K11n32

K11n31

K11n33

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n32 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,12,6,13 X2837 X16,9,17,10 X11,6,12,7 X20,14,21,13 X10,15,11,16 X22,17,1,18 X14,20,15,19 X18,21,19,22
Gauss code	1, -4, 2, -1, -3, 6, 4, -2, 5, -8, -6, 3, 7, -10, 8, -5, 9, -11, 10, -7, 11, -9
Dowker-Thistlethwaite code	4 8 -12 2 16 -6 20 10 22 14 18

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

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n32/ThurstonBennequinNumber Hyperbolic Volume 13.939 A-Polynomial See Data:K11n32/A-polynomial

[edit Notes for K11n32's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for K11n32's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+6t^{2}-16t+23-16t^{-1}+6t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 69, 0 }
Jones polynomial	$-q^{5}+3q^{4}-6q^{3}+10q^{2}-11q+12-11q^{-1}+8q^{-2}-5q^{-3}+2q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+a^{2}z^{4}+2z^{4}a^{-2}-3z^{4}+4z^{2}a^{-2}-z^{2}a^{-4}-4z^{2}+a^{4}-a^{2}+3a^{-2}-a^{-4}-1$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+3z^{8}a^{-2}+5z^{8}+a^{3}z^{7}+3az^{7}+6z^{7}a^{-1}+4z^{7}a^{-3}-a^{2}z^{6}+z^{6}a^{-2}+3z^{6}a^{-4}-3z^{6}+3a^{3}z^{5}-9z^{5}a^{-1}-5z^{5}a^{-3}+z^{5}a^{-5}+3a^{4}z^{4}+4a^{2}z^{4}-10z^{4}a^{-2}-6z^{4}a^{-4}-3z^{4}-6a^{3}z^{3}-8az^{3}-z^{3}a^{-1}-z^{3}a^{-3}-2z^{3}a^{-5}-4a^{4}z^{2}-6a^{2}z^{2}+8z^{2}a^{-2}+4z^{2}a^{-4}+2z^{2}+3a^{3}z+5az+3za^{-1}+2za^{-3}+za^{-5}+a^{4}+a^{2}-3a^{-2}-a^{-4}-1$
The A2 invariant	Data:K11n32/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n32/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n32 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["K11n32"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+6t^{2}-16t+23-16t^{-1}+6t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 69, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+3q^{4}-6q^{3}+10q^{2}-11q+12-11q^{-1}+8q^{-2}-5q^{-3}+2q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}+a^{2}z^{4}+2z^{4}a^{-2}-3z^{4}+4z^{2}a^{-2}-z^{2}a^{-4}-4z^{2}+a^{4}-a^{2}+3a^{-2}-a^{-4}-1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+3z^{8}a^{-2}+5z^{8}+a^{3}z^{7}+3az^{7}+6z^{7}a^{-1}+4z^{7}a^{-3}-a^{2}z^{6}+z^{6}a^{-2}+3z^{6}a^{-4}-3z^{6}+3a^{3}z^{5}-9z^{5}a^{-1}-5z^{5}a^{-3}+z^{5}a^{-5}+3a^{4}z^{4}+4a^{2}z^{4}-10z^{4}a^{-2}-6z^{4}a^{-4}-3z^{4}-6a^{3}z^{3}-8az^{3}-z^{3}a^{-1}-z^{3}a^{-3}-2z^{3}a^{-5}-4a^{4}z^{2}-6a^{2}z^{2}+8z^{2}a^{-2}+4z^{2}a^{-4}+2z^{2}+3a^{3}z+5az+3za^{-1}+2za^{-3}+za^{-5}+a^{4}+a^{2}-3a^{-2}-a^{-4}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_34, K11n119,}

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["K11n32"];

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]=

{ $-t^{3}+6t^{2}-16t+23-16t^{-1}+6t^{-2}-t^{-3}$, $-q^{5}+3q^{4}-6q^{3}+10q^{2}-11q+12-11q^{-1}+8q^{-2}-5q^{-3}+2q^{-4}$ }

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]=

{9_34, K11n119,}

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, 2)

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 $-4$ $16$ $8$ ${\frac {34}{3}}$ ${\frac {14}{3}}$ $-64$ $-{\frac {320}{3}}$ $-{\frac {128}{3}}$ $-16$ $-{\frac {32}{3}}$ $128$ $-{\frac {136}{3}}$ $-{\frac {56}{3}}$ ${\frac {5489}{30}}$ ${\frac {862}{15}}$ $-{\frac {302}{45}}$ ${\frac {367}{18}}$ $-{\frac {751}{30}}$

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=$0 is the signature of K11n32. 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 11                   1 -1 9                 2   2 7               4 1   -3 5             6 2     4 3           5 4       -1 1         7 6         1 -1       5 6           1 -3     3 6             -3 -5   2 5               3 -7   3                 -3 -9 2                   2

Integral Khovanov Homology (db, data source)

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

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.

K11n31 K11n33