K11n119

K11n118

K11n120

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n119 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X14,5,15,6 X7,20,8,21 X2,10,3,9 X16,11,17,12 X18,14,19,13 X8,15,9,16 X22,17,1,18 X19,6,20,7 X12,22,13,21
Gauss code	1, -5, 2, -1, 3, 10, -4, -8, 5, -2, 6, -11, 7, -3, 8, -6, 9, -7, -10, 4, 11, -9
Dowker-Thistlethwaite code	4 10 14 -20 2 16 18 8 22 -6 12

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n119/ThurstonBennequinNumber Hyperbolic Volume 14.3682 A-Polynomial See Data:K11n119/A-polynomial

[edit Notes for K11n119'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 K11n119'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	$-2q^{3}+6q^{2}-8q+11-12q^{-1}+11q^{-2}-9q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{6}+a^{4}z^{4}-4a^{2}z^{4}+3z^{4}+2a^{4}z^{2}-8a^{2}z^{2}-2z^{2}a^{-2}+7z^{2}+2a^{4}-5a^{2}-a^{-2}+5$
Kauffman polynomial (db, data sources)	$2a^{3}z^{9}+2az^{9}+4a^{4}z^{8}+9a^{2}z^{8}+5z^{8}+3a^{5}z^{7}+a^{3}z^{7}+2az^{7}+4z^{7}a^{-1}+a^{6}z^{6}-11a^{4}z^{6}-26a^{2}z^{6}+z^{6}a^{-2}-13z^{6}-9a^{5}z^{5}-14a^{3}z^{5}-10az^{5}-5z^{5}a^{-1}-3a^{6}z^{4}+8a^{4}z^{4}+29a^{2}z^{4}+6z^{4}a^{-2}+24z^{4}+7a^{5}z^{3}+12a^{3}z^{3}+8az^{3}+6z^{3}a^{-1}+3z^{3}a^{-3}+2a^{6}z^{2}-4a^{4}z^{2}-19a^{2}z^{2}-7z^{2}a^{-2}-20z^{2}-2a^{5}z-3a^{3}z-2az-2za^{-1}-za^{-3}+2a^{4}+5a^{2}+a^{-2}+5$
The A2 invariant	$q^{18}-q^{16}+2q^{14}+q^{12}-3q^{10}+q^{8}-3q^{6}+q^{2}+4q^{-2}-q^{-4}+2q^{-6}+q^{-8}-2q^{-10}$
The G2 invariant	Data:K11n119/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n119 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["K11n119"];

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]=

$-2q^{3}+6q^{2}-8q+11-12q^{-1}+11q^{-2}-9q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{6}+a^{4}z^{4}-4a^{2}z^{4}+3z^{4}+2a^{4}z^{2}-8a^{2}z^{2}-2z^{2}a^{-2}+7z^{2}+2a^{4}-5a^{2}-a^{-2}+5$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{3}z^{9}+2az^{9}+4a^{4}z^{8}+9a^{2}z^{8}+5z^{8}+3a^{5}z^{7}+a^{3}z^{7}+2az^{7}+4z^{7}a^{-1}+a^{6}z^{6}-11a^{4}z^{6}-26a^{2}z^{6}+z^{6}a^{-2}-13z^{6}-9a^{5}z^{5}-14a^{3}z^{5}-10az^{5}-5z^{5}a^{-1}-3a^{6}z^{4}+8a^{4}z^{4}+29a^{2}z^{4}+6z^{4}a^{-2}+24z^{4}+7a^{5}z^{3}+12a^{3}z^{3}+8az^{3}+6z^{3}a^{-1}+3z^{3}a^{-3}+2a^{6}z^{2}-4a^{4}z^{2}-19a^{2}z^{2}-7z^{2}a^{-2}-20z^{2}-2a^{5}z-3a^{3}z-2az-2za^{-1}-za^{-3}+2a^{4}+5a^{2}+a^{-2}+5$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_34, K11n32,}

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["K11n119"];

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]=

{ $-t^{3}+6t^{2}-16t+23-16t^{-1}+6t^{-2}-t^{-3}$, $-2q^{3}+6q^{2}-8q+11-12q^{-1}+11q^{-2}-9q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$ }

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, K11n32,}

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 $-4$ $16$ $8$ $-{\frac {14}{3}}$ ${\frac {14}{3}}$ $-64$ $-{\frac {416}{3}}$ $-{\frac {128}{3}}$ $-48$ $-{\frac {32}{3}}$ $128$ ${\frac {56}{3}}$ $-{\frac {56}{3}}$ ${\frac {11009}{30}}$ $-{\frac {418}{15}}$ ${\frac {7618}{45}}$ ${\frac {1087}{18}}$ ${\frac {449}{30}}$

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 K11n119. 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 χ 7                   2 -2 5                 4   4 3               4 2   -2 1             7 4     3 -1           6 5       -1 -3         5 6         -1 -5       4 6           2 -7     2 5             -3 -9   1 4               3 -11   2                 -2 -13 1                   1

Integral Khovanov Homology (db, data source)

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

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.

K11n118 K11n120