K11a226

K11a225

K11a227

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a226 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X12,3,13,4 X18,5,19,6 X20,7,21,8 X16,10,17,9 X14,12,15,11 X2,13,3,14 X10,16,11,15 X22,17,1,18 X8,19,9,20 X6,21,7,22
Gauss code	1, -7, 2, -1, 3, -11, 4, -10, 5, -8, 6, -2, 7, -6, 8, -5, 9, -3, 10, -4, 11, -9
Dowker-Thistlethwaite code	4 12 18 20 16 14 2 10 22 8 6

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a226/ThurstonBennequinNumber Hyperbolic Volume 11.5144 A-Polynomial See Data:K11a226/A-polynomial

[edit Notes for K11a226's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a226's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-4t^{2}+18t-27+18t^{-1}-4t^{-2}$
Conway polynomial	$-4z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 71, -2 }
Jones polynomial	$q^{3}-3q^{2}+5q-7+10q^{-1}-11q^{-2}+11q^{-3}-9q^{-4}+7q^{-5}-4q^{-6}+2q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{8}+2z^{2}a^{6}+a^{6}-z^{4}a^{4}+z^{2}a^{4}+a^{4}-2z^{4}a^{2}-2z^{2}a^{2}-a^{2}-z^{4}+1+z^{2}a^{-2}$
Kauffman polynomial (db, data sources)	$a^{4}z^{10}+a^{2}z^{10}+2a^{5}z^{9}+5a^{3}z^{9}+3az^{9}+2a^{6}z^{8}+2a^{2}z^{8}+4z^{8}+2a^{7}z^{7}-5a^{5}z^{7}-19a^{3}z^{7}-9az^{7}+3z^{7}a^{-1}+2a^{8}z^{6}-2a^{6}z^{6}-5a^{4}z^{6}-16a^{2}z^{6}+z^{6}a^{-2}-14z^{6}+a^{9}z^{5}-2a^{7}z^{5}+10a^{5}z^{5}+31a^{3}z^{5}+8az^{5}-10z^{5}a^{-1}-5a^{8}z^{4}-a^{6}z^{4}+13a^{4}z^{4}+25a^{2}z^{4}-3z^{4}a^{-2}+13z^{4}-3a^{9}z^{3}-3a^{7}z^{3}-8a^{5}z^{3}-19a^{3}z^{3}-5az^{3}+6z^{3}a^{-1}+3a^{8}z^{2}+2a^{6}z^{2}-7a^{4}z^{2}-12a^{2}z^{2}+z^{2}a^{-2}-5z^{2}+2a^{9}z+2a^{7}z+2a^{5}z+4a^{3}z+2az-a^{8}-a^{6}+a^{4}+a^{2}+1$
The A2 invariant	Data:K11a226/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a226/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a226 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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a226"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-4t^{2}+18t-27+18t^{-1}-4t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-4z^{4}+2z^{2}+1$

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]=

{ 71, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-3q^{2}+5q-7+10q^{-1}-11q^{-2}+11q^{-3}-9q^{-4}+7q^{-5}-4q^{-6}+2q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{8}+2z^{2}a^{6}+a^{6}-z^{4}a^{4}+z^{2}a^{4}+a^{4}-2z^{4}a^{2}-2z^{2}a^{2}-a^{2}-z^{4}+1+z^{2}a^{-2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{4}z^{10}+a^{2}z^{10}+2a^{5}z^{9}+5a^{3}z^{9}+3az^{9}+2a^{6}z^{8}+2a^{2}z^{8}+4z^{8}+2a^{7}z^{7}-5a^{5}z^{7}-19a^{3}z^{7}-9az^{7}+3z^{7}a^{-1}+2a^{8}z^{6}-2a^{6}z^{6}-5a^{4}z^{6}-16a^{2}z^{6}+z^{6}a^{-2}-14z^{6}+a^{9}z^{5}-2a^{7}z^{5}+10a^{5}z^{5}+31a^{3}z^{5}+8az^{5}-10z^{5}a^{-1}-5a^{8}z^{4}-a^{6}z^{4}+13a^{4}z^{4}+25a^{2}z^{4}-3z^{4}a^{-2}+13z^{4}-3a^{9}z^{3}-3a^{7}z^{3}-8a^{5}z^{3}-19a^{3}z^{3}-5az^{3}+6z^{3}a^{-1}+3a^{8}z^{2}+2a^{6}z^{2}-7a^{4}z^{2}-12a^{2}z^{2}+z^{2}a^{-2}-5z^{2}+2a^{9}z+2a^{7}z+2a^{5}z+4a^{3}z+2az-a^{8}-a^{6}+a^{4}+a^{2}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a229,}

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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a226"];

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]=

{ $-4t^{2}+18t-27+18t^{-1}-4t^{-2}$, $q^{3}-3q^{2}+5q-7+10q^{-1}-11q^{-2}+11q^{-3}-9q^{-4}+7q^{-5}-4q^{-6}+2q^{-7}-q^{-8}$ }

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]=

{K11a229,}

In[6]:=

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

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

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(2, -5)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $8$ $-40$ $32$ ${\frac {556}{3}}$ ${\frac {140}{3}}$ $-320$ $-{\frac {2608}{3}}$ $-{\frac {256}{3}}$ $-264$ ${\frac {256}{3}}$ $800$ ${\frac {4448}{3}}$ ${\frac {1120}{3}}$ ${\frac {63031}{15}}$ $-{\frac {11924}{15}}$ ${\frac {122884}{45}}$ ${\frac {1337}{9}}$ ${\frac {6871}{15}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$-2 is the signature of K11a226. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     2   -2 3                   3 1   2 1                 4 2     -2 -1               6 3       3 -3             6 5         -1 -5           5 5           0 -7         4 6             2 -9       3 5               -2 -11     1 4                 3 -13   1 3                   -2 -15   1                     1 -17 1                       -1

Integral Khovanov Homology (db, data source)

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages. See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top.

K11a225 K11a227