K11a21

K11a20

K11a22

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a21 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a21/ThurstonBennequinNumber Hyperbolic Volume 12.6394 A-Polynomial See Data:K11a21/A-polynomial

[edit Notes for K11a21's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a21's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-5t^{2}+19t-27+19t^{-1}-5t^{-2}$
Conway polynomial	$-5z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 75, 2 }
Jones polynomial	$-q^{10}+3q^{9}-5q^{8}+8q^{7}-10q^{6}+11q^{5}-12q^{4}+10q^{3}-7q^{2}+5q-2+q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{-2}-2z^{4}a^{-4}-2z^{4}a^{-6}-2z^{2}a^{-4}-3z^{2}a^{-6}+3z^{2}a^{-8}+z^{2}+a^{-2}-a^{-4}-2a^{-6}+3a^{-8}-a^{-10}+1$
Kauffman polynomial (db, data sources)	$z^{10}a^{-6}+z^{10}a^{-8}+3z^{9}a^{-5}+6z^{9}a^{-7}+3z^{9}a^{-9}+3z^{8}a^{-4}+4z^{8}a^{-6}+4z^{8}a^{-8}+3z^{8}a^{-10}+3z^{7}a^{-3}-7z^{7}a^{-5}-21z^{7}a^{-7}-10z^{7}a^{-9}+z^{7}a^{-11}+3z^{6}a^{-2}-2z^{6}a^{-4}-21z^{6}a^{-6}-29z^{6}a^{-8}-13z^{6}a^{-10}+2z^{5}a^{-1}-z^{5}a^{-3}+12z^{5}a^{-5}+24z^{5}a^{-7}+5z^{5}a^{-9}-4z^{5}a^{-11}-3z^{4}a^{-2}-2z^{4}a^{-4}+28z^{4}a^{-6}+42z^{4}a^{-8}+16z^{4}a^{-10}+z^{4}-2z^{3}a^{-1}-2z^{3}a^{-3}-15z^{3}a^{-5}-16z^{3}a^{-7}+3z^{3}a^{-9}+4z^{3}a^{-11}+2z^{2}a^{-2}+3z^{2}a^{-4}-16z^{2}a^{-6}-22z^{2}a^{-8}-7z^{2}a^{-10}-2z^{2}+2za^{-3}+7za^{-5}+6za^{-7}-za^{-11}-a^{-2}-a^{-4}+2a^{-6}+3a^{-8}+a^{-10}+1$
The A2 invariant	Data:K11a21/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a21/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a21 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["K11a21"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-5t^{2}+19t-27+19t^{-1}-5t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-5z^{4}-z^{2}+1$

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]=

{ 75, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{10}+3q^{9}-5q^{8}+8q^{7}-10q^{6}+11q^{5}-12q^{4}+10q^{3}-7q^{2}+5q-2+q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}a^{-2}-2z^{4}a^{-4}-2z^{4}a^{-6}-2z^{2}a^{-4}-3z^{2}a^{-6}+3z^{2}a^{-8}+z^{2}+a^{-2}-a^{-4}-2a^{-6}+3a^{-8}-a^{-10}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-6}+z^{10}a^{-8}+3z^{9}a^{-5}+6z^{9}a^{-7}+3z^{9}a^{-9}+3z^{8}a^{-4}+4z^{8}a^{-6}+4z^{8}a^{-8}+3z^{8}a^{-10}+3z^{7}a^{-3}-7z^{7}a^{-5}-21z^{7}a^{-7}-10z^{7}a^{-9}+z^{7}a^{-11}+3z^{6}a^{-2}-2z^{6}a^{-4}-21z^{6}a^{-6}-29z^{6}a^{-8}-13z^{6}a^{-10}+2z^{5}a^{-1}-z^{5}a^{-3}+12z^{5}a^{-5}+24z^{5}a^{-7}+5z^{5}a^{-9}-4z^{5}a^{-11}-3z^{4}a^{-2}-2z^{4}a^{-4}+28z^{4}a^{-6}+42z^{4}a^{-8}+16z^{4}a^{-10}+z^{4}-2z^{3}a^{-1}-2z^{3}a^{-3}-15z^{3}a^{-5}-16z^{3}a^{-7}+3z^{3}a^{-9}+4z^{3}a^{-11}+2z^{2}a^{-2}+3z^{2}a^{-4}-16z^{2}a^{-6}-22z^{2}a^{-8}-7z^{2}a^{-10}-2z^{2}+2za^{-3}+7za^{-5}+6za^{-7}-za^{-11}-a^{-2}-a^{-4}+2a^{-6}+3a^{-8}+a^{-10}+1$

"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["K11a21"];

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]=

{ $-5t^{2}+19t-27+19t^{-1}-5t^{-2}$, $-q^{10}+3q^{9}-5q^{8}+8q^{7}-10q^{6}+11q^{5}-12q^{4}+10q^{3}-7q^{2}+5q-2+q^{-1}$ }

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:

(-1, -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 $-4$ $-8$ $8$ ${\frac {226}{3}}$ ${\frac {134}{3}}$ $32$ ${\frac {1360}{3}}$ ${\frac {352}{3}}$ $152$ $-{\frac {32}{3}}$ $32$ $-{\frac {904}{3}}$ $-{\frac {536}{3}}$ ${\frac {27089}{30}}$ ${\frac {422}{15}}$ ${\frac {16378}{45}}$ ${\frac {2863}{18}}$ $-{\frac {1231}{30}}$

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=$2 is the signature of K11a21. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a20 K11a22