K11n79

K11n78

K11n80

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n79 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,14,6,15 X2837 X9,21,10,20 X11,19,12,18 X13,6,14,7 X15,22,16,1 X17,13,18,12 X19,11,20,10 X21,16,22,17
Gauss code	1, -4, 2, -1, -3, 7, 4, -2, -5, 10, -6, 9, -7, 3, -8, 11, -9, 6, -10, 5, -11, 8
Dowker-Thistlethwaite code	4 8 -14 2 -20 -18 -6 -22 -12 -10 -16

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:K11n79/ThurstonBennequinNumber Hyperbolic Volume 6.75197 A-Polynomial See Data:K11n79/A-polynomial

[edit Notes for K11n79'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 K11n79's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{2}+4t-3+4t^{-1}-2t^{-2}$
Conway polynomial	$-2z^{4}-4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 15, 2 }
Jones polynomial	$q^{5}-q^{4}+2q^{3}-3q^{2}+2q-2+2q^{-1}-q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{-2}-z^{4}+a^{2}z^{2}-3z^{2}a^{-2}+z^{2}a^{-4}-3z^{2}+2a^{2}-2a^{-2}+2a^{-4}-1$
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+2z^{8}a^{-2}+z^{8}a^{-4}+z^{8}+az^{7}-6z^{7}a^{-1}-7z^{7}a^{-3}+a^{2}z^{6}-12z^{6}a^{-2}-7z^{6}a^{-4}-4z^{6}-4az^{5}+13z^{5}a^{-1}+17z^{5}a^{-3}-5a^{2}z^{4}+23z^{4}a^{-2}+16z^{4}a^{-4}+2z^{4}+2az^{3}-15z^{3}a^{-1}-17z^{3}a^{-3}+6a^{2}z^{2}-16z^{2}a^{-2}-12z^{2}a^{-4}+2z^{2}+az+7za^{-1}+7za^{-3}+za^{-5}-2a^{2}+2a^{-2}+2a^{-4}-1$
The A2 invariant	Data:K11n79/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n79/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n79 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["K11n79"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{2}+4t-3+4t^{-1}-2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 15, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-1}+z^{9}a^{-3}+2z^{8}a^{-2}+z^{8}a^{-4}+z^{8}+az^{7}-6z^{7}a^{-1}-7z^{7}a^{-3}+a^{2}z^{6}-12z^{6}a^{-2}-7z^{6}a^{-4}-4z^{6}-4az^{5}+13z^{5}a^{-1}+17z^{5}a^{-3}-5a^{2}z^{4}+23z^{4}a^{-2}+16z^{4}a^{-4}+2z^{4}+2az^{3}-15z^{3}a^{-1}-17z^{3}a^{-3}+6a^{2}z^{2}-16z^{2}a^{-2}-12z^{2}a^{-4}+2z^{2}+az+7za^{-1}+7za^{-3}+za^{-5}-2a^{2}+2a^{-2}+2a^{-4}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n138,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n138,}

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["K11n79"];

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]=

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

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]=

{K11n138,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11n138,}

Vassiliev invariants

V2 and V3:

(-4, -2)

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 $-16$ $-16$ $128$ ${\frac {568}{3}}$ ${\frac {248}{3}}$ $256$ ${\frac {1280}{3}}$ ${\frac {320}{3}}$ $80$ $-{\frac {2048}{3}}$ $128$ $-{\frac {9088}{3}}$ $-{\frac {3968}{3}}$ $-{\frac {37862}{15}}$ ${\frac {3336}{5}}$ $-{\frac {119528}{45}}$ ${\frac {3350}{9}}$ $-{\frac {9302}{15}}$

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 K11n79. 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 11                 1 1 9                   0 7             2 1   1 5           1       -1 3         1 2       -1 1       2 2         0 -1       1 1         0 -3   1 2             1 -5                   0 -7 1                 1

Integral Khovanov Homology (db, data source)

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

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.

K11n78 K11n80