K11n22

K11n21

K11n23

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n22 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,13,6,12 X2837 X14,9,15,10 X18,11,19,12 X13,7,14,6 X20,16,21,15 X10,17,11,18 X22,20,1,19 X16,22,17,21
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
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n22/ThurstonBennequinNumber Hyperbolic Volume 13.189 A-Polynomial See Data:K11n22/A-polynomial

[edit Notes for K11n22's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -2

[edit Notes for K11n22's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-5t^{2}+13t-17+13t^{-1}-5t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 55, 2 }
Jones polynomial	$q^{9}-3q^{8}+5q^{7}-8q^{6}+9q^{5}-9q^{4}+9q^{3}-6q^{2}+4q-1$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-4}-z^{4}a^{-2}+4z^{4}a^{-4}-2z^{4}a^{-6}-z^{2}a^{-2}+7z^{2}a^{-4}-5z^{2}a^{-6}+z^{2}a^{-8}+4a^{-4}-4a^{-6}+a^{-8}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-5}+z^{9}a^{-7}+3z^{8}a^{-4}+6z^{8}a^{-6}+3z^{8}a^{-8}+2z^{7}a^{-3}+4z^{7}a^{-5}+5z^{7}a^{-7}+3z^{7}a^{-9}-9z^{6}a^{-4}-17z^{6}a^{-6}-7z^{6}a^{-8}+z^{6}a^{-10}-4z^{5}a^{-3}-18z^{5}a^{-5}-24z^{5}a^{-7}-10z^{5}a^{-9}+4z^{4}a^{-2}+17z^{4}a^{-4}+17z^{4}a^{-6}+z^{4}a^{-8}-3z^{4}a^{-10}+z^{3}a^{-1}+9z^{3}a^{-3}+24z^{3}a^{-5}+25z^{3}a^{-7}+9z^{3}a^{-9}-3z^{2}a^{-2}-12z^{2}a^{-4}-11z^{2}a^{-6}+2z^{2}a^{-10}-za^{-1}-4za^{-3}-10za^{-5}-10za^{-7}-3za^{-9}+4a^{-4}+4a^{-6}+a^{-8}$
The A2 invariant	$-1+2q^{-2}-q^{-4}+q^{-6}+3q^{-8}+3q^{-12}-q^{-14}-q^{-18}-3q^{-20}+q^{-22}-q^{-24}+q^{-28}$
The G2 invariant	Data:K11n22/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n22 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["K11n22"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-5t^{2}+13t-17+13t^{-1}-5t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 55, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{9}-3q^{8}+5q^{7}-8q^{6}+9q^{5}-9q^{4}+9q^{3}-6q^{2}+4q-1$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-5}+z^{9}a^{-7}+3z^{8}a^{-4}+6z^{8}a^{-6}+3z^{8}a^{-8}+2z^{7}a^{-3}+4z^{7}a^{-5}+5z^{7}a^{-7}+3z^{7}a^{-9}-9z^{6}a^{-4}-17z^{6}a^{-6}-7z^{6}a^{-8}+z^{6}a^{-10}-4z^{5}a^{-3}-18z^{5}a^{-5}-24z^{5}a^{-7}-10z^{5}a^{-9}+4z^{4}a^{-2}+17z^{4}a^{-4}+17z^{4}a^{-6}+z^{4}a^{-8}-3z^{4}a^{-10}+z^{3}a^{-1}+9z^{3}a^{-3}+24z^{3}a^{-5}+25z^{3}a^{-7}+9z^{3}a^{-9}-3z^{2}a^{-2}-12z^{2}a^{-4}-11z^{2}a^{-6}+2z^{2}a^{-10}-za^{-1}-4za^{-3}-10za^{-5}-10za^{-7}-3za^{-9}+4a^{-4}+4a^{-6}+a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_31, K11n11, K11n112, K11n127,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n127,}

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["K11n22"];

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]=

{ $t^{3}-5t^{2}+13t-17+13t^{-1}-5t^{-2}+t^{-3}$, $q^{9}-3q^{8}+5q^{7}-8q^{6}+9q^{5}-9q^{4}+9q^{3}-6q^{2}+4q-1$ }

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]=

{9_31, K11n11, K11n112, K11n127,}

In[6]:=

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

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

Out[6]=

{K11n127,}

Vassiliev invariants

V2 and V3:

(2, 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 $8$ $16$ $32$ ${\frac {124}{3}}$ ${\frac {20}{3}}$ $128$ ${\frac {448}{3}}$ ${\frac {64}{3}}$ $48$ ${\frac {256}{3}}$ $128$ ${\frac {992}{3}}$ ${\frac {160}{3}}$ ${\frac {7471}{15}}$ $-{\frac {2044}{15}}$ ${\frac {19084}{45}}$ ${\frac {161}{9}}$ ${\frac {991}{15}}$

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=$2 is the signature of K11n22. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -1 0 1 2 3 4 5 6 7 8 χ 19                   1 1 17                 2   -2 15               3 1   2 13             5 2     -3 11           4 3       1 9         5 5         0 7       4 4           0 5     2 5             3 3   2 4               -2 1   3                 3 -1 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-1$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11n21 K11n23