K11n67

K11n66

K11n68

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n67 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X14,5,15,6 X2837 X9,19,10,18 X11,17,12,16 X13,20,14,21 X6,15,7,16 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	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n67/ThurstonBennequinNumber Hyperbolic Volume 10.6503 A-Polynomial See Data:K11n67/A-polynomial

[edit Notes for K11n67's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n67's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t+5-2t^{-1}$
Conway polynomial	$1-2z^{2}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 9, 0 }
Jones polynomial	$-q^{7}+2q^{6}-2q^{5}+3q^{4}-2q^{3}+q^{2}-q+q^{-1}-q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-4}-z^{4}+a^{2}z^{2}-z^{2}a^{-2}+3z^{2}a^{-4}-z^{2}a^{-6}-4z^{2}+2a^{2}-a^{-2}+3a^{-4}-a^{-6}-2$
Kauffman polynomial (db, data sources)	$z^{9}a^{-3}+z^{9}a^{-5}+z^{8}a^{-2}+3z^{8}a^{-4}+2z^{8}a^{-6}+az^{7}-6z^{7}a^{-3}-4z^{7}a^{-5}+z^{7}a^{-7}+a^{2}z^{6}-7z^{6}a^{-2}-18z^{6}a^{-4}-11z^{6}a^{-6}+z^{6}-5az^{5}-z^{5}a^{-1}+10z^{5}a^{-3}+z^{5}a^{-5}-5z^{5}a^{-7}-5a^{2}z^{4}+12z^{4}a^{-2}+31z^{4}a^{-4}+17z^{4}a^{-6}-7z^{4}+5az^{3}-z^{3}a^{-1}-9z^{3}a^{-3}+3z^{3}a^{-5}+6z^{3}a^{-7}+6a^{2}z^{2}-8z^{2}a^{-2}-20z^{2}a^{-4}-9z^{2}a^{-6}+9z^{2}-az+2za^{-1}+4za^{-3}-za^{-7}-2a^{2}+a^{-2}+3a^{-4}+a^{-6}-2$
The A2 invariant	Data:K11n67/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n67/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n67 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["K11n67"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$1-2z^{2}$

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]=

{ 9, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-3}+z^{9}a^{-5}+z^{8}a^{-2}+3z^{8}a^{-4}+2z^{8}a^{-6}+az^{7}-6z^{7}a^{-3}-4z^{7}a^{-5}+z^{7}a^{-7}+a^{2}z^{6}-7z^{6}a^{-2}-18z^{6}a^{-4}-11z^{6}a^{-6}+z^{6}-5az^{5}-z^{5}a^{-1}+10z^{5}a^{-3}+z^{5}a^{-5}-5z^{5}a^{-7}-5a^{2}z^{4}+12z^{4}a^{-2}+31z^{4}a^{-4}+17z^{4}a^{-6}-7z^{4}+5az^{3}-z^{3}a^{-1}-9z^{3}a^{-3}+3z^{3}a^{-5}+6z^{3}a^{-7}+6a^{2}z^{2}-8z^{2}a^{-2}-20z^{2}a^{-4}-9z^{2}a^{-6}+9z^{2}-az+2za^{-1}+4za^{-3}-za^{-7}-2a^{2}+a^{-2}+3a^{-4}+a^{-6}-2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {6_1, 9_46, K11n97, K11n139,}

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["K11n67"];

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]=

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

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]=

{6_1, 9_46, K11n97, K11n139,}

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:

(-2, 1)

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 $-8$ $8$ $32$ ${\frac {212}{3}}$ ${\frac {52}{3}}$ $-64$ $-{\frac {304}{3}}$ $-{\frac {160}{3}}$ $-24$ $-{\frac {256}{3}}$ $32$ $-{\frac {1696}{3}}$ $-{\frac {416}{3}}$ $-{\frac {9751}{15}}$ $-{\frac {372}{5}}$ $-{\frac {15844}{45}}$ ${\frac {343}{9}}$ $-{\frac {871}{15}}$

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 K11n67. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n66 K11n68