K11a225

K11a224

K11a226

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a225 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 X6,19,7,20 X8,21,9,22
Gauss code	1, -7, 2, -1, 3, -10, 4, -11, 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 6 8

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:K11a225/ThurstonBennequinNumber Hyperbolic Volume 9.36224 A-Polynomial See Data:K11a225/A-polynomial

[edit Notes for K11a225'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 K11a225's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+8t^{2}-11t+11-11t^{-1}+8t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-4z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 53, -4 }
Jones polynomial	$q^{2}-2q+3-5q^{-1}+7q^{-2}-7q^{-3}+8q^{-4}-7q^{-5}+6q^{-6}-4q^{-7}+2q^{-8}-q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{8}-2a^{8}+2z^{4}a^{6}+6z^{2}a^{6}+3a^{6}-z^{6}a^{4}-3z^{4}a^{4}-z^{2}a^{4}-z^{6}a^{2}-4z^{4}a^{2}-4z^{2}a^{2}-a^{2}+z^{4}+3z^{2}+1$
Kauffman polynomial (db, data sources)	$z^{3}a^{11}-za^{11}+2z^{4}a^{10}-z^{2}a^{10}+3z^{5}a^{9}-2z^{3}a^{9}+za^{9}+4z^{6}a^{8}-6z^{4}a^{8}+5z^{2}a^{8}-2a^{8}+4z^{7}a^{7}-7z^{5}a^{7}+2z^{3}a^{7}+za^{7}+3z^{8}a^{6}-4z^{6}a^{6}-7z^{4}a^{6}+9z^{2}a^{6}-3a^{6}+2z^{9}a^{5}-3z^{7}a^{5}-5z^{5}a^{5}+4z^{3}a^{5}+z^{10}a^{4}-z^{8}a^{4}-5z^{6}a^{4}+3z^{4}a^{4}+z^{2}a^{4}+4z^{9}a^{3}-19z^{7}a^{3}+28z^{5}a^{3}-16z^{3}a^{3}+3za^{3}+z^{10}a^{2}-3z^{8}a^{2}-3z^{6}a^{2}+13z^{4}a^{2}-8z^{2}a^{2}+a^{2}+2z^{9}a-12z^{7}a+23z^{5}a-15z^{3}a+2za+z^{8}-6z^{6}+11z^{4}-6z^{2}+1$
The A2 invariant	Data:K11a225/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a225/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a225 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["K11a225"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+8t^{2}-11t+11-11t^{-1}+8t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-2z^{6}-4z^{4}+3z^{2}+1$

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]=

{ 53, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{2}a^{8}-2a^{8}+2z^{4}a^{6}+6z^{2}a^{6}+3a^{6}-z^{6}a^{4}-3z^{4}a^{4}-z^{2}a^{4}-z^{6}a^{2}-4z^{4}a^{2}-4z^{2}a^{2}-a^{2}+z^{4}+3z^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{3}a^{11}-za^{11}+2z^{4}a^{10}-z^{2}a^{10}+3z^{5}a^{9}-2z^{3}a^{9}+za^{9}+4z^{6}a^{8}-6z^{4}a^{8}+5z^{2}a^{8}-2a^{8}+4z^{7}a^{7}-7z^{5}a^{7}+2z^{3}a^{7}+za^{7}+3z^{8}a^{6}-4z^{6}a^{6}-7z^{4}a^{6}+9z^{2}a^{6}-3a^{6}+2z^{9}a^{5}-3z^{7}a^{5}-5z^{5}a^{5}+4z^{3}a^{5}+z^{10}a^{4}-z^{8}a^{4}-5z^{6}a^{4}+3z^{4}a^{4}+z^{2}a^{4}+4z^{9}a^{3}-19z^{7}a^{3}+28z^{5}a^{3}-16z^{3}a^{3}+3za^{3}+z^{10}a^{2}-3z^{8}a^{2}-3z^{6}a^{2}+13z^{4}a^{2}-8z^{2}a^{2}+a^{2}+2z^{9}a-12z^{7}a+23z^{5}a-15z^{3}a+2za+z^{8}-6z^{6}+11z^{4}-6z^{2}+1$

"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["K11a225"];

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]=

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

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:

(3, -8)

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 $12$ $-64$ $72$ $318$ $66$ $-768$ $-{\frac {5248}{3}}$ $-{\frac {736}{3}}$ $-384$ $288$ $2048$ $3816$ $792$ ${\frac {98991}{10}}$ $-{\frac {3822}{5}}$ ${\frac {77182}{15}}$ ${\frac {1201}{6}}$ ${\frac {7631}{10}}$

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 K11a225. 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 5                       1 1 3                     1   -1 1                   2 1   1 -1                 3 1     -2 -3               4 2       2 -5             4 4         0 -7           4 3           1 -9         3 4             1 -11       3 4               -1 -13     1 3                 2 -15   1 3                   -2 -17   1                     1 -19 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

K11a224 K11a226