K11n50

K11n49

K11n51

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n50 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X5,12,6,13 X7,18,8,19 X2,9,3,10 X11,16,12,17 X13,21,14,20 X15,6,16,7 X17,10,18,11 X19,1,20,22 X21,15,22,14
Gauss code	1, -5, 2, -1, -3, 8, -4, -2, 5, 9, -6, 3, -7, 11, -8, 6, -9, 4, -10, 7, -11, 10
Dowker-Thistlethwaite code	4 8 -12 -18 2 -16 -20 -6 -10 -22 -14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n50/ThurstonBennequinNumber Hyperbolic Volume 10.2068 A-Polynomial See Data:K11n50/A-polynomial

[edit Notes for K11n50's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n50's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{2}-6t+9-6t^{-1}+2t^{-2}$
Conway polynomial	$2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 25, 0 }
Jones polynomial	$-q+3-3q^{-1}+4q^{-2}-4q^{-3}+4q^{-4}-3q^{-5}+2q^{-6}-q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{6}-a^{6}+z^{4}a^{4}+2z^{2}a^{4}+a^{4}+z^{4}a^{2}+2z^{2}a^{2}+a^{2}-z^{2}$
Kauffman polynomial (db, data sources)	$a^{5}z^{9}+a^{3}z^{9}+2a^{6}z^{8}+4a^{4}z^{8}+2a^{2}z^{8}+a^{7}z^{7}-2a^{5}z^{7}-2a^{3}z^{7}+az^{7}-10a^{6}z^{6}-19a^{4}z^{6}-9a^{2}z^{6}-5a^{7}z^{5}-7a^{5}z^{5}-6a^{3}z^{5}-4az^{5}+14a^{6}z^{4}+24a^{4}z^{4}+10a^{2}z^{4}+7a^{7}z^{3}+13a^{5}z^{3}+10a^{3}z^{3}+4az^{3}-7a^{6}z^{2}-10a^{4}z^{2}-2a^{2}z^{2}+z^{2}-3a^{7}z-5a^{5}z-3a^{3}z-az+a^{6}+a^{4}-a^{2}$
The A2 invariant	Data:K11n50/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n50/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n50 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["K11n50"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{2}-6t+9-6t^{-1}+2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 25, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{5}z^{9}+a^{3}z^{9}+2a^{6}z^{8}+4a^{4}z^{8}+2a^{2}z^{8}+a^{7}z^{7}-2a^{5}z^{7}-2a^{3}z^{7}+az^{7}-10a^{6}z^{6}-19a^{4}z^{6}-9a^{2}z^{6}-5a^{7}z^{5}-7a^{5}z^{5}-6a^{3}z^{5}-4az^{5}+14a^{6}z^{4}+24a^{4}z^{4}+10a^{2}z^{4}+7a^{7}z^{3}+13a^{5}z^{3}+10a^{3}z^{3}+4az^{3}-7a^{6}z^{2}-10a^{4}z^{2}-2a^{2}z^{2}+z^{2}-3a^{7}z-5a^{5}z-3a^{3}z-az+a^{6}+a^{4}-a^{2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_8, 10_129, K11n39, K11n45, K11n132,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n132, K11n133,}

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["K11n50"];

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]=

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

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]=

{8_8, 10_129, K11n39, K11n45, K11n132,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11n132, K11n133,}

Vassiliev invariants

V2 and V3:

(2, -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 $8$ $-24$ $32$ ${\frac {220}{3}}$ $-{\frac {4}{3}}$ $-192$ $-304$ $-32$ $-24$ ${\frac {256}{3}}$ $288$ ${\frac {1760}{3}}$ $-{\frac {32}{3}}$ ${\frac {18991}{15}}$ ${\frac {2156}{15}}$ ${\frac {10564}{45}}$ ${\frac {257}{9}}$ $-{\frac {209}{15}}$

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 K11n50. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 χ 3                 1 -1 1               2   2 -1             2 2   0 -3           2 1     1 -5         2 2       0 -7       2 2         0 -9     1 2           1 -11   1 2             -1 -13   1               1 -15 1                 -1

Integral Khovanov Homology (db, data source)

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

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.

K11n49 K11n51