K11n8

K11n7

K11n9

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n8 at Knotilus!

Knot K11n8.

A graph, knot K11n8.

A part of a knot and a part of a graph.

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n8/ThurstonBennequinNumber Hyperbolic Volume 13.098 A-Polynomial See Data:K11n8/A-polynomial

[edit Notes for K11n8'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 K11n8's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+6t^{2}-12t+15-12t^{-1}+6t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 53, -4 }
Jones polynomial	$1-3q^{-1}+6q^{-2}-7q^{-3}+9q^{-4}-9q^{-5}+8q^{-6}-6q^{-7}+3q^{-8}-q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{8}-a^{8}+2z^{4}a^{6}+4z^{2}a^{6}+a^{6}-z^{6}a^{4}-3z^{4}a^{4}-2z^{2}a^{4}+z^{4}a^{2}+2z^{2}a^{2}+a^{2}$
Kauffman polynomial (db, data sources)	$z^{3}a^{11}-za^{11}+3z^{4}a^{10}-2z^{2}a^{10}+z^{7}a^{9}+3z^{3}a^{9}-za^{9}+2z^{8}a^{8}-4z^{6}a^{8}+6z^{4}a^{8}-a^{8}+z^{9}a^{7}+2z^{7}a^{7}-9z^{5}a^{7}+10z^{3}a^{7}-3za^{7}+5z^{8}a^{6}-11z^{6}a^{6}+3z^{4}a^{6}+3z^{2}a^{6}-a^{6}+z^{9}a^{5}+4z^{7}a^{5}-18z^{5}a^{5}+14z^{3}a^{5}-4za^{5}+3z^{8}a^{4}-6z^{6}a^{4}-3z^{4}a^{4}+4z^{2}a^{4}+3z^{7}a^{3}-9z^{5}a^{3}+6z^{3}a^{3}-za^{3}+z^{6}a^{2}-3z^{4}a^{2}+3z^{2}a^{2}-a^{2}$
The A2 invariant	Data:K11n8/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n8/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n8 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["K11n8"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+6t^{2}-12t+15-12t^{-1}+6t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{6}+3z^{2}+1$

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]=

{ 53, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{3}a^{11}-za^{11}+3z^{4}a^{10}-2z^{2}a^{10}+z^{7}a^{9}+3z^{3}a^{9}-za^{9}+2z^{8}a^{8}-4z^{6}a^{8}+6z^{4}a^{8}-a^{8}+z^{9}a^{7}+2z^{7}a^{7}-9z^{5}a^{7}+10z^{3}a^{7}-3za^{7}+5z^{8}a^{6}-11z^{6}a^{6}+3z^{4}a^{6}+3z^{2}a^{6}-a^{6}+z^{9}a^{5}+4z^{7}a^{5}-18z^{5}a^{5}+14z^{3}a^{5}-4za^{5}+3z^{8}a^{4}-6z^{6}a^{4}-3z^{4}a^{4}+4z^{2}a^{4}+3z^{7}a^{3}-9z^{5}a^{3}+6z^{3}a^{3}-za^{3}+z^{6}a^{2}-3z^{4}a^{2}+3z^{2}a^{2}-a^{2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n59,}

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["K11n8"];

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]=

{ $-t^{3}+6t^{2}-12t+15-12t^{-1}+6t^{-2}-t^{-3}$, $1-3q^{-1}+6q^{-2}-7q^{-3}+9q^{-4}-9q^{-5}+8q^{-6}-6q^{-7}+3q^{-8}-q^{-9}$ }

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]=

{K11n59,}

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:

(3, -6)

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 $12$ $-48$ $72$ $222$ $34$ $-576$ $-1120$ $-192$ $-144$ $288$ $1152$ $2664$ $408$ ${\frac {58191}{10}}$ ${\frac {3734}{15}}$ ${\frac {31742}{15}}$ ${\frac {209}{6}}$ ${\frac {2671}{10}}$

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 K11n8. 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 χ 1                   1 1 -1                 2   -2 -3               4 1   3 -5             4 3     -1 -7           5 3       2 -9         4 4         0 -11       4 5           -1 -13     2 4             2 -15   1 4               -3 -17   2                 2 -19 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

K11n7 K11n9