K11a328

K11a327

K11a329

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a328 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a328/ThurstonBennequinNumber Hyperbolic Volume 17.6919 A-Polynomial See Data:K11a328/A-polynomial

[edit Notes for K11a328'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 K11a328's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{3}+16t^{2}-34t+43-34t^{-1}+16t^{-2}-3t^{-3}$
Conway polynomial	$-3z^{6}-2z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 149, 4 }
Jones polynomial	$-q^{11}+4q^{10}-9q^{9}+15q^{8}-21q^{7}+24q^{6}-24q^{5}+21q^{4}-15q^{3}+10q^{2}-4q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}-2z^{6}a^{-6}+z^{4}a^{-2}-6z^{4}a^{-6}+3z^{4}a^{-8}+z^{2}a^{-2}+5z^{2}a^{-4}-8z^{2}a^{-6}+6z^{2}a^{-8}-z^{2}a^{-10}+4a^{-4}-5a^{-6}+3a^{-8}-a^{-10}$
Kauffman polynomial (db, data sources)	$3z^{10}a^{-6}+3z^{10}a^{-8}+8z^{9}a^{-5}+16z^{9}a^{-7}+8z^{9}a^{-9}+8z^{8}a^{-4}+11z^{8}a^{-6}+13z^{8}a^{-8}+10z^{8}a^{-10}+4z^{7}a^{-3}-15z^{7}a^{-5}-33z^{7}a^{-7}-6z^{7}a^{-9}+8z^{7}a^{-11}+z^{6}a^{-2}-19z^{6}a^{-4}-43z^{6}a^{-6}-42z^{6}a^{-8}-15z^{6}a^{-10}+4z^{6}a^{-12}-8z^{5}a^{-3}+6z^{5}a^{-5}+19z^{5}a^{-7}-8z^{5}a^{-9}-12z^{5}a^{-11}+z^{5}a^{-13}-2z^{4}a^{-2}+15z^{4}a^{-4}+46z^{4}a^{-6}+43z^{4}a^{-8}+9z^{4}a^{-10}-5z^{4}a^{-12}+3z^{3}a^{-3}+2z^{3}a^{-7}+12z^{3}a^{-9}+6z^{3}a^{-11}-z^{3}a^{-13}+z^{2}a^{-2}-10z^{2}a^{-4}-23z^{2}a^{-6}-17z^{2}a^{-8}-4z^{2}a^{-10}+z^{2}a^{-12}-2za^{-5}-3za^{-7}-3za^{-9}-2za^{-11}+4a^{-4}+5a^{-6}+3a^{-8}+a^{-10}$
The A2 invariant	Data:K11a328/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a328/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a328 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["K11a328"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3t^{3}+16t^{2}-34t+43-34t^{-1}+16t^{-2}-3t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 149, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{11}+4q^{10}-9q^{9}+15q^{8}-21q^{7}+24q^{6}-24q^{5}+21q^{4}-15q^{3}+10q^{2}-4q+1$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-4}-2z^{6}a^{-6}+z^{4}a^{-2}-6z^{4}a^{-6}+3z^{4}a^{-8}+z^{2}a^{-2}+5z^{2}a^{-4}-8z^{2}a^{-6}+6z^{2}a^{-8}-z^{2}a^{-10}+4a^{-4}-5a^{-6}+3a^{-8}-a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3z^{10}a^{-6}+3z^{10}a^{-8}+8z^{9}a^{-5}+16z^{9}a^{-7}+8z^{9}a^{-9}+8z^{8}a^{-4}+11z^{8}a^{-6}+13z^{8}a^{-8}+10z^{8}a^{-10}+4z^{7}a^{-3}-15z^{7}a^{-5}-33z^{7}a^{-7}-6z^{7}a^{-9}+8z^{7}a^{-11}+z^{6}a^{-2}-19z^{6}a^{-4}-43z^{6}a^{-6}-42z^{6}a^{-8}-15z^{6}a^{-10}+4z^{6}a^{-12}-8z^{5}a^{-3}+6z^{5}a^{-5}+19z^{5}a^{-7}-8z^{5}a^{-9}-12z^{5}a^{-11}+z^{5}a^{-13}-2z^{4}a^{-2}+15z^{4}a^{-4}+46z^{4}a^{-6}+43z^{4}a^{-8}+9z^{4}a^{-10}-5z^{4}a^{-12}+3z^{3}a^{-3}+2z^{3}a^{-7}+12z^{3}a^{-9}+6z^{3}a^{-11}-z^{3}a^{-13}+z^{2}a^{-2}-10z^{2}a^{-4}-23z^{2}a^{-6}-17z^{2}a^{-8}-4z^{2}a^{-10}+z^{2}a^{-12}-2za^{-5}-3za^{-7}-3za^{-9}-2za^{-11}+4a^{-4}+5a^{-6}+3a^{-8}+a^{-10}$

"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["K11a328"];

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]=

{ $-3t^{3}+16t^{2}-34t+43-34t^{-1}+16t^{-2}-3t^{-3}$, $-q^{11}+4q^{10}-9q^{9}+15q^{8}-21q^{7}+24q^{6}-24q^{5}+21q^{4}-15q^{3}+10q^{2}-4q+1$ }

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, 6)

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 $12$ $48$ $72$ $254$ $50$ $576$ $1376$ $224$ $272$ $288$ $1152$ $3048$ $600$ ${\frac {73711}{10}}$ $-{\frac {7306}{15}}$ ${\frac {18874}{5}}$ ${\frac {347}{2}}$ ${\frac {5231}{10}}$

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 K11a328. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a327 K11a329