K11a323

K11a322

K11a324

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a323 at Knotilus!

Knot presentations

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

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:K11a323/ThurstonBennequinNumber Hyperbolic Volume 13.4473 A-Polynomial See Data:K11a323/A-polynomial

[edit Notes for K11a323'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 K11a323's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-9t^{2}+19t-23+19t^{-1}-9t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+3z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 83, -2 }
Jones polynomial	$q^{5}-3q^{4}+5q^{3}-8q^{2}+11q-12+13q^{-1}-11q^{-2}+9q^{-3}-6q^{-4}+3q^{-5}-q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$a^{2}z^{6}+z^{6}-a^{4}z^{4}+3a^{2}z^{4}-2z^{4}a^{-2}+3z^{4}-2a^{4}z^{2}+3a^{2}z^{2}-5z^{2}a^{-2}+z^{2}a^{-4}+4z^{2}-a^{4}+a^{2}-3a^{-2}+a^{-4}+3$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}+6az^{9}+9z^{9}a^{-1}+3z^{9}a^{-3}+9a^{2}z^{8}-5z^{8}a^{-2}+z^{8}a^{-4}+3z^{8}+10a^{3}z^{7}-14az^{7}-40z^{7}a^{-1}-16z^{7}a^{-3}+9a^{4}z^{6}-21a^{2}z^{6}-9z^{6}a^{-2}-5z^{6}a^{-4}-34z^{6}+6a^{5}z^{5}-19a^{3}z^{5}+52z^{5}a^{-1}+27z^{5}a^{-3}+3a^{6}z^{4}-13a^{4}z^{4}+8a^{2}z^{4}+29z^{4}a^{-2}+8z^{4}a^{-4}+45z^{4}+a^{7}z^{3}-4a^{5}z^{3}+6a^{3}z^{3}+6az^{3}-20z^{3}a^{-1}-15z^{3}a^{-3}+5a^{4}z^{2}-18z^{2}a^{-2}-5z^{2}a^{-4}-18z^{2}+a^{5}z-az+za^{-1}+za^{-3}-a^{4}-a^{2}+3a^{-2}+a^{-4}+3$
The A2 invariant	Data:K11a323/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a323/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a323 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["K11a323"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-9t^{2}+19t-23+19t^{-1}-9t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 83, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{2}z^{6}+z^{6}-a^{4}z^{4}+3a^{2}z^{4}-2z^{4}a^{-2}+3z^{4}-2a^{4}z^{2}+3a^{2}z^{2}-5z^{2}a^{-2}+z^{2}a^{-4}+4z^{2}-a^{4}+a^{2}-3a^{-2}+a^{-4}+3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}+6az^{9}+9z^{9}a^{-1}+3z^{9}a^{-3}+9a^{2}z^{8}-5z^{8}a^{-2}+z^{8}a^{-4}+3z^{8}+10a^{3}z^{7}-14az^{7}-40z^{7}a^{-1}-16z^{7}a^{-3}+9a^{4}z^{6}-21a^{2}z^{6}-9z^{6}a^{-2}-5z^{6}a^{-4}-34z^{6}+6a^{5}z^{5}-19a^{3}z^{5}+52z^{5}a^{-1}+27z^{5}a^{-3}+3a^{6}z^{4}-13a^{4}z^{4}+8a^{2}z^{4}+29z^{4}a^{-2}+8z^{4}a^{-4}+45z^{4}+a^{7}z^{3}-4a^{5}z^{3}+6a^{3}z^{3}+6az^{3}-20z^{3}a^{-1}-15z^{3}a^{-3}+5a^{4}z^{2}-18z^{2}a^{-2}-5z^{2}a^{-4}-18z^{2}+a^{5}z-az+za^{-1}+za^{-3}-a^{4}-a^{2}+3a^{-2}+a^{-4}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_83, K11a307,}

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["K11a323"];

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]=

{ $2t^{3}-9t^{2}+19t-23+19t^{-1}-9t^{-2}+2t^{-3}$, $q^{5}-3q^{4}+5q^{3}-8q^{2}+11q-12+13q^{-1}-11q^{-2}+9q^{-3}-6q^{-4}+3q^{-5}-q^{-6}$ }

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_83, K11a307,}

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:

(1, -2)

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 $4$ $-16$ $8$ $-{\frac {34}{3}}$ $-{\frac {62}{3}}$ $-64$ $-{\frac {448}{3}}$ ${\frac {320}{3}}$ $-176$ ${\frac {32}{3}}$ $128$ $-{\frac {136}{3}}$ $-{\frac {248}{3}}$ ${\frac {5071}{30}}$ ${\frac {646}{5}}$ $-{\frac {2338}{45}}$ $-{\frac {1135}{18}}$ $-{\frac {2129}{30}}$

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 K11a323. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a322 K11a324