K11n80

K11n79

K11n81

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n80 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X14,6,15,5 X10,8,11,7 X2,9,3,10 X11,18,12,19 X6,14,7,13 X15,20,16,21 X17,22,18,1 X19,12,20,13 X21,16,22,17
Gauss code	1, -5, 2, -1, 3, -7, 4, -2, 5, -4, -6, 10, 7, -3, -8, 11, -9, 6, -10, 8, -11, 9
Dowker-Thistlethwaite code	4 8 14 10 2 -18 6 -20 -22 -12 -16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n80/ThurstonBennequinNumber Hyperbolic Volume 10.8343 A-Polynomial See Data:K11n80/A-polynomial

[edit Notes for K11n80'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 K11n80's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+t^{2}+5t-9+5t^{-1}+t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-5z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 15, -2 }
Jones polynomial	$q^{2}-q+1-2q^{-2}+3q^{-3}-3q^{-4}+4q^{-5}-3q^{-6}+2q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{8}+2z^{2}a^{6}+a^{6}+3z^{2}a^{4}+3a^{4}-z^{6}a^{2}-6z^{4}a^{2}-9z^{2}a^{2}-5a^{2}+z^{4}+4z^{2}+3$
Kauffman polynomial (db, data sources)	$z^{5}a^{9}-3z^{3}a^{9}+2za^{9}+2z^{6}a^{8}-6z^{4}a^{8}+4z^{2}a^{8}-a^{8}+z^{7}a^{7}-6z^{3}a^{7}+3za^{7}+3z^{6}a^{6}-7z^{4}a^{6}+3z^{2}a^{6}-a^{6}+3z^{5}a^{5}-6z^{3}a^{5}+3za^{5}+z^{8}a^{4}-8z^{6}a^{4}+20z^{4}a^{4}-15z^{2}a^{4}+3a^{4}+z^{9}a^{3}-8z^{7}a^{3}+18z^{5}a^{3}-13z^{3}a^{3}+5za^{3}+2z^{8}a^{2}-16z^{6}a^{2}+36z^{4}a^{2}-26z^{2}a^{2}+5a^{2}+z^{9}a-7z^{7}a+14z^{5}a-10z^{3}a+3za+z^{8}-7z^{6}+15z^{4}-12z^{2}+3$
The A2 invariant	Data:K11n80/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n80/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n80 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["K11n80"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+t^{2}+5t-9+5t^{-1}+t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 15, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{9}-3z^{3}a^{9}+2za^{9}+2z^{6}a^{8}-6z^{4}a^{8}+4z^{2}a^{8}-a^{8}+z^{7}a^{7}-6z^{3}a^{7}+3za^{7}+3z^{6}a^{6}-7z^{4}a^{6}+3z^{2}a^{6}-a^{6}+3z^{5}a^{5}-6z^{3}a^{5}+3za^{5}+z^{8}a^{4}-8z^{6}a^{4}+20z^{4}a^{4}-15z^{2}a^{4}+3a^{4}+z^{9}a^{3}-8z^{7}a^{3}+18z^{5}a^{3}-13z^{3}a^{3}+5za^{3}+2z^{8}a^{2}-16z^{6}a^{2}+36z^{4}a^{2}-26z^{2}a^{2}+5a^{2}+z^{9}a-7z^{7}a+14z^{5}a-10z^{3}a+3za+z^{8}-7z^{6}+15z^{4}-12z^{2}+3$

"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["K11n80"];

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]=

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

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, -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 $0$ $-24$ $0$ $128$ $40$ $0$ $-304$ $-64$ $-56$ $0$ $288$ $0$ $0$ $1008$ $56$ ${\frac {920}{3}}$ ${\frac {224}{3}}$ $16$

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 $s=$-2 is the signature of K11n80. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n79 K11n81