K11a22

K11a21

K11a23

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a22 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X12,5,13,6 X2837 X18,10,19,9 X14,11,15,12 X6,13,7,14 X20,16,21,15 X22,18,1,17 X10,20,11,19 X16,22,17,21
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -3, 7, -6, 8, -11, 9, -5, 10, -8, 11, -9
Dowker-Thistlethwaite code	4 8 12 2 18 14 6 20 22 10 16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a22/ThurstonBennequinNumber Hyperbolic Volume 14.6533 A-Polynomial See Data:K11a22/A-polynomial

[edit Notes for K11a22's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a22's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+13t^{2}-20t+23-20t^{-1}+13t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+3z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 101, 4 }
Jones polynomial	$q^{10}-3q^{9}+6q^{8}-11q^{7}+14q^{6}-16q^{5}+16q^{4}-13q^{3}+11q^{2}-6q+3-q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-4}-z^{6}a^{-2}+6z^{6}a^{-4}-2z^{6}a^{-6}-4z^{4}a^{-2}+15z^{4}a^{-4}-9z^{4}a^{-6}+z^{4}a^{-8}-5z^{2}a^{-2}+19z^{2}a^{-4}-14z^{2}a^{-6}+3z^{2}a^{-8}-2a^{-2}+9a^{-4}-8a^{-6}+2a^{-8}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-4}+z^{10}a^{-6}+3z^{9}a^{-3}+7z^{9}a^{-5}+4z^{9}a^{-7}+3z^{8}a^{-2}+8z^{8}a^{-4}+12z^{8}a^{-6}+7z^{8}a^{-8}+z^{7}a^{-1}-6z^{7}a^{-3}-11z^{7}a^{-5}+3z^{7}a^{-7}+7z^{7}a^{-9}-12z^{6}a^{-2}-38z^{6}a^{-4}-41z^{6}a^{-6}-10z^{6}a^{-8}+5z^{6}a^{-10}-4z^{5}a^{-1}-7z^{5}a^{-3}-17z^{5}a^{-5}-26z^{5}a^{-7}-9z^{5}a^{-9}+3z^{5}a^{-11}+16z^{4}a^{-2}+48z^{4}a^{-4}+43z^{4}a^{-6}+6z^{4}a^{-8}-4z^{4}a^{-10}+z^{4}a^{-12}+5z^{3}a^{-1}+17z^{3}a^{-3}+32z^{3}a^{-5}+29z^{3}a^{-7}+6z^{3}a^{-9}-3z^{3}a^{-11}-9z^{2}a^{-2}-29z^{2}a^{-4}-25z^{2}a^{-6}-3z^{2}a^{-8}+z^{2}a^{-10}-z^{2}a^{-12}-2za^{-1}-7za^{-3}-14za^{-5}-11za^{-7}-za^{-9}+za^{-11}+2a^{-2}+9a^{-4}+8a^{-6}+2a^{-8}$
The A2 invariant	$-q^{2}+1-2q^{-2}+q^{-4}+2q^{-6}+6q^{-10}-q^{-12}+3q^{-14}-q^{-16}-3q^{-18}-4q^{-22}+q^{-24}+q^{-30}$
The G2 invariant	$q^{12}-2q^{10}+6q^{8}-11q^{6}+14q^{4}-16q^{2}+5+17q^{-2}-49q^{-4}+82q^{-6}-97q^{-8}+72q^{-10}-7q^{-12}-91q^{-14}+185q^{-16}-233q^{-18}+204q^{-20}-95q^{-22}-72q^{-24}+227q^{-26}-309q^{-28}+285q^{-30}-147q^{-32}-33q^{-34}+193q^{-36}-259q^{-38}+215q^{-40}-72q^{-42}-87q^{-44}+211q^{-46}-220q^{-48}+128q^{-50}+45q^{-52}-215q^{-54}+321q^{-56}-304q^{-58}+173q^{-60}+35q^{-62}-247q^{-64}+384q^{-66}-395q^{-68}+272q^{-70}-61q^{-72}-166q^{-74}+309q^{-76}-336q^{-78}+227q^{-80}-51q^{-82}-122q^{-84}+211q^{-86}-194q^{-88}+76q^{-90}+69q^{-92}-183q^{-94}+208q^{-96}-141q^{-98}+7q^{-100}+127q^{-102}-216q^{-104}+231q^{-106}-168q^{-108}+63q^{-110}+49q^{-112}-134q^{-114}+170q^{-116}-158q^{-118}+115q^{-120}-46q^{-122}-13q^{-124}+59q^{-126}-83q^{-128}+81q^{-130}-64q^{-132}+40q^{-134}-11q^{-136}-11q^{-138}+25q^{-140}-30q^{-142}+26q^{-144}-18q^{-146}+10q^{-148}-2q^{-150}-4q^{-152}+5q^{-154}-6q^{-156}+4q^{-158}-2q^{-160}+q^{-162}$

Further Quantum Invariants

K11a22 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["K11a22"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+13t^{2}-20t+23-20t^{-1}+13t^{-2}-5t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+3z^{6}+3z^{4}+3z^{2}+1$

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]=

{ 101, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{10}-3q^{9}+6q^{8}-11q^{7}+14q^{6}-16q^{5}+16q^{4}-13q^{3}+11q^{2}-6q+3-q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-4}-z^{6}a^{-2}+6z^{6}a^{-4}-2z^{6}a^{-6}-4z^{4}a^{-2}+15z^{4}a^{-4}-9z^{4}a^{-6}+z^{4}a^{-8}-5z^{2}a^{-2}+19z^{2}a^{-4}-14z^{2}a^{-6}+3z^{2}a^{-8}-2a^{-2}+9a^{-4}-8a^{-6}+2a^{-8}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-4}+z^{10}a^{-6}+3z^{9}a^{-3}+7z^{9}a^{-5}+4z^{9}a^{-7}+3z^{8}a^{-2}+8z^{8}a^{-4}+12z^{8}a^{-6}+7z^{8}a^{-8}+z^{7}a^{-1}-6z^{7}a^{-3}-11z^{7}a^{-5}+3z^{7}a^{-7}+7z^{7}a^{-9}-12z^{6}a^{-2}-38z^{6}a^{-4}-41z^{6}a^{-6}-10z^{6}a^{-8}+5z^{6}a^{-10}-4z^{5}a^{-1}-7z^{5}a^{-3}-17z^{5}a^{-5}-26z^{5}a^{-7}-9z^{5}a^{-9}+3z^{5}a^{-11}+16z^{4}a^{-2}+48z^{4}a^{-4}+43z^{4}a^{-6}+6z^{4}a^{-8}-4z^{4}a^{-10}+z^{4}a^{-12}+5z^{3}a^{-1}+17z^{3}a^{-3}+32z^{3}a^{-5}+29z^{3}a^{-7}+6z^{3}a^{-9}-3z^{3}a^{-11}-9z^{2}a^{-2}-29z^{2}a^{-4}-25z^{2}a^{-6}-3z^{2}a^{-8}+z^{2}a^{-10}-z^{2}a^{-12}-2za^{-1}-7za^{-3}-14za^{-5}-11za^{-7}-za^{-9}+za^{-11}+2a^{-2}+9a^{-4}+8a^{-6}+2a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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["K11a22"];

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]=

{ $t^{4}-5t^{3}+13t^{2}-20t+23-20t^{-1}+13t^{-2}-5t^{-3}+t^{-4}$, $q^{10}-3q^{9}+6q^{8}-11q^{7}+14q^{6}-16q^{5}+16q^{4}-13q^{3}+11q^{2}-6q+3-q^{-1}$ }

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]=

{}

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, 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 $12$ $24$ $72$ $78$ $10$ $288$ $304$ $64$ $56$ $288$ $288$ $936$ $120$ ${\frac {11871}{10}}$ $-{\frac {1186}{15}}$ ${\frac {10502}{15}}$ ${\frac {65}{6}}$ ${\frac {991}{10}}$

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=$4 is the signature of K11a22. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 21                       1 1 19                     2   -2 17                   4 1   3 15                 7 2     -5 13               7 4       3 11             9 7         -2 9           7 7           0 7         6 9             3 5       5 7               -2 3     2 7                 5 1   1 4                   -3 -1   2                     2 -3 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a21 K11a23