K11a322

K11a321

K11a323

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a322 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X12,4,13,3 X16,5,17,6 X22,8,1,7 X20,9,21,10 X4,12,5,11 X18,13,19,14 X8,15,9,16 X2,17,3,18 X10,19,11,20 X14,21,15,22
Gauss code	1, -9, 2, -6, 3, -1, 4, -8, 5, -10, 6, -2, 7, -11, 8, -3, 9, -7, 10, -5, 11, -4
Dowker-Thistlethwaite code	6 12 16 22 20 4 18 8 2 10 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a322/ThurstonBennequinNumber Hyperbolic Volume 17.6972 A-Polynomial See Data:K11a322/A-polynomial

[edit Notes for K11a322's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for K11a322's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-13t^{2}+36t-49+36t^{-1}-13t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}-z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^{2}+t+1\right\}$
Determinant and Signature	{ 151, -2 }
Jones polynomial	$q^{3}-4q^{2}+9q-15+21q^{-1}-24q^{-2}+25q^{-3}-21q^{-4}+16q^{-5}-10q^{-6}+4q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{6}-z^{2}a^{6}-2a^{6}+z^{6}a^{4}+2z^{4}a^{4}+5z^{2}a^{4}+4a^{4}+z^{6}a^{2}-2z^{2}a^{2}-2a^{2}-2z^{4}-z^{2}+1+z^{2}a^{-2}$
Kauffman polynomial (db, data sources)	$3a^{4}z^{10}+3a^{2}z^{10}+9a^{5}z^{9}+16a^{3}z^{9}+7az^{9}+12a^{6}z^{8}+14a^{4}z^{8}+9a^{2}z^{8}+7z^{8}+9a^{7}z^{7}-9a^{5}z^{7}-33a^{3}z^{7}-11az^{7}+4z^{7}a^{-1}+4a^{8}z^{6}-22a^{6}z^{6}-42a^{4}z^{6}-32a^{2}z^{6}+z^{6}a^{-2}-15z^{6}+a^{9}z^{5}-14a^{7}z^{5}-3a^{5}z^{5}+24a^{3}z^{5}+3az^{5}-9z^{5}a^{-1}-4a^{8}z^{4}+18a^{6}z^{4}+40a^{4}z^{4}+29a^{2}z^{4}-2z^{4}a^{-2}+9z^{4}-a^{9}z^{3}+8a^{7}z^{3}+8a^{5}z^{3}-8a^{3}z^{3}-2az^{3}+5z^{3}a^{-1}-8a^{6}z^{2}-17a^{4}z^{2}-13a^{2}z^{2}+z^{2}a^{-2}-3z^{2}-3a^{7}z-3a^{5}z+a^{3}z+az+2a^{6}+4a^{4}+2a^{2}+1$
The A2 invariant	Data:K11a322/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a322/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a322 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["K11a322"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-13t^{2}+36t-49+36t^{-1}-13t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\left\{2,t^{2}+t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 151, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-4q^{2}+9q-15+21q^{-1}-24q^{-2}+25q^{-3}-21q^{-4}+16q^{-5}-10q^{-6}+4q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}a^{6}-z^{2}a^{6}-2a^{6}+z^{6}a^{4}+2z^{4}a^{4}+5z^{2}a^{4}+4a^{4}+z^{6}a^{2}-2z^{2}a^{2}-2a^{2}-2z^{4}-z^{2}+1+z^{2}a^{-2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3a^{4}z^{10}+3a^{2}z^{10}+9a^{5}z^{9}+16a^{3}z^{9}+7az^{9}+12a^{6}z^{8}+14a^{4}z^{8}+9a^{2}z^{8}+7z^{8}+9a^{7}z^{7}-9a^{5}z^{7}-33a^{3}z^{7}-11az^{7}+4z^{7}a^{-1}+4a^{8}z^{6}-22a^{6}z^{6}-42a^{4}z^{6}-32a^{2}z^{6}+z^{6}a^{-2}-15z^{6}+a^{9}z^{5}-14a^{7}z^{5}-3a^{5}z^{5}+24a^{3}z^{5}+3az^{5}-9z^{5}a^{-1}-4a^{8}z^{4}+18a^{6}z^{4}+40a^{4}z^{4}+29a^{2}z^{4}-2z^{4}a^{-2}+9z^{4}-a^{9}z^{3}+8a^{7}z^{3}+8a^{5}z^{3}-8a^{3}z^{3}-2az^{3}+5z^{3}a^{-1}-8a^{6}z^{2}-17a^{4}z^{2}-13a^{2}z^{2}+z^{2}a^{-2}-3z^{2}-3a^{7}z-3a^{5}z+a^{3}z+az+2a^{6}+4a^{4}+2a^{2}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a147,}

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["K11a322"];

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]=

{ $2t^{3}-13t^{2}+36t-49+36t^{-1}-13t^{-2}+2t^{-3}$, $q^{3}-4q^{2}+9q-15+21q^{-1}-24q^{-2}+25q^{-3}-21q^{-4}+16q^{-5}-10q^{-6}+4q^{-7}-q^{-8}$ }

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]=

{}

In[6]:=

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

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

Out[6]=

{K11a147,}

Vassiliev invariants

V2 and V3:

(2, -4)

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 $8$ $-32$ $32$ ${\frac {364}{3}}$ ${\frac {68}{3}}$ $-256$ $-{\frac {1664}{3}}$ $-{\frac {32}{3}}$ $-192$ ${\frac {256}{3}}$ $512$ ${\frac {2912}{3}}$ ${\frac {544}{3}}$ ${\frac {37351}{15}}$ $-{\frac {9484}{15}}$ ${\frac {77164}{45}}$ ${\frac {761}{9}}$ ${\frac {4231}{15}}$

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 K11a322. 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                     3   -3 3                   6 1   5 1                 9 3     -6 -1               12 6       6 -3             13 10         -3 -5           12 11           1 -7         9 13             4 -9       7 12               -5 -11     3 9                 6 -13   1 7                   -6 -15   3                     3 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a321 K11a323