K11n106

K11n105

K11n107

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n106 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X5,15,6,14 X7,16,8,17 X2,10,3,9 X20,11,21,12 X22,13,1,14 X15,18,16,19 X17,8,18,9 X19,7,20,6 X12,21,13,22
Gauss code	1, -5, 2, -1, -3, 10, -4, 9, 5, -2, 6, -11, 7, 3, -8, 4, -9, 8, -10, -6, 11, -7
Dowker-Thistlethwaite code	4 10 -14 -16 2 20 22 -18 -8 -6 12

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n106/ThurstonBennequinNumber Hyperbolic Volume 9.99629 A-Polynomial See Data:K11n106/A-polynomial

[edit Notes for K11n106's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $1$ Rasmussen s-Invariant 2

[edit Notes for K11n106's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-3t^{2}+6t-7+6t^{-1}-3t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+3z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 27, -2 }
Jones polynomial	$-q^{4}+2q^{3}-3q^{2}+4q-4+5q^{-1}-3q^{-2}+3q^{-3}-2q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}-a^{2}z^{4}-z^{4}a^{-2}+5z^{4}-2a^{2}z^{2}-3z^{2}a^{-2}+8z^{2}-a^{4}-2a^{-2}+4$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+2z^{8}a^{-2}+4z^{8}+a^{3}z^{7}-2az^{7}-2z^{7}a^{-1}+z^{7}a^{-3}-9a^{2}z^{6}-10z^{6}a^{-2}-19z^{6}-3a^{3}z^{5}-5az^{5}-7z^{5}a^{-1}-5z^{5}a^{-3}+a^{4}z^{4}+13a^{2}z^{4}+14z^{4}a^{-2}+26z^{4}+2a^{3}z^{3}+9az^{3}+14z^{3}a^{-1}+7z^{3}a^{-3}-8a^{2}z^{2}-7z^{2}a^{-2}-15z^{2}+2a^{5}z+a^{3}z-4az-6za^{-1}-3za^{-3}-a^{4}+2a^{-2}+4$
The A2 invariant	$-q^{14}-2q^{12}+2q^{6}+2q^{4}+q^{2}+2+q^{-4}-q^{-8}-q^{-12}$
The G2 invariant	Data:K11n106/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n106 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["K11n106"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-3t^{2}+6t-7+6t^{-1}-3t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 27, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+2z^{8}a^{-2}+4z^{8}+a^{3}z^{7}-2az^{7}-2z^{7}a^{-1}+z^{7}a^{-3}-9a^{2}z^{6}-10z^{6}a^{-2}-19z^{6}-3a^{3}z^{5}-5az^{5}-7z^{5}a^{-1}-5z^{5}a^{-3}+a^{4}z^{4}+13a^{2}z^{4}+14z^{4}a^{-2}+26z^{4}+2a^{3}z^{3}+9az^{3}+14z^{3}a^{-1}+7z^{3}a^{-3}-8a^{2}z^{2}-7z^{2}a^{-2}-15z^{2}+2a^{5}z+a^{3}z-4az-6za^{-1}-3za^{-3}-a^{4}+2a^{-2}+4$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_10, 10_143,}

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["K11n106"];

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]=

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

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]=

{8_10, 10_143,}

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, -1)

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 $12$ $-8$ $72$ $94$ $10$ $-96$ $-{\frac {464}{3}}$ $-{\frac {128}{3}}$ $-8$ $288$ $32$ $1128$ $120$ ${\frac {12591}{10}}$ ${\frac {1534}{15}}$ ${\frac {5942}{15}}$ ${\frac {17}{6}}$ ${\frac {271}{10}}$

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 $s=$-2 is the signature of K11n106. 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 9                 1 -1 7               1   1 5             2 1   -1 3           2 1     1 1         2 2       0 -1       3 2         1 -3     1 3           2 -5   2 2             0 -7   1               1 -9 2                 -2

Integral Khovanov Homology (db, data source)

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

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.

K11n105 K11n107