K11n20

K11n19

K11n21

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n20 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X7,17,8,16 X2,9,3,10 X11,18,12,19 X13,20,14,21 X15,22,16,1 X17,7,18,6 X19,14,20,15 X21,12,22,13
Gauss code	1, -5, 2, -1, 3, 9, -4, -2, 5, -3, -6, 11, -7, 10, -8, 4, -9, 6, -10, 7, -11, 8
Dowker-Thistlethwaite code	4 8 10 -16 2 -18 -20 -22 -6 -14 -12

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

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n20/ThurstonBennequinNumber Hyperbolic Volume 9.41057 A-Polynomial See Data:K11n20/A-polynomial

[edit Notes for K11n20's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for K11n20's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{2}+6t-7+6t^{-1}-2t^{-2}$
Conway polynomial	$-2z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 23, -2 }
Jones polynomial	$q^{3}-2q^{2}+3q-3+4q^{-1}-4q^{-2}+3q^{-3}-2q^{-4}+q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{4}+a^{4}-z^{4}a^{2}-2z^{2}a^{2}-a^{2}-z^{4}-2z^{2}+z^{2}a^{-2}+a^{-2}$
Kauffman polynomial (db, data sources)	$a^{3}z^{9}+az^{9}+a^{4}z^{8}+3a^{2}z^{8}+2z^{8}-5a^{3}z^{7}-3az^{7}+2z^{7}a^{-1}-6a^{4}z^{6}-15a^{2}z^{6}+z^{6}a^{-2}-8z^{6}+7a^{3}z^{5}-az^{5}-8z^{5}a^{-1}+12a^{4}z^{4}+22a^{2}z^{4}-4z^{4}a^{-2}+6z^{4}+a^{5}z^{3}-a^{3}z^{3}+4az^{3}+6z^{3}a^{-1}-7a^{4}z^{2}-10a^{2}z^{2}+3z^{2}a^{-2}-a^{5}z-a^{3}z-az-za^{-1}+a^{4}+a^{2}-a^{-2}$
The A2 invariant	Data:K11n20/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n20/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n20 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["K11n20"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 23, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{2}a^{4}+a^{4}-z^{4}a^{2}-2z^{2}a^{2}-a^{2}-z^{4}-2z^{2}+z^{2}a^{-2}+a^{-2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{3}z^{9}+az^{9}+a^{4}z^{8}+3a^{2}z^{8}+2z^{8}-5a^{3}z^{7}-3az^{7}+2z^{7}a^{-1}-6a^{4}z^{6}-15a^{2}z^{6}+z^{6}a^{-2}-8z^{6}+7a^{3}z^{5}-az^{5}-8z^{5}a^{-1}+12a^{4}z^{4}+22a^{2}z^{4}-4z^{4}a^{-2}+6z^{4}+a^{5}z^{3}-a^{3}z^{3}+4az^{3}+6z^{3}a^{-1}-7a^{4}z^{2}-10a^{2}z^{2}+3z^{2}a^{-2}-a^{5}z-a^{3}z-az-za^{-1}+a^{4}+a^{2}-a^{-2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_6, K11n151, K11n152,}

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["K11n20"];

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]=

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

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_6, K11n151, K11n152,}

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:

(-2, 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 $-8$ $8$ $32$ ${\frac {164}{3}}$ ${\frac {100}{3}}$ $-64$ $-{\frac {400}{3}}$ $-{\frac {64}{3}}$ $-56$ $-{\frac {256}{3}}$ $32$ $-{\frac {1312}{3}}$ $-{\frac {800}{3}}$ $-{\frac {4111}{15}}$ ${\frac {268}{5}}$ $-{\frac {18244}{45}}$ ${\frac {1231}{9}}$ $-{\frac {1471}{15}}$

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 K11n20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 χ 7                 1 1 5               1   -1 3             2 1   1 1         1 2 1     0 -1         3 2       1 -3       2 2         0 -5     1 2           -1 -7   1 2             1 -9   1               -1 -11 1                 1

Integral Khovanov Homology (db, data source)

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

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.

K11n19 K11n21