K11n6

K11n5

K11n7

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n6 at Knotilus!

Knot K11n6.

A graph, knot K11n6.

A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X11,18,12,19 X6,14,7,13 X15,20,16,21 X17,12,18,13 X19,22,20,1 X21,16,22,17
Gauss code	1, -5, 2, -1, 3, -7, 4, -2, 5, -3, -6, 9, 7, -4, -8, 11, -9, 6, -10, 8, -11, 10
Dowker-Thistlethwaite code	4 8 10 14 2 -18 6 -20 -12 -22 -16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n6/ThurstonBennequinNumber Hyperbolic Volume 11.407 A-Polynomial See Data:K11n6/A-polynomial

[edit Notes for K11n6'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 K11n6's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+3t^{2}-3t+3-3t^{-1}+3t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-3z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 17, 0 }
Jones polynomial	$q^{3}-q^{2}+1-2q^{-1}+3q^{-2}-3q^{-3}+4q^{-4}-3q^{-5}+2q^{-6}-q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{6}-2a^{6}+2z^{4}a^{4}+7z^{2}a^{4}+5a^{4}-z^{6}a^{2}-5z^{4}a^{2}-7z^{2}a^{2}-3a^{2}+z^{2}a^{-2}+a^{-2}$
Kauffman polynomial (db, data sources)	$a^{5}z^{9}+a^{3}z^{9}+2a^{6}z^{8}+4a^{4}z^{8}+2a^{2}z^{8}+a^{7}z^{7}-2a^{5}z^{7}-3a^{3}z^{7}+az^{7}+z^{7}a^{-1}-10a^{6}z^{6}-21a^{4}z^{6}-12a^{2}z^{6}+z^{6}a^{-2}-5a^{7}z^{5}-8a^{5}z^{5}-5a^{3}z^{5}-8az^{5}-6z^{5}a^{-1}+14a^{6}z^{4}+31a^{4}z^{4}+19a^{2}z^{4}-5z^{4}a^{-2}-3z^{4}+7a^{7}z^{3}+15a^{5}z^{3}+13a^{3}z^{3}+13az^{3}+8z^{3}a^{-1}-8a^{6}z^{2}-19a^{4}z^{2}-12a^{2}z^{2}+5z^{2}a^{-2}+4z^{2}-3a^{7}z-6a^{5}z-6a^{3}z-6az-3za^{-1}+2a^{6}+5a^{4}+3a^{2}-a^{-2}$
The A2 invariant	Data:K11n6/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n6/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n6 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["K11n6"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 17, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{5}z^{9}+a^{3}z^{9}+2a^{6}z^{8}+4a^{4}z^{8}+2a^{2}z^{8}+a^{7}z^{7}-2a^{5}z^{7}-3a^{3}z^{7}+az^{7}+z^{7}a^{-1}-10a^{6}z^{6}-21a^{4}z^{6}-12a^{2}z^{6}+z^{6}a^{-2}-5a^{7}z^{5}-8a^{5}z^{5}-5a^{3}z^{5}-8az^{5}-6z^{5}a^{-1}+14a^{6}z^{4}+31a^{4}z^{4}+19a^{2}z^{4}-5z^{4}a^{-2}-3z^{4}+7a^{7}z^{3}+15a^{5}z^{3}+13a^{3}z^{3}+13az^{3}+8z^{3}a^{-1}-8a^{6}z^{2}-19a^{4}z^{2}-12a^{2}z^{2}+5z^{2}a^{-2}+4z^{2}-3a^{7}z-6a^{5}z-6a^{3}z-6az-3za^{-1}+2a^{6}+5a^{4}+3a^{2}-a^{-2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_2,}

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["K11n6"];

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]=

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

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]=

{8_2,}

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:

(0, -3)

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 $0$ $-24$ $0$ $80$ $24$ $0$ $-112$ $0$ $-24$ $0$ $288$ $0$ $0$ $632$ $-{\frac {40}{3}}$ $200$ $56$ $8$

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 K11n6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                         0 3                 1 1 1   -1 1               2 1       1 -1             2 2 1       -1 -3           2 2 1         1 -5         2 2             0 -7       2 2 1             1 -9     1 2                 1 -11   1 2                   -1 -13   1                     1 -15 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

K11n5 K11n7