K11n23

K11n22

K11n24

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n23 at Knotilus!

Knot presentations

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

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

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{2,3\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n23/ThurstonBennequinNumber Hyperbolic Volume 10.8083 A-Polynomial See Data:K11n23/A-polynomial

[edit Notes for K11n23's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n23's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-3t^{3}+5t^{2}-4t+3-4t^{-1}+5t^{-2}-3t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+5z^{6}+7z^{4}+5z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 29, 4 }
Jones polynomial	$-2q^{7}+3q^{6}-4q^{5}+5q^{4}-4q^{3}+5q^{2}-3q+2-q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-4}-z^{6}a^{-2}+7z^{6}a^{-4}-z^{6}a^{-6}-5z^{4}a^{-2}+18z^{4}a^{-4}-6z^{4}a^{-6}-7z^{2}a^{-2}+22z^{2}a^{-4}-11z^{2}a^{-6}+z^{2}a^{-8}-3a^{-2}+10a^{-4}-7a^{-6}+a^{-8}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+5z^{8}a^{-4}+3z^{8}a^{-6}+z^{7}a^{-1}-z^{7}a^{-3}+2z^{7}a^{-7}-10z^{6}a^{-2}-25z^{6}a^{-4}-15z^{6}a^{-6}-5z^{5}a^{-1}-12z^{5}a^{-3}-16z^{5}a^{-5}-9z^{5}a^{-7}+15z^{4}a^{-2}+39z^{4}a^{-4}+25z^{4}a^{-6}+z^{4}a^{-8}+7z^{3}a^{-1}+21z^{3}a^{-3}+27z^{3}a^{-5}+13z^{3}a^{-7}-10z^{2}a^{-2}-29z^{2}a^{-4}-20z^{2}a^{-6}-z^{2}a^{-8}-3za^{-1}-9za^{-3}-13za^{-5}-6za^{-7}+za^{-9}+3a^{-2}+10a^{-4}+7a^{-6}+a^{-8}$
The A2 invariant	$-q^{2}-q^{-2}+q^{-6}+2q^{-8}+4q^{-10}+q^{-12}+3q^{-14}-q^{-16}-q^{-18}-2q^{-20}-2q^{-22}-q^{-26}+q^{-28}$
The G2 invariant	Data:K11n23/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n23 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["K11n23"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-3t^{3}+5t^{2}-4t+3-4t^{-1}+5t^{-2}-3t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+5z^{6}+7z^{4}+5z^{2}+1$

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]=

{ 29, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-4}-z^{6}a^{-2}+7z^{6}a^{-4}-z^{6}a^{-6}-5z^{4}a^{-2}+18z^{4}a^{-4}-6z^{4}a^{-6}-7z^{2}a^{-2}+22z^{2}a^{-4}-11z^{2}a^{-6}+z^{2}a^{-8}-3a^{-2}+10a^{-4}-7a^{-6}+a^{-8}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+5z^{8}a^{-4}+3z^{8}a^{-6}+z^{7}a^{-1}-z^{7}a^{-3}+2z^{7}a^{-7}-10z^{6}a^{-2}-25z^{6}a^{-4}-15z^{6}a^{-6}-5z^{5}a^{-1}-12z^{5}a^{-3}-16z^{5}a^{-5}-9z^{5}a^{-7}+15z^{4}a^{-2}+39z^{4}a^{-4}+25z^{4}a^{-6}+z^{4}a^{-8}+7z^{3}a^{-1}+21z^{3}a^{-3}+27z^{3}a^{-5}+13z^{3}a^{-7}-10z^{2}a^{-2}-29z^{2}a^{-4}-20z^{2}a^{-6}-z^{2}a^{-8}-3za^{-1}-9za^{-3}-13za^{-5}-6za^{-7}+za^{-9}+3a^{-2}+10a^{-4}+7a^{-6}+a^{-8}$

"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["K11n23"];

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]=

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

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:

(5, 8)

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 $20$ $64$ $200$ ${\frac {1030}{3}}$ ${\frac {122}{3}}$ $1280$ ${\frac {5728}{3}}$ ${\frac {928}{3}}$ $224$ ${\frac {4000}{3}}$ $2048$ ${\frac {20600}{3}}$ ${\frac {2440}{3}}$ ${\frac {67135}{6}}$ ${\frac {1666}{3}}$ ${\frac {33662}{9}}$ ${\frac {1669}{18}}$ ${\frac {2623}{6}}$

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 K11n23. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n22 K11n24