K11a358

K11a357

K11a359

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a358 at Knotilus!

Knot presentations

Planar diagram presentation	X8291 X14,4,15,3 X16,6,17,5 X18,8,19,7 X22,10,1,9 X20,12,21,11 X2,14,3,13 X4,16,5,15 X6,18,7,17 X12,20,13,19 X10,22,11,21
Gauss code	1, -7, 2, -8, 3, -9, 4, -1, 5, -11, 6, -10, 7, -2, 8, -3, 9, -4, 10, -6, 11, -5
Dowker-Thistlethwaite code	8 14 16 18 22 20 2 4 6 12 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a358/ThurstonBennequinNumber Hyperbolic Volume 6.1028 A-Polynomial See Data:K11a358/A-polynomial

[edit Notes for K11a358's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -6

[edit Notes for K11a358's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{3}-5t^{2}+5t-5+5t^{-1}-5t^{-2}+3t^{-3}$
Conway polynomial	$3z^{6}+13z^{4}+12z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 31, 6 }
Jones polynomial	$-q^{14}+q^{13}-2q^{12}+3q^{11}-4q^{10}+5q^{9}-4q^{8}+4q^{7}-3q^{6}+2q^{5}-q^{4}+q^{3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-6}+z^{6}a^{-8}+z^{6}a^{-10}+5z^{4}a^{-6}+4z^{4}a^{-8}+5z^{4}a^{-10}-z^{4}a^{-12}+6z^{2}a^{-6}+3z^{2}a^{-8}+7z^{2}a^{-10}-4z^{2}a^{-12}+a^{-6}+3a^{-10}-3a^{-12}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-10}+z^{10}a^{-12}+z^{9}a^{-9}+2z^{9}a^{-11}+z^{9}a^{-13}+z^{8}a^{-8}-7z^{8}a^{-10}-7z^{8}a^{-12}+z^{8}a^{-14}+z^{7}a^{-7}-5z^{7}a^{-9}-12z^{7}a^{-11}-5z^{7}a^{-13}+z^{7}a^{-15}+z^{6}a^{-6}-4z^{6}a^{-8}+20z^{6}a^{-10}+20z^{6}a^{-12}-4z^{6}a^{-14}+z^{6}a^{-16}-4z^{5}a^{-7}+9z^{5}a^{-9}+26z^{5}a^{-11}+9z^{5}a^{-13}-3z^{5}a^{-15}+z^{5}a^{-17}-5z^{4}a^{-6}+4z^{4}a^{-8}-26z^{4}a^{-10}-26z^{4}a^{-12}+6z^{4}a^{-14}-3z^{4}a^{-16}+3z^{3}a^{-7}-7z^{3}a^{-9}-20z^{3}a^{-11}-4z^{3}a^{-13}+2z^{3}a^{-15}-4z^{3}a^{-17}+6z^{2}a^{-6}-2z^{2}a^{-8}+14z^{2}a^{-10}+18z^{2}a^{-12}-3z^{2}a^{-14}+z^{2}a^{-16}+5za^{-11}+za^{-13}-za^{-15}+3za^{-17}-a^{-6}-3a^{-10}-3a^{-12}$
The A2 invariant	Data:K11a358/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a358/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a358 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["K11a358"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{3}-5t^{2}+5t-5+5t^{-1}-5t^{-2}+3t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$3z^{6}+13z^{4}+12z^{2}+1$

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]=

{ 31, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-6}+z^{6}a^{-8}+z^{6}a^{-10}+5z^{4}a^{-6}+4z^{4}a^{-8}+5z^{4}a^{-10}-z^{4}a^{-12}+6z^{2}a^{-6}+3z^{2}a^{-8}+7z^{2}a^{-10}-4z^{2}a^{-12}+a^{-6}+3a^{-10}-3a^{-12}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-10}+z^{10}a^{-12}+z^{9}a^{-9}+2z^{9}a^{-11}+z^{9}a^{-13}+z^{8}a^{-8}-7z^{8}a^{-10}-7z^{8}a^{-12}+z^{8}a^{-14}+z^{7}a^{-7}-5z^{7}a^{-9}-12z^{7}a^{-11}-5z^{7}a^{-13}+z^{7}a^{-15}+z^{6}a^{-6}-4z^{6}a^{-8}+20z^{6}a^{-10}+20z^{6}a^{-12}-4z^{6}a^{-14}+z^{6}a^{-16}-4z^{5}a^{-7}+9z^{5}a^{-9}+26z^{5}a^{-11}+9z^{5}a^{-13}-3z^{5}a^{-15}+z^{5}a^{-17}-5z^{4}a^{-6}+4z^{4}a^{-8}-26z^{4}a^{-10}-26z^{4}a^{-12}+6z^{4}a^{-14}-3z^{4}a^{-16}+3z^{3}a^{-7}-7z^{3}a^{-9}-20z^{3}a^{-11}-4z^{3}a^{-13}+2z^{3}a^{-15}-4z^{3}a^{-17}+6z^{2}a^{-6}-2z^{2}a^{-8}+14z^{2}a^{-10}+18z^{2}a^{-12}-3z^{2}a^{-14}+z^{2}a^{-16}+5za^{-11}+za^{-13}-za^{-15}+3za^{-17}-a^{-6}-3a^{-10}-3a^{-12}$

"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["K11a358"];

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]=

{ $3t^{3}-5t^{2}+5t-5+5t^{-1}-5t^{-2}+3t^{-3}$, $-q^{14}+q^{13}-2q^{12}+3q^{11}-4q^{10}+5q^{9}-4q^{8}+4q^{7}-3q^{6}+2q^{5}-q^{4}+q^{3}$ }

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:

(12, 41)

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 $48$ $328$ $1152$ $2904$ $464$ $15744$ ${\frac {85168}{3}}$ ${\frac {15136}{3}}$ $3944$ $18432$ $53792$ $139392$ $22272$ ${\frac {1408222}{5}}$ ${\frac {76816}{15}}$ ${\frac {569456}{5}}$ $1942$ ${\frac {75822}{5}}$

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=$6 is the signature of K11a358. 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 29                       1 -1 27                         0 25                   2 1   -1 23                 1       1 21               3 2       -1 19             2 1         1 17           2 3           1 15         2 2             0 13       1 2               1 11     1 2                 -1 9     1                   1 7 1 1                     0 5 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a357 K11a359