K11a59

K11a58

K11a60

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a59 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X16,6,17,5 X2837 X22,9,1,10 X20,11,21,12 X18,13,19,14 X6,16,7,15 X14,17,15,18 X12,19,13,20 X10,21,11,22
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -9, 8, -3, 9, -7, 10, -6, 11, -5
Dowker-Thistlethwaite code	4 8 16 2 22 20 18 6 14 12 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a59/ThurstonBennequinNumber Hyperbolic Volume 8.56716 A-Polynomial See Data:K11a59/A-polynomial

[edit Notes for K11a59's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $2$ Rasmussen s-Invariant -2

[edit Notes for K11a59's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{2}+11t-15+11t^{-1}-3t^{-2}$
Conway polynomial	$-3z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 43, 2 }
Jones polynomial	$-q^{6}+2q^{5}-3q^{4}+5q^{3}-5q^{2}+6q-6+5q^{-1}-4q^{-2}+3q^{-3}-2q^{-4}+q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{4}+a^{4}-z^{4}a^{2}-2z^{2}a^{2}-a^{2}-z^{4}-z^{2}-z^{4}a^{-2}-z^{2}a^{-2}+2z^{2}a^{-4}+2a^{-4}-a^{-6}$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+2a^{3}z^{9}+4az^{9}+2z^{9}a^{-1}+a^{4}z^{8}-3a^{2}z^{8}+2z^{8}a^{-2}-2z^{8}-12a^{3}z^{7}-21az^{7}-7z^{7}a^{-1}+2z^{7}a^{-3}-6a^{4}z^{6}-5a^{2}z^{6}-4z^{6}a^{-2}+2z^{6}a^{-4}-5z^{6}+22a^{3}z^{5}+33az^{5}+7z^{5}a^{-1}-2z^{5}a^{-3}+2z^{5}a^{-5}+11a^{4}z^{4}+18a^{2}z^{4}+2z^{4}a^{-6}+9z^{4}-14a^{3}z^{3}-19az^{3}-5z^{3}a^{-1}-z^{3}a^{-3}+z^{3}a^{-7}-7a^{4}z^{2}-11a^{2}z^{2}-3z^{2}a^{-4}-2z^{2}a^{-6}-3z^{2}+3a^{3}z+4az+za^{-1}-za^{-5}-za^{-7}+a^{4}+a^{2}+2a^{-4}+a^{-6}$
The A2 invariant	Data:K11a59/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a59/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a59 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["K11a59"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3t^{2}+11t-15+11t^{-1}-3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-3z^{4}-z^{2}+1$

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]=

{ 43, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{10}+z^{10}+2a^{3}z^{9}+4az^{9}+2z^{9}a^{-1}+a^{4}z^{8}-3a^{2}z^{8}+2z^{8}a^{-2}-2z^{8}-12a^{3}z^{7}-21az^{7}-7z^{7}a^{-1}+2z^{7}a^{-3}-6a^{4}z^{6}-5a^{2}z^{6}-4z^{6}a^{-2}+2z^{6}a^{-4}-5z^{6}+22a^{3}z^{5}+33az^{5}+7z^{5}a^{-1}-2z^{5}a^{-3}+2z^{5}a^{-5}+11a^{4}z^{4}+18a^{2}z^{4}+2z^{4}a^{-6}+9z^{4}-14a^{3}z^{3}-19az^{3}-5z^{3}a^{-1}-z^{3}a^{-3}+z^{3}a^{-7}-7a^{4}z^{2}-11a^{2}z^{2}-3z^{2}a^{-4}-2z^{2}a^{-6}-3z^{2}+3a^{3}z+4az+za^{-1}-za^{-5}-za^{-7}+a^{4}+a^{2}+2a^{-4}+a^{-6}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_7, K11n3,}

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["K11a59"];

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]=

{ $-3t^{2}+11t-15+11t^{-1}-3t^{-2}$, $-q^{6}+2q^{5}-3q^{4}+5q^{3}-5q^{2}+6q-6+5q^{-1}-4q^{-2}+3q^{-3}-2q^{-4}+q^{-5}$ }

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_7, K11n3,}

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, 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 $-4$ $24$ $8$ ${\frac {178}{3}}$ ${\frac {86}{3}}$ $-96$ $-80$ $-64$ $24$ $-{\frac {32}{3}}$ $288$ $-{\frac {712}{3}}$ $-{\frac {344}{3}}$ ${\frac {9569}{30}}$ ${\frac {1222}{15}}$ $-{\frac {2822}{45}}$ ${\frac {1375}{18}}$ $-{\frac {1471}{30}}$

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=$2 is the signature of K11a59. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a58 K11a60