K11a219

K11a218

K11a220

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a219 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a219/ThurstonBennequinNumber Hyperbolic Volume 13.4578 A-Polynomial See Data:K11a219/A-polynomial

[edit Notes for K11a219'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 K11a219's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-4t^{2}+22t-35+22t^{-1}-4t^{-2}$
Conway polynomial	$-4z^{4}+6z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 87, -2 }
Jones polynomial	$q-3+6q^{-1}-10q^{-2}+13q^{-3}-13q^{-4}+14q^{-5}-11q^{-6}+8q^{-7}-5q^{-8}+2q^{-9}-q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$-a^{10}+2z^{2}a^{8}-z^{4}a^{6}+2z^{2}a^{6}+a^{6}-2z^{4}a^{4}+a^{4}-z^{4}a^{2}+z^{2}a^{2}+z^{2}$
Kauffman polynomial (db, data sources)	$z^{7}a^{11}-5z^{5}a^{11}+8z^{3}a^{11}-4za^{11}+2z^{8}a^{10}-8z^{6}a^{10}+9z^{4}a^{10}-3z^{2}a^{10}+a^{10}+2z^{9}a^{9}-5z^{7}a^{9}+3z^{3}a^{9}+z^{10}a^{8}+2z^{8}a^{8}-14z^{6}a^{8}+14z^{4}a^{8}-5z^{2}a^{8}+6z^{9}a^{7}-18z^{7}a^{7}+20z^{5}a^{7}-17z^{3}a^{7}+6za^{7}+z^{10}a^{6}+7z^{8}a^{6}-27z^{6}a^{6}+30z^{4}a^{6}-13z^{2}a^{6}-a^{6}+4z^{9}a^{5}-5z^{7}a^{5}+2za^{5}+7z^{8}a^{4}-16z^{6}a^{4}+19z^{4}a^{4}-9z^{2}a^{4}+a^{4}+7z^{7}a^{3}-12z^{5}a^{3}+9z^{3}a^{3}+5z^{6}a^{2}-5z^{4}a^{2}+z^{2}a^{2}+3z^{5}a-3z^{3}a+z^{4}-z^{2}$
The A2 invariant	Data:K11a219/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a219/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a219 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["K11a219"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-4t^{2}+22t-35+22t^{-1}-4t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 87, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q-3+6q^{-1}-10q^{-2}+13q^{-3}-13q^{-4}+14q^{-5}-11q^{-6}+8q^{-7}-5q^{-8}+2q^{-9}-q^{-10}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{10}+2z^{2}a^{8}-z^{4}a^{6}+2z^{2}a^{6}+a^{6}-2z^{4}a^{4}+a^{4}-z^{4}a^{2}+z^{2}a^{2}+z^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{7}a^{11}-5z^{5}a^{11}+8z^{3}a^{11}-4za^{11}+2z^{8}a^{10}-8z^{6}a^{10}+9z^{4}a^{10}-3z^{2}a^{10}+a^{10}+2z^{9}a^{9}-5z^{7}a^{9}+3z^{3}a^{9}+z^{10}a^{8}+2z^{8}a^{8}-14z^{6}a^{8}+14z^{4}a^{8}-5z^{2}a^{8}+6z^{9}a^{7}-18z^{7}a^{7}+20z^{5}a^{7}-17z^{3}a^{7}+6za^{7}+z^{10}a^{6}+7z^{8}a^{6}-27z^{6}a^{6}+30z^{4}a^{6}-13z^{2}a^{6}-a^{6}+4z^{9}a^{5}-5z^{7}a^{5}+2za^{5}+7z^{8}a^{4}-16z^{6}a^{4}+19z^{4}a^{4}-9z^{2}a^{4}+a^{4}+7z^{7}a^{3}-12z^{5}a^{3}+9z^{3}a^{3}+5z^{6}a^{2}-5z^{4}a^{2}+z^{2}a^{2}+3z^{5}a-3z^{3}a+z^{4}-z^{2}$

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

In[3]:=

K = Knot["K11a219"];

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]=

{ $-4t^{2}+22t-35+22t^{-1}-4t^{-2}$, $q-3+6q^{-1}-10q^{-2}+13q^{-3}-13q^{-4}+14q^{-5}-11q^{-6}+8q^{-7}-5q^{-8}+2q^{-9}-q^{-10}$ }

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]=

{}

Vassiliev invariants

V2 and V3:

(6, -15)

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 $24$ $-120$ $288$ $812$ $172$ $-2880$ $-5776$ $-960$ $-1176$ $2304$ $7200$ $19488$ $4128$ ${\frac {205431}{5}}$ $-{\frac {15924}{5}}$ ${\frac {110988}{5}}$ $499$ ${\frac {17271}{5}}$

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 K11a219. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                       1 1 1                     2   -2 -1                   4 1   3 -3                 7 3     -4 -5               6 3       3 -7             7 7         0 -9           7 6           1 -11         4 7             3 -13       4 7               -3 -15     1 4                 3 -17   1 4                   -3 -19   1                     1 -21 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

K11a218 K11a220