K11a64

K11a63

K11a65

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a64 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{2,3\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a64/ThurstonBennequinNumber Hyperbolic Volume 14.0131 A-Polynomial See Data:K11a64/A-polynomial

[edit Notes for K11a64'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 K11a64's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+11t^{2}-22t+27-22t^{-1}+11t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 97, -4 }
Jones polynomial	$1-2q^{-1}+5q^{-2}-9q^{-3}+13q^{-4}-15q^{-5}+16q^{-6}-14q^{-7}+11q^{-8}-7q^{-9}+3q^{-10}-q^{-11}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{10}-a^{10}+2z^{4}a^{8}+3z^{2}a^{8}-z^{6}a^{6}-z^{4}a^{6}+3z^{2}a^{6}+3a^{6}-z^{6}a^{4}-3z^{4}a^{4}-4z^{2}a^{4}-3a^{4}+z^{4}a^{2}+3z^{2}a^{2}+2a^{2}$
Kauffman polynomial (db, data sources)	$z^{5}a^{13}-2z^{3}a^{13}+za^{13}+3z^{6}a^{12}-5z^{4}a^{12}+2z^{2}a^{12}+5z^{7}a^{11}-8z^{5}a^{11}+4z^{3}a^{11}-za^{11}+5z^{8}a^{10}-6z^{6}a^{10}+3z^{4}a^{10}-3z^{2}a^{10}+a^{10}+3z^{9}a^{9}+z^{7}a^{9}-5z^{5}a^{9}+2z^{3}a^{9}+z^{10}a^{8}+5z^{8}a^{8}-7z^{6}a^{8}+2z^{4}a^{8}+5z^{9}a^{7}-7z^{7}a^{7}+7z^{5}a^{7}-7z^{3}a^{7}+2za^{7}+z^{10}a^{6}+2z^{8}a^{6}-11z^{4}a^{6}+11z^{2}a^{6}-3a^{6}+2z^{9}a^{5}-z^{7}a^{5}-3z^{5}a^{5}+2z^{3}a^{5}-za^{5}+2z^{8}a^{4}-z^{6}a^{4}-9z^{4}a^{4}+11z^{2}a^{4}-3a^{4}+2z^{7}a^{3}-6z^{5}a^{3}+5z^{3}a^{3}-za^{3}+z^{6}a^{2}-4z^{4}a^{2}+5z^{2}a^{2}-2a^{2}$
The A2 invariant	Data:K11a64/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a64/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a64 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["K11a64"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+11t^{2}-22t+27-22t^{-1}+11t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 97, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$1-2q^{-1}+5q^{-2}-9q^{-3}+13q^{-4}-15q^{-5}+16q^{-6}-14q^{-7}+11q^{-8}-7q^{-9}+3q^{-10}-q^{-11}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{13}-2z^{3}a^{13}+za^{13}+3z^{6}a^{12}-5z^{4}a^{12}+2z^{2}a^{12}+5z^{7}a^{11}-8z^{5}a^{11}+4z^{3}a^{11}-za^{11}+5z^{8}a^{10}-6z^{6}a^{10}+3z^{4}a^{10}-3z^{2}a^{10}+a^{10}+3z^{9}a^{9}+z^{7}a^{9}-5z^{5}a^{9}+2z^{3}a^{9}+z^{10}a^{8}+5z^{8}a^{8}-7z^{6}a^{8}+2z^{4}a^{8}+5z^{9}a^{7}-7z^{7}a^{7}+7z^{5}a^{7}-7z^{3}a^{7}+2za^{7}+z^{10}a^{6}+2z^{8}a^{6}-11z^{4}a^{6}+11z^{2}a^{6}-3a^{6}+2z^{9}a^{5}-z^{7}a^{5}-3z^{5}a^{5}+2z^{3}a^{5}-za^{5}+2z^{8}a^{4}-z^{6}a^{4}-9z^{4}a^{4}+11z^{2}a^{4}-3a^{4}+2z^{7}a^{3}-6z^{5}a^{3}+5z^{3}a^{3}-za^{3}+z^{6}a^{2}-4z^{4}a^{2}+5z^{2}a^{2}-2a^{2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n174,}

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["K11a64"];

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]=

{ $-2t^{3}+11t^{2}-22t+27-22t^{-1}+11t^{-2}-2t^{-3}$, $1-2q^{-1}+5q^{-2}-9q^{-3}+13q^{-4}-15q^{-5}+16q^{-6}-14q^{-7}+11q^{-8}-7q^{-9}+3q^{-10}-q^{-11}$ }

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]=

{K11n174,}

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:

(4, -11)

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 $16$ $-88$ $128$ ${\frac {1448}{3}}$ ${\frac {208}{3}}$ $-1408$ $-{\frac {8944}{3}}$ $-{\frac {1408}{3}}$ $-440$ ${\frac {2048}{3}}$ $3872$ ${\frac {23168}{3}}$ ${\frac {3328}{3}}$ ${\frac {285542}{15}}$ ${\frac {112}{15}}$ ${\frac {347648}{45}}$ ${\frac {2602}{9}}$ ${\frac {14822}{15}}$

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 $s=$-4 is the signature of K11a64. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                       1 1 -1                     1   -1 -3                   4 1   3 -5                 6 2     -4 -7               7 3       4 -9             8 6         -2 -11           8 7           1 -13         6 8             2 -15       5 8               -3 -17     2 6                 4 -19   1 5                   -4 -21   2                     2 -23 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-5$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a63 K11a65