K11a20

K11a19

K11a21

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a20 at Knotilus!

Knot presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{2,3\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a20/ThurstonBennequinNumber Hyperbolic Volume 15.3455 A-Polynomial See Data:K11a20/A-polynomial

[edit Notes for K11a20's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a20's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{3}+13t^{2}-25t+31-25t^{-1}+13t^{-2}-3t^{-3}$
Conway polynomial	$-3z^{6}-5z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 113, 4 }
Jones polynomial	$-q^{11}+4q^{10}-8q^{9}+13q^{8}-17q^{7}+18q^{6}-18q^{5}+15q^{4}-10q^{3}+6q^{2}-2q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}-2z^{6}a^{-6}+z^{4}a^{-2}-2z^{4}a^{-4}-7z^{4}a^{-6}+3z^{4}a^{-8}+3z^{2}a^{-2}-9z^{2}a^{-6}+7z^{2}a^{-8}-z^{2}a^{-10}+2a^{-2}+a^{-4}-5a^{-6}+4a^{-8}-a^{-10}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-6}+z^{10}a^{-8}+3z^{9}a^{-5}+7z^{9}a^{-7}+4z^{9}a^{-9}+3z^{8}a^{-4}+8z^{8}a^{-6}+12z^{8}a^{-8}+7z^{8}a^{-10}+2z^{7}a^{-3}-4z^{7}a^{-5}-8z^{7}a^{-7}+5z^{7}a^{-9}+7z^{7}a^{-11}+z^{6}a^{-2}-6z^{6}a^{-4}-26z^{6}a^{-6}-30z^{6}a^{-8}-7z^{6}a^{-10}+4z^{6}a^{-12}-5z^{5}a^{-3}+z^{5}a^{-5}-5z^{5}a^{-7}-23z^{5}a^{-9}-11z^{5}a^{-11}+z^{5}a^{-13}-4z^{4}a^{-2}+3z^{4}a^{-4}+34z^{4}a^{-6}+30z^{4}a^{-8}-3z^{4}a^{-10}-6z^{4}a^{-12}+2z^{3}a^{-3}+14z^{3}a^{-7}+22z^{3}a^{-9}+5z^{3}a^{-11}-z^{3}a^{-13}+5z^{2}a^{-2}-2z^{2}a^{-4}-22z^{2}a^{-6}-15z^{2}a^{-8}+2z^{2}a^{-10}+2z^{2}a^{-12}+za^{-3}-za^{-5}-7za^{-7}-7za^{-9}-2za^{-11}-2a^{-2}+a^{-4}+5a^{-6}+4a^{-8}+a^{-10}$
The A2 invariant	Data:K11a20/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a20/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a20 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["K11a20"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3t^{3}+13t^{2}-25t+31-25t^{-1}+13t^{-2}-3t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-3z^{6}-5z^{4}+1$

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]=

{ 113, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{11}+4q^{10}-8q^{9}+13q^{8}-17q^{7}+18q^{6}-18q^{5}+15q^{4}-10q^{3}+6q^{2}-2q+1$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-4}-2z^{6}a^{-6}+z^{4}a^{-2}-2z^{4}a^{-4}-7z^{4}a^{-6}+3z^{4}a^{-8}+3z^{2}a^{-2}-9z^{2}a^{-6}+7z^{2}a^{-8}-z^{2}a^{-10}+2a^{-2}+a^{-4}-5a^{-6}+4a^{-8}-a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-6}+z^{10}a^{-8}+3z^{9}a^{-5}+7z^{9}a^{-7}+4z^{9}a^{-9}+3z^{8}a^{-4}+8z^{8}a^{-6}+12z^{8}a^{-8}+7z^{8}a^{-10}+2z^{7}a^{-3}-4z^{7}a^{-5}-8z^{7}a^{-7}+5z^{7}a^{-9}+7z^{7}a^{-11}+z^{6}a^{-2}-6z^{6}a^{-4}-26z^{6}a^{-6}-30z^{6}a^{-8}-7z^{6}a^{-10}+4z^{6}a^{-12}-5z^{5}a^{-3}+z^{5}a^{-5}-5z^{5}a^{-7}-23z^{5}a^{-9}-11z^{5}a^{-11}+z^{5}a^{-13}-4z^{4}a^{-2}+3z^{4}a^{-4}+34z^{4}a^{-6}+30z^{4}a^{-8}-3z^{4}a^{-10}-6z^{4}a^{-12}+2z^{3}a^{-3}+14z^{3}a^{-7}+22z^{3}a^{-9}+5z^{3}a^{-11}-z^{3}a^{-13}+5z^{2}a^{-2}-2z^{2}a^{-4}-22z^{2}a^{-6}-15z^{2}a^{-8}+2z^{2}a^{-10}+2z^{2}a^{-12}+za^{-3}-za^{-5}-7za^{-7}-7za^{-9}-2za^{-11}-2a^{-2}+a^{-4}+5a^{-6}+4a^{-8}+a^{-10}$

"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["K11a20"];

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]=

{ $-3t^{3}+13t^{2}-25t+31-25t^{-1}+13t^{-2}-3t^{-3}$, $-q^{11}+4q^{10}-8q^{9}+13q^{8}-17q^{7}+18q^{6}-18q^{5}+15q^{4}-10q^{3}+6q^{2}-2q+1$ }

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:

(0, -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 $0$ $-8$ $0$ $32$ $40$ $0$ ${\frac {976}{3}}$ ${\frac {352}{3}}$ $152$ $0$ $32$ $0$ $0$ $1184$ $8$ ${\frac {2264}{3}}$ ${\frac {416}{3}}$ $16$

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 K11a20. 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 23                       1 -1 21                     3   3 19                   5 1   -4 17                 8 3     5 15               9 5       -4 13             9 8         1 11           9 9           0 9         6 9             -3 7       4 9               5 5     2 6                 -4 3   1 5                   4 1   1                     -1 -1 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a19 K11a21