -4 q 5 + 9 q 4 -16 q 3 + 22 q 2 -25 q + 26-22 q -1 + 17 q -2 -10 q -3 + 4 q -4 - q -5$ }

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]= {K11a24, K11a26,}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]= {K11a24, K11a26,}

[edit] Vassiliev invariants

V₂ and V₃:

(1, -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 =

0 is the signature of K11a315. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

-3

-7

-3

-1

-3

-5

-7

-6

-9

-11

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{13}$
$r = 1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$