K11n64

K11n63

K11n65

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n64 at Knotilus!

Knot presentations

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

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

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n64/ThurstonBennequinNumber Hyperbolic Volume 7.23965 A-Polynomial See Data:K11n64/A-polynomial

[edit Notes for K11n64's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n64's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+5t^{2}-3t+1-3t^{-1}+5t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-7z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 21, 4 }
Jones polynomial	$-q^{7}+2q^{6}-2q^{5}+3q^{4}-3q^{3}+3q^{2}-3q+2-q^{-1}+q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-z^{6}a^{-4}-5z^{4}a^{-2}-4z^{4}a^{-4}+z^{4}a^{-6}+z^{4}-7z^{2}a^{-2}-2z^{2}a^{-4}+4z^{2}a^{-6}+4z^{2}-4a^{-2}+a^{-4}+2a^{-6}-a^{-8}+3$
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+3z^{8}a^{-2}+2z^{8}a^{-4}+z^{8}-6z^{7}a^{-1}-4z^{7}a^{-3}+2z^{7}a^{-5}-18z^{6}a^{-2}-9z^{6}a^{-4}+2z^{6}a^{-6}-7z^{6}+11z^{5}a^{-1}+3z^{5}a^{-3}-7z^{5}a^{-5}+z^{5}a^{-7}+34z^{4}a^{-2}+11z^{4}a^{-4}-7z^{4}a^{-6}+16z^{4}-7z^{3}a^{-1}-z^{3}a^{-3}+4z^{3}a^{-5}-2z^{3}a^{-7}-23z^{2}a^{-2}-6z^{2}a^{-4}+6z^{2}a^{-6}+2z^{2}a^{-8}-13z^{2}+2za^{-1}+2za^{-3}+za^{-7}+za^{-9}+4a^{-2}+a^{-4}-2a^{-6}-a^{-8}+3$
The A2 invariant	Data:K11n64/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n64/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n64 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["K11n64"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$-2z^{6}-7z^{4}-z^{2}+1$

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]=

{ 21, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-1}+z^{9}a^{-3}+3z^{8}a^{-2}+2z^{8}a^{-4}+z^{8}-6z^{7}a^{-1}-4z^{7}a^{-3}+2z^{7}a^{-5}-18z^{6}a^{-2}-9z^{6}a^{-4}+2z^{6}a^{-6}-7z^{6}+11z^{5}a^{-1}+3z^{5}a^{-3}-7z^{5}a^{-5}+z^{5}a^{-7}+34z^{4}a^{-2}+11z^{4}a^{-4}-7z^{4}a^{-6}+16z^{4}-7z^{3}a^{-1}-z^{3}a^{-3}+4z^{3}a^{-5}-2z^{3}a^{-7}-23z^{2}a^{-2}-6z^{2}a^{-4}+6z^{2}a^{-6}+2z^{2}a^{-8}-13z^{2}+2za^{-1}+2za^{-3}+za^{-7}+za^{-9}+4a^{-2}+a^{-4}-2a^{-6}-a^{-8}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n28,}

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["K11n64"];

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]=

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

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]=

{K11n28,}

Vassiliev invariants

V2 and V3:

(-1, 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 $-4$ $8$ $8$ ${\frac {370}{3}}$ ${\frac {182}{3}}$ $-32$ ${\frac {848}{3}}$ ${\frac {320}{3}}$ $72$ $-{\frac {32}{3}}$ $32$ $-{\frac {1480}{3}}$ $-{\frac {728}{3}}$ ${\frac {929}{30}}$ ${\frac {5542}{15}}$ $-{\frac {23462}{45}}$ ${\frac {2911}{18}}$ $-{\frac {5311}{30}}$

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 K11n64. 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 χ 15                   1 -1 13                 1   1 11               1 1   0 9             2 1     1 7           1 1       0 5         2 2         0 3       2 2           0 1       1             -1 -1   1 2               1 -3                     0 -5 1                   1

Integral Khovanov Homology (db, data source)

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

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.

K11n63 K11n65