K11a224

K11a223

K11a225

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a224 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a224/ThurstonBennequinNumber Hyperbolic Volume 13.0759 A-Polynomial See Data:K11a224/A-polynomial

[edit Notes for K11a224's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -4

[edit Notes for K11a224's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+10t^{2}-20t+25-20t^{-1}+10t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 89, 4 }
Jones polynomial	$-q^{11}+3q^{10}-6q^{9}+10q^{8}-13q^{7}+14q^{6}-14q^{5}+12q^{4}-8q^{3}+5q^{2}-2q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-3z^{4}a^{-4}-2z^{4}a^{-6}+2z^{4}a^{-8}+3z^{2}a^{-2}-3z^{2}a^{-4}-z^{2}a^{-6}+4z^{2}a^{-8}-z^{2}a^{-10}+2a^{-2}-a^{-4}-a^{-6}+2a^{-8}-a^{-10}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-6}+z^{10}a^{-8}+2z^{9}a^{-5}+5z^{9}a^{-7}+3z^{9}a^{-9}+2z^{8}a^{-4}+z^{8}a^{-6}+4z^{8}a^{-8}+5z^{8}a^{-10}+2z^{7}a^{-3}-3z^{7}a^{-5}-12z^{7}a^{-7}-2z^{7}a^{-9}+5z^{7}a^{-11}+z^{6}a^{-2}-2z^{6}a^{-4}-z^{6}a^{-6}-11z^{6}a^{-8}-10z^{6}a^{-10}+3z^{6}a^{-12}-6z^{5}a^{-3}+4z^{5}a^{-5}+20z^{5}a^{-7}-2z^{5}a^{-9}-11z^{5}a^{-11}+z^{5}a^{-13}-4z^{4}a^{-2}-6z^{4}a^{-4}-z^{4}a^{-6}+16z^{4}a^{-8}+9z^{4}a^{-10}-6z^{4}a^{-12}+4z^{3}a^{-3}-8z^{3}a^{-5}-17z^{3}a^{-7}+5z^{3}a^{-9}+8z^{3}a^{-11}-2z^{3}a^{-13}+5z^{2}a^{-2}+7z^{2}a^{-4}-3z^{2}a^{-6}-10z^{2}a^{-8}-3z^{2}a^{-10}+2z^{2}a^{-12}+za^{-3}+3za^{-5}+3za^{-7}-za^{-9}-2za^{-11}-2a^{-2}-a^{-4}+a^{-6}+2a^{-8}+a^{-10}$
The A2 invariant	Data:K11a224/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a224/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a224 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["K11a224"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+10t^{2}-20t+25-20t^{-1}+10t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 89, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{11}+3q^{10}-6q^{9}+10q^{8}-13q^{7}+14q^{6}-14q^{5}+12q^{4}-8q^{3}+5q^{2}-2q+1$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-3z^{4}a^{-4}-2z^{4}a^{-6}+2z^{4}a^{-8}+3z^{2}a^{-2}-3z^{2}a^{-4}-z^{2}a^{-6}+4z^{2}a^{-8}-z^{2}a^{-10}+2a^{-2}-a^{-4}-a^{-6}+2a^{-8}-a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_92, K11a153, K11n35, K11n43,}

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["K11a224"];

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]=

{ $-2t^{3}+10t^{2}-20t+25-20t^{-1}+10t^{-2}-2t^{-3}$, $-q^{11}+3q^{10}-6q^{9}+10q^{8}-13q^{7}+14q^{6}-14q^{5}+12q^{4}-8q^{3}+5q^{2}-2q+1$ }

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]=

{10_92, K11a153, K11n35, K11n43,}

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 $8$ $40$ $32$ ${\frac {604}{3}}$ ${\frac {92}{3}}$ $320$ ${\frac {2992}{3}}$ ${\frac {256}{3}}$ $232$ ${\frac {256}{3}}$ $800$ ${\frac {4832}{3}}$ ${\frac {736}{3}}$ ${\frac {75151}{15}}$ $-{\frac {13124}{15}}$ ${\frac {126724}{45}}$ ${\frac {2321}{9}}$ ${\frac {5791}{15}}$

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=$4 is the signature of K11a224. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 23                       1 -1 21                     2   2 19                   4 1   -3 17                 6 2     4 15               7 4       -3 13             7 6         1 11           7 7           0 9         5 7             -2 7       3 7               4 5     2 5                 -3 3   1 4                   3 1   1                     -1 -1 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a223 K11a225