K11a13

K11a12

K11a14

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a13 at Knotilus!

Knot presentations

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

A Braid Representative	{{{braid_table}}}
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:K11a13/ThurstonBennequinNumber Hyperbolic Volume 10.7354 A-Polynomial See Data:K11a13/A-polynomial

[edit Notes for K11a13's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for K11a13's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{2}-15t+25-15t^{-1}+3t^{-2}$
Conway polynomial	$3z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 61, 0 }
Jones polynomial	$-q^{7}+3q^{6}-4q^{5}+6q^{4}-8q^{3}+9q^{2}-9q+8-6q^{-1}+4q^{-2}-2q^{-3}+q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$a^{4}-2z^{2}a^{2}-a^{2}+z^{4}+1+z^{4}a^{-2}-z^{2}a^{-2}-a^{-2}+z^{4}a^{-4}+z^{2}a^{-4}+a^{-4}-z^{2}a^{-6}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+5z^{9}a^{-3}+3z^{9}a^{-5}+z^{8}a^{-4}+3z^{8}a^{-6}+2z^{8}+2az^{7}-5z^{7}a^{-1}-21z^{7}a^{-3}-13z^{7}a^{-5}+z^{7}a^{-7}+2a^{2}z^{6}-8z^{6}a^{-2}-18z^{6}a^{-4}-14z^{6}a^{-6}-2z^{6}+2a^{3}z^{5}+8z^{5}a^{-1}+29z^{5}a^{-3}+15z^{5}a^{-5}-4z^{5}a^{-7}+a^{4}z^{4}+13z^{4}a^{-2}+27z^{4}a^{-4}+17z^{4}a^{-6}+2z^{4}-3a^{3}z^{3}-2az^{3}-8z^{3}a^{-1}-18z^{3}a^{-3}-6z^{3}a^{-5}+3z^{3}a^{-7}-2a^{4}z^{2}-3a^{2}z^{2}-8z^{2}a^{-2}-12z^{2}a^{-4}-5z^{2}a^{-6}-2z^{2}+a^{3}z+az+3za^{-1}+4za^{-3}+za^{-5}+a^{4}+a^{2}+a^{-2}+a^{-4}+1$
The A2 invariant	Data:K11a13/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a13/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a13 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["K11a13"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{2}-15t+25-15t^{-1}+3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 61, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{7}+3q^{6}-4q^{5}+6q^{4}-8q^{3}+9q^{2}-9q+8-6q^{-1}+4q^{-2}-2q^{-3}+q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+5z^{9}a^{-3}+3z^{9}a^{-5}+z^{8}a^{-4}+3z^{8}a^{-6}+2z^{8}+2az^{7}-5z^{7}a^{-1}-21z^{7}a^{-3}-13z^{7}a^{-5}+z^{7}a^{-7}+2a^{2}z^{6}-8z^{6}a^{-2}-18z^{6}a^{-4}-14z^{6}a^{-6}-2z^{6}+2a^{3}z^{5}+8z^{5}a^{-1}+29z^{5}a^{-3}+15z^{5}a^{-5}-4z^{5}a^{-7}+a^{4}z^{4}+13z^{4}a^{-2}+27z^{4}a^{-4}+17z^{4}a^{-6}+2z^{4}-3a^{3}z^{3}-2az^{3}-8z^{3}a^{-1}-18z^{3}a^{-3}-6z^{3}a^{-5}+3z^{3}a^{-7}-2a^{4}z^{2}-3a^{2}z^{2}-8z^{2}a^{-2}-12z^{2}a^{-4}-5z^{2}a^{-6}-2z^{2}+a^{3}z+az+3za^{-1}+4za^{-3}+za^{-5}+a^{4}+a^{2}+a^{-2}+a^{-4}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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["K11a13"];

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]=

{ $3t^{2}-15t+25-15t^{-1}+3t^{-2}$, $-q^{7}+3q^{6}-4q^{5}+6q^{4}-8q^{3}+9q^{2}-9q+8-6q^{-1}+4q^{-2}-2q^{-3}+q^{-4}$ }

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]=

{}

Vassiliev invariants

V2 and V3:

(-3, 0)

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 $-12$ $0$ $72$ $66$ $14$ $0$ $-32$ $-64$ $32$ $-288$ $0$ $-792$ $-168$ $-{\frac {6511}{10}}$ $-{\frac {3974}{15}}$ $-{\frac {782}{15}}$ ${\frac {79}{6}}$ ${\frac {49}{10}}$

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 K11a13. 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                     2   2 11                   2 1   -1 9                 4 2     2 7               4 2       -2 5             5 4         1 3           4 4           0 1         4 5             -1 -1       3 5               2 -3     1 3                 -2 -5   1 3                   2 -7   1                     -1 -9 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a12 K11a14