K11n69

K11n68

K11n70

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n69 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,15,6,14 X2837 X9,19,10,18 X11,17,12,16 X13,20,14,21 X15,7,16,6 X17,11,18,10 X19,1,20,22 X21,12,22,13
Gauss code	1, -4, 2, -1, -3, 8, 4, -2, -5, 9, -6, 11, -7, 3, -8, 6, -9, 5, -10, 7, -11, 10
Dowker-Thistlethwaite code	4 8 -14 2 -18 -16 -20 -6 -10 -22 -12

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:K11n69/ThurstonBennequinNumber Hyperbolic Volume 11.956 A-Polynomial See Data:K11n69/A-polynomial

[edit Notes for K11n69's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $[2,3]$ Rasmussen s-Invariant -4

[edit Notes for K11n69's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+7t^{2}-9t+9-9t^{-1}+7t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-5z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 45, 4 }
Jones polynomial	$-q^{9}+3q^{8}-5q^{7}+7q^{6}-8q^{5}+7q^{4}-6q^{3}+5q^{2}-2q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-3z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}a^{-8}+3z^{2}a^{-2}-z^{2}a^{-4}-5z^{2}a^{-6}+4z^{2}a^{-8}+2a^{-2}-3a^{-6}+3a^{-8}-a^{-10}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-5}+z^{9}a^{-7}+2z^{8}a^{-4}+4z^{8}a^{-6}+2z^{8}a^{-8}+2z^{7}a^{-3}-z^{7}a^{-7}+z^{7}a^{-9}+z^{6}a^{-2}-4z^{6}a^{-4}-12z^{6}a^{-6}-7z^{6}a^{-8}-6z^{5}a^{-3}-4z^{5}a^{-5}+z^{5}a^{-7}-z^{5}a^{-9}-4z^{4}a^{-2}-2z^{4}a^{-4}+15z^{4}a^{-6}+16z^{4}a^{-8}+3z^{4}a^{-10}+3z^{3}a^{-3}-z^{3}a^{-5}-z^{3}a^{-7}+4z^{3}a^{-9}+z^{3}a^{-11}+5z^{2}a^{-2}+2z^{2}a^{-4}-12z^{2}a^{-6}-12z^{2}a^{-8}-3z^{2}a^{-10}+za^{-3}+2za^{-5}+za^{-7}-za^{-9}-za^{-11}-2a^{-2}+3a^{-6}+3a^{-8}+a^{-10}$
The A2 invariant	Data:K11n69/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n69/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n69 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["K11n69"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+7t^{2}-9t+9-9t^{-1}+7t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-2z^{6}-5z^{4}+z^{2}+1$

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]=

{ 45, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{9}+3q^{8}-5q^{7}+7q^{6}-8q^{5}+7q^{4}-6q^{3}+5q^{2}-2q+1$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-5}+z^{9}a^{-7}+2z^{8}a^{-4}+4z^{8}a^{-6}+2z^{8}a^{-8}+2z^{7}a^{-3}-z^{7}a^{-7}+z^{7}a^{-9}+z^{6}a^{-2}-4z^{6}a^{-4}-12z^{6}a^{-6}-7z^{6}a^{-8}-6z^{5}a^{-3}-4z^{5}a^{-5}+z^{5}a^{-7}-z^{5}a^{-9}-4z^{4}a^{-2}-2z^{4}a^{-4}+15z^{4}a^{-6}+16z^{4}a^{-8}+3z^{4}a^{-10}+3z^{3}a^{-3}-z^{3}a^{-5}-z^{3}a^{-7}+4z^{3}a^{-9}+z^{3}a^{-11}+5z^{2}a^{-2}+2z^{2}a^{-4}-12z^{2}a^{-6}-12z^{2}a^{-8}-3z^{2}a^{-10}+za^{-3}+2za^{-5}+za^{-7}-za^{-9}-za^{-11}-2a^{-2}+3a^{-6}+3a^{-8}+a^{-10}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_21,}

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["K11n69"];

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]=

{ $-2t^{3}+7t^{2}-9t+9-9t^{-1}+7t^{-2}-2t^{-3}$, $-q^{9}+3q^{8}-5q^{7}+7q^{6}-8q^{5}+7q^{4}-6q^{3}+5q^{2}-2q+1$ }

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]=

{10_21,}

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, 2)

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 $4$ $16$ $8$ ${\frac {350}{3}}$ ${\frac {130}{3}}$ $64$ ${\frac {1792}{3}}$ ${\frac {448}{3}}$ $176$ ${\frac {32}{3}}$ $128$ ${\frac {1400}{3}}$ ${\frac {520}{3}}$ ${\frac {72271}{30}}$ $-{\frac {954}{5}}$ ${\frac {71582}{45}}$ ${\frac {1937}{18}}$ ${\frac {5551}{30}}$

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=$4 is the signature of K11n69. 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 χ 19                   1 -1 17                 2   2 15               3 1   -2 13             4 2     2 11           4 3       -1 9         3 4         -1 7       3 4           1 5     2 3             -1 3   1 4               3 1   1                 -1 -1 1                   1

Integral Khovanov Homology (db, data source)

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

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.

K11n68 K11n70