K11a69

K11a68

K11a70

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a69 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X16,5,17,6 X14,8,15,7 X2,9,3,10 X18,12,19,11 X20,14,21,13 X22,15,1,16 X12,18,13,17 X10,20,11,19 X6,21,7,22
Gauss code	1, -5, 2, -1, 3, -11, 4, -2, 5, -10, 6, -9, 7, -4, 8, -3, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code	4 8 16 14 2 18 20 22 12 10 6

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

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a69/ThurstonBennequinNumber Hyperbolic Volume 16.7403 A-Polynomial See Data:K11a69/A-polynomial

[edit Notes for K11a69's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for K11a69's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+13t^{2}-33t+45-33t^{-1}+13t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}+z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 141, 0 }
Jones polynomial	$q^{6}-4q^{5}+8q^{4}-14q^{3}+20q^{2}-22q+23-20q^{-1}+15q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-z^{6}+2a^{2}z^{4}-2z^{4}a^{-2}+z^{4}a^{-4}-a^{4}z^{2}+a^{2}z^{2}-3z^{2}a^{-2}+z^{2}a^{-4}+3z^{2}-a^{2}-a^{-2}+3$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}+6az^{9}+12z^{9}a^{-1}+6z^{9}a^{-3}+9a^{2}z^{8}+12z^{8}a^{-2}+7z^{8}a^{-4}+14z^{8}+8a^{3}z^{7}+3az^{7}-17z^{7}a^{-1}-8z^{7}a^{-3}+4z^{7}a^{-5}+4a^{4}z^{6}-10a^{2}z^{6}-37z^{6}a^{-2}-18z^{6}a^{-4}+z^{6}a^{-6}-32z^{6}+a^{5}z^{5}-12a^{3}z^{5}-18az^{5}+z^{5}a^{-1}-4z^{5}a^{-3}-10z^{5}a^{-5}-5a^{4}z^{4}+2a^{2}z^{4}+34z^{4}a^{-2}+15z^{4}a^{-4}-2z^{4}a^{-6}+24z^{4}-a^{5}z^{3}+7a^{3}z^{3}+11az^{3}+5z^{3}a^{-1}+8z^{3}a^{-3}+6z^{3}a^{-5}+2a^{4}z^{2}-a^{2}z^{2}-13z^{2}a^{-2}-4z^{2}a^{-4}-12z^{2}-a^{3}z-2az-2za^{-1}-za^{-3}+a^{2}+a^{-2}+3$
The A2 invariant	Data:K11a69/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a69/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a69 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["K11a69"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+13t^{2}-33t+45-33t^{-1}+13t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 141, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-4q^{5}+8q^{4}-14q^{3}+20q^{2}-22q+23-20q^{-1}+15q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}+6az^{9}+12z^{9}a^{-1}+6z^{9}a^{-3}+9a^{2}z^{8}+12z^{8}a^{-2}+7z^{8}a^{-4}+14z^{8}+8a^{3}z^{7}+3az^{7}-17z^{7}a^{-1}-8z^{7}a^{-3}+4z^{7}a^{-5}+4a^{4}z^{6}-10a^{2}z^{6}-37z^{6}a^{-2}-18z^{6}a^{-4}+z^{6}a^{-6}-32z^{6}+a^{5}z^{5}-12a^{3}z^{5}-18az^{5}+z^{5}a^{-1}-4z^{5}a^{-3}-10z^{5}a^{-5}-5a^{4}z^{4}+2a^{2}z^{4}+34z^{4}a^{-2}+15z^{4}a^{-4}-2z^{4}a^{-6}+24z^{4}-a^{5}z^{3}+7a^{3}z^{3}+11az^{3}+5z^{3}a^{-1}+8z^{3}a^{-3}+6z^{3}a^{-5}+2a^{4}z^{2}-a^{2}z^{2}-13z^{2}a^{-2}-4z^{2}a^{-4}-12z^{2}-a^{3}z-2az-2za^{-1}-za^{-3}+a^{2}+a^{-2}+3$

"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["K11a69"];

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]=

{ $-2t^{3}+13t^{2}-33t+45-33t^{-1}+13t^{-2}-2t^{-3}$, $q^{6}-4q^{5}+8q^{4}-14q^{3}+20q^{2}-22q+23-20q^{-1}+15q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$ }

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:

(1, 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 $4$ $0$ $8$ ${\frac {14}{3}}$ $-{\frac {14}{3}}$ $0$ $0$ $64$ $-64$ ${\frac {32}{3}}$ $0$ ${\frac {56}{3}}$ $-{\frac {56}{3}}$ ${\frac {511}{30}}$ ${\frac {1858}{15}}$ $-{\frac {7618}{45}}$ ${\frac {641}{18}}$ $-{\frac {1409}{30}}$

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=$0 is the signature of K11a69. 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                     3   -3 9                   5 1   4 7                 9 3     -6 5               11 5       6 3             11 9         -2 1           12 11           1 -1         9 12             3 -3       6 11               -5 -5     3 9                 6 -7   1 6                   -5 -9   3                     3 -11 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

K11a68 K11a70