K11a326

K11a325

K11a327

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a326 at Knotilus!

Knot presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a326/ThurstonBennequinNumber Hyperbolic Volume 18.6893 A-Polynomial See Data:K11a326/A-polynomial

[edit Notes for K11a326's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $[0,4]$ Rasmussen s-Invariant 0

[edit Notes for K11a326's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+19t^{2}-36t+45-36t^{-1}+19t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+2z^{6}+3z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 169, 0 }
Jones polynomial	$q^{6}-5q^{5}+11q^{4}-18q^{3}+24q^{2}-27q+28-23q^{-1}+17q^{-2}-10q^{-3}+4q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+5z^{6}-3a^{2}z^{4}-6z^{4}a^{-2}+z^{4}a^{-4}+11z^{4}-4a^{2}z^{2}-6z^{2}a^{-2}+z^{2}a^{-4}+11z^{2}-2a^{2}-2a^{-2}+5$
Kauffman polynomial (db, data sources)	$4z^{10}a^{-2}+4z^{10}+11az^{9}+21z^{9}a^{-1}+10z^{9}a^{-3}+13a^{2}z^{8}+13z^{8}a^{-2}+10z^{8}a^{-4}+16z^{8}+9a^{3}z^{7}-14az^{7}-44z^{7}a^{-1}-16z^{7}a^{-3}+5z^{7}a^{-5}+4a^{4}z^{6}-24a^{2}z^{6}-47z^{6}a^{-2}-21z^{6}a^{-4}+z^{6}a^{-6}-53z^{6}+a^{5}z^{5}-13a^{3}z^{5}+3az^{5}+28z^{5}a^{-1}+2z^{5}a^{-3}-9z^{5}a^{-5}-4a^{4}z^{4}+22a^{2}z^{4}+39z^{4}a^{-2}+11z^{4}a^{-4}-z^{4}a^{-6}+53z^{4}-a^{5}z^{3}+7a^{3}z^{3}+6az^{3}-5z^{3}a^{-1}+3z^{3}a^{-5}-10a^{2}z^{2}-14z^{2}a^{-2}-2z^{2}a^{-4}-22z^{2}-2a^{3}z-3az-za^{-1}+za^{-3}+za^{-5}+2a^{2}+2a^{-2}+5$
The A2 invariant	$-q^{14}+2q^{12}-4q^{10}+2q^{8}+q^{6}-4q^{4}+7q^{2}-3+5q^{-2}-2q^{-6}+4q^{-8}-5q^{-10}+2q^{-12}-2q^{-16}+q^{-18}$
The G2 invariant	Data:K11a326/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a326 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["K11a326"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-6t^{3}+19t^{2}-36t+45-36t^{-1}+19t^{-2}-6t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 169, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-5q^{5}+11q^{4}-18q^{3}+24q^{2}-27q+28-23q^{-1}+17q^{-2}-10q^{-3}+4q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+5z^{6}-3a^{2}z^{4}-6z^{4}a^{-2}+z^{4}a^{-4}+11z^{4}-4a^{2}z^{2}-6z^{2}a^{-2}+z^{2}a^{-4}+11z^{2}-2a^{2}-2a^{-2}+5$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$4z^{10}a^{-2}+4z^{10}+11az^{9}+21z^{9}a^{-1}+10z^{9}a^{-3}+13a^{2}z^{8}+13z^{8}a^{-2}+10z^{8}a^{-4}+16z^{8}+9a^{3}z^{7}-14az^{7}-44z^{7}a^{-1}-16z^{7}a^{-3}+5z^{7}a^{-5}+4a^{4}z^{6}-24a^{2}z^{6}-47z^{6}a^{-2}-21z^{6}a^{-4}+z^{6}a^{-6}-53z^{6}+a^{5}z^{5}-13a^{3}z^{5}+3az^{5}+28z^{5}a^{-1}+2z^{5}a^{-3}-9z^{5}a^{-5}-4a^{4}z^{4}+22a^{2}z^{4}+39z^{4}a^{-2}+11z^{4}a^{-4}-z^{4}a^{-6}+53z^{4}-a^{5}z^{3}+7a^{3}z^{3}+6az^{3}-5z^{3}a^{-1}+3z^{3}a^{-5}-10a^{2}z^{2}-14z^{2}a^{-2}-2z^{2}a^{-4}-22z^{2}-2a^{3}z-3az-za^{-1}+za^{-3}+za^{-5}+2a^{2}+2a^{-2}+5$

"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["K11a326"];

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]=

{ $t^{4}-6t^{3}+19t^{2}-36t+45-36t^{-1}+19t^{-2}-6t^{-3}+t^{-4}$, $q^{6}-5q^{5}+11q^{4}-18q^{3}+24q^{2}-27q+28-23q^{-1}+17q^{-2}-10q^{-3}+4q^{-4}-q^{-5}$ }

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:

(2, 0)

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 $8$ $0$ $32$ ${\frac {76}{3}}$ $-{\frac {28}{3}}$ $0$ $-32$ $64$ $-96$ ${\frac {256}{3}}$ $0$ ${\frac {608}{3}}$ $-{\frac {224}{3}}$ ${\frac {2551}{15}}$ ${\frac {3076}{15}}$ $-{\frac {8756}{45}}$ $-{\frac {295}{9}}$ $-{\frac {1049}{15}}$

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=$0 is the signature of K11a326. 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 13                       1 1 11                     4   -4 9                   7 1   6 7                 11 4     -7 5               13 7       6 3             14 11         -3 1           14 13           1 -1         10 15             5 -3       7 13               -6 -5     3 10                 7 -7   1 7                   -6 -9   3                     3 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a325 K11a327