K11n83

K11n82

K11n84

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n83 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X5,17,6,16 X7,12,8,13 X2,9,3,10 X11,18,12,19 X13,1,14,22 X15,21,16,20 X17,10,18,11 X19,7,20,6 X21,15,22,14
Gauss code	1, -5, 2, -1, -3, 10, -4, -2, 5, 9, -6, 4, -7, 11, -8, 3, -9, 6, -10, 8, -11, 7
Dowker-Thistlethwaite code	4 8 -16 -12 2 -18 -22 -20 -10 -6 -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:K11n83/ThurstonBennequinNumber Hyperbolic Volume 12.0122 A-Polynomial See Data:K11n83/A-polynomial

[edit Notes for K11n83's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n83's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{2}-12t+19-12t^{-1}+3t^{-2}$
Conway polynomial	$3z^{4}+1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^{2}+t+1\right\}$
Determinant and Signature	{ 49, 0 }
Jones polynomial	$q^{4}-3q^{3}+5q^{2}-7q+9-8q^{-1}+7q^{-2}-5q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{4}+z^{4}a^{2}-a^{2}+2z^{4}+4z^{2}+4-3z^{2}a^{-2}-3a^{-2}+a^{-4}$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+3a^{2}z^{8}+z^{8}a^{-2}+4z^{8}+4a^{3}z^{7}+az^{7}-3z^{7}a^{-1}+3a^{4}z^{6}-7a^{2}z^{6}-3z^{6}a^{-2}-13z^{6}+a^{5}z^{5}-10a^{3}z^{5}-5az^{5}+9z^{5}a^{-1}+3z^{5}a^{-3}-7a^{4}z^{4}+7a^{2}z^{4}+10z^{4}a^{-2}+z^{4}a^{-4}+23z^{4}-2a^{5}z^{3}+6a^{3}z^{3}+6az^{3}-7z^{3}a^{-1}-5z^{3}a^{-3}+2a^{4}z^{2}-4a^{2}z^{2}-11z^{2}a^{-2}-2z^{2}a^{-4}-15z^{2}-a^{3}z-az+za^{-1}+za^{-3}+a^{2}+3a^{-2}+a^{-4}+4$
The A2 invariant	Data:K11n83/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n83/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n83 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["K11n83"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{2}-12t+19-12t^{-1}+3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$3z^{4}+1$

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]=

$\left\{2,t^{2}+t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 49, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{9}+z^{9}a^{-1}+3a^{2}z^{8}+z^{8}a^{-2}+4z^{8}+4a^{3}z^{7}+az^{7}-3z^{7}a^{-1}+3a^{4}z^{6}-7a^{2}z^{6}-3z^{6}a^{-2}-13z^{6}+a^{5}z^{5}-10a^{3}z^{5}-5az^{5}+9z^{5}a^{-1}+3z^{5}a^{-3}-7a^{4}z^{4}+7a^{2}z^{4}+10z^{4}a^{-2}+z^{4}a^{-4}+23z^{4}-2a^{5}z^{3}+6a^{3}z^{3}+6az^{3}-7z^{3}a^{-1}-5z^{3}a^{-3}+2a^{4}z^{2}-4a^{2}z^{2}-11z^{2}a^{-2}-2z^{2}a^{-4}-15z^{2}-a^{3}z-az+za^{-1}+za^{-3}+a^{2}+3a^{-2}+a^{-4}+4$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_41,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {9_27,}

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["K11n83"];

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]=

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

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]=

{9_41,}

In[6]:=

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

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

Out[6]=

{9_27,}

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 $0$ $-8$ $0$ $-32$ $-24$ $0$ $-{\frac {80}{3}}$ ${\frac {160}{3}}$ $-72$ $0$ $32$ $0$ $0$ $32$ ${\frac {232}{3}}$ $-24$ $-48$ $-32$

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=$0 is the signature of K11n83. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n82 K11n84