K11a329

K11a328

K11a330

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a329 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{2,3\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a329/ThurstonBennequinNumber Hyperbolic Volume 17.4305 A-Polynomial See Data:K11a329/A-polynomial

[edit Notes for K11a329's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a329's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$11t^{2}-36t+51-36t^{-1}+11t^{-2}$
Conway polynomial	$11z^{4}+8z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^{2}+t+1\right\}$
Determinant and Signature	{ 145, 4 }
Jones polynomial	$-q^{13}+4q^{12}-9q^{11}+14q^{10}-20q^{9}+23q^{8}-23q^{7}+21q^{6}-15q^{5}+10q^{4}-4q^{3}+q^{2}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-4}+4z^{4}a^{-6}+5z^{4}a^{-8}+z^{4}a^{-10}+5z^{2}a^{-6}+8z^{2}a^{-8}-4z^{2}a^{-10}-z^{2}a^{-12}+a^{-6}+4a^{-8}-5a^{-10}+a^{-12}$
Kauffman polynomial (db, data sources)	$3z^{10}a^{-10}+3z^{10}a^{-12}+10z^{9}a^{-9}+16z^{9}a^{-11}+6z^{9}a^{-13}+16z^{8}a^{-8}+17z^{8}a^{-10}+5z^{8}a^{-12}+4z^{8}a^{-14}+15z^{7}a^{-7}-5z^{7}a^{-9}-38z^{7}a^{-11}-17z^{7}a^{-13}+z^{7}a^{-15}+10z^{6}a^{-6}-26z^{6}a^{-8}-59z^{6}a^{-10}-36z^{6}a^{-12}-13z^{6}a^{-14}+4z^{5}a^{-5}-19z^{5}a^{-7}-24z^{5}a^{-9}+13z^{5}a^{-11}+11z^{5}a^{-13}-3z^{5}a^{-15}+z^{4}a^{-4}-9z^{4}a^{-6}+16z^{4}a^{-8}+49z^{4}a^{-10}+36z^{4}a^{-12}+13z^{4}a^{-14}+7z^{3}a^{-7}+21z^{3}a^{-9}+13z^{3}a^{-11}+2z^{3}a^{-13}+3z^{3}a^{-15}+5z^{2}a^{-6}-11z^{2}a^{-8}-20z^{2}a^{-10}-8z^{2}a^{-12}-4z^{2}a^{-14}-7za^{-9}-7za^{-11}-za^{-13}-za^{-15}-a^{-6}+4a^{-8}+5a^{-10}+a^{-12}$
The A2 invariant	Data:K11a329/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a329/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a329 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["K11a329"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$11t^{2}-36t+51-36t^{-1}+11t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$11z^{4}+8z^{2}+1$

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]=

{ 145, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{13}+4q^{12}-9q^{11}+14q^{10}-20q^{9}+23q^{8}-23q^{7}+21q^{6}-15q^{5}+10q^{4}-4q^{3}+q^{2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{4}a^{-4}+4z^{4}a^{-6}+5z^{4}a^{-8}+z^{4}a^{-10}+5z^{2}a^{-6}+8z^{2}a^{-8}-4z^{2}a^{-10}-z^{2}a^{-12}+a^{-6}+4a^{-8}-5a^{-10}+a^{-12}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3z^{10}a^{-10}+3z^{10}a^{-12}+10z^{9}a^{-9}+16z^{9}a^{-11}+6z^{9}a^{-13}+16z^{8}a^{-8}+17z^{8}a^{-10}+5z^{8}a^{-12}+4z^{8}a^{-14}+15z^{7}a^{-7}-5z^{7}a^{-9}-38z^{7}a^{-11}-17z^{7}a^{-13}+z^{7}a^{-15}+10z^{6}a^{-6}-26z^{6}a^{-8}-59z^{6}a^{-10}-36z^{6}a^{-12}-13z^{6}a^{-14}+4z^{5}a^{-5}-19z^{5}a^{-7}-24z^{5}a^{-9}+13z^{5}a^{-11}+11z^{5}a^{-13}-3z^{5}a^{-15}+z^{4}a^{-4}-9z^{4}a^{-6}+16z^{4}a^{-8}+49z^{4}a^{-10}+36z^{4}a^{-12}+13z^{4}a^{-14}+7z^{3}a^{-7}+21z^{3}a^{-9}+13z^{3}a^{-11}+2z^{3}a^{-13}+3z^{3}a^{-15}+5z^{2}a^{-6}-11z^{2}a^{-8}-20z^{2}a^{-10}-8z^{2}a^{-12}-4z^{2}a^{-14}-7za^{-9}-7za^{-11}-za^{-13}-za^{-15}-a^{-6}+4a^{-8}+5a^{-10}+a^{-12}$

"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["K11a329"];

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]=

{ $11t^{2}-36t+51-36t^{-1}+11t^{-2}$, $-q^{13}+4q^{12}-9q^{11}+14q^{10}-20q^{9}+23q^{8}-23q^{7}+21q^{6}-15q^{5}+10q^{4}-4q^{3}+q^{2}$ }

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:

(8, 21)

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 $32$ $168$ $512$ ${\frac {3472}{3}}$ ${\frac {488}{3}}$ $5376$ $8880$ $1504$ $1064$ ${\frac {16384}{3}}$ $14112$ ${\frac {111104}{3}}$ ${\frac {15616}{3}}$ ${\frac {1049884}{15}}$ ${\frac {46424}{15}}$ ${\frac {1106776}{45}}$ ${\frac {4916}{9}}$ ${\frac {46204}{15}}$

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 K11a329. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 10 11 χ 27                       1 -1 25                     3   3 23                   6 1   -5 21                 8 3     5 19               12 6       -6 17             11 8         3 15           12 12           0 13         9 11             -2 11       6 12               6 9     4 9                 -5 7     6                   6 5 1 4                     -3 3 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }^{4}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=5$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=6$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=7$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=8$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=9$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=10$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=11$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a328 K11a330