K11n108

K11n107

K11n109

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n108 at Knotilus!

Knot K11n108.

A graph, K11n108.

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

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X5,14,6,15 X7,16,8,17 X18,9,19,10 X2,11,3,12 X20,13,21,14 X15,6,16,7 X22,18,1,17 X12,19,13,20 X8,21,9,22
Gauss code	1, -6, 2, -1, -3, 8, -4, -11, 5, -2, 6, -10, 7, 3, -8, 4, 9, -5, 10, -7, 11, -9
Dowker-Thistlethwaite code	4 10 -14 -16 18 2 20 -6 22 12 8

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:K11n108/ThurstonBennequinNumber Hyperbolic Volume 14.6102 A-Polynomial See Data:K11n108/A-polynomial

[edit Notes for K11n108'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 K11n108's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+8t^{2}-17t+21-17t^{-1}+8t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+2z^{4}+6z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 73, -4 }
Jones polynomial	$2q^{-2}-5q^{-3}+9q^{-4}-11q^{-5}+13q^{-6}-12q^{-7}+10q^{-8}-7q^{-9}+3q^{-10}-q^{-11}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{10}-a^{10}+2z^{4}a^{8}+3z^{2}a^{8}-z^{6}a^{6}-2z^{4}a^{6}+a^{6}+2z^{4}a^{4}+4z^{2}a^{4}+a^{4}$
Kauffman polynomial (db, data sources)	$z^{5}a^{13}-2z^{3}a^{13}+za^{13}+3z^{6}a^{12}-5z^{4}a^{12}+2z^{2}a^{12}+5z^{7}a^{11}-9z^{5}a^{11}+6z^{3}a^{11}-3za^{11}+4z^{8}a^{10}-3z^{6}a^{10}-2z^{4}a^{10}-z^{2}a^{10}+a^{10}+z^{9}a^{9}+9z^{7}a^{9}-20z^{5}a^{9}+13z^{3}a^{9}-3za^{9}+6z^{8}a^{8}-5z^{6}a^{8}-3z^{4}a^{8}+3z^{2}a^{8}+z^{9}a^{7}+5z^{7}a^{7}-7z^{5}a^{7}+z^{3}a^{7}+za^{7}+2z^{8}a^{6}+z^{6}a^{6}-3z^{4}a^{6}+2z^{2}a^{6}-a^{6}+z^{7}a^{5}+3z^{5}a^{5}-4z^{3}a^{5}+3z^{4}a^{4}-4z^{2}a^{4}+a^{4}$
The A2 invariant	$-q^{34}+q^{30}-3q^{28}+q^{26}-q^{24}-q^{22}+3q^{20}-q^{18}+4q^{16}-q^{14}+2q^{10}-2q^{8}+2q^{6}$
The G2 invariant	Data:K11n108/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n108 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["K11n108"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+8t^{2}-17t+21-17t^{-1}+8t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{6}+2z^{4}+6z^{2}+1$

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]=

{ 73, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$2q^{-2}-5q^{-3}+9q^{-4}-11q^{-5}+13q^{-6}-12q^{-7}+10q^{-8}-7q^{-9}+3q^{-10}-q^{-11}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{2}a^{10}-a^{10}+2z^{4}a^{8}+3z^{2}a^{8}-z^{6}a^{6}-2z^{4}a^{6}+a^{6}+2z^{4}a^{4}+4z^{2}a^{4}+a^{4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{13}-2z^{3}a^{13}+za^{13}+3z^{6}a^{12}-5z^{4}a^{12}+2z^{2}a^{12}+5z^{7}a^{11}-9z^{5}a^{11}+6z^{3}a^{11}-3za^{11}+4z^{8}a^{10}-3z^{6}a^{10}-2z^{4}a^{10}-z^{2}a^{10}+a^{10}+z^{9}a^{9}+9z^{7}a^{9}-20z^{5}a^{9}+13z^{3}a^{9}-3za^{9}+6z^{8}a^{8}-5z^{6}a^{8}-3z^{4}a^{8}+3z^{2}a^{8}+z^{9}a^{7}+5z^{7}a^{7}-7z^{5}a^{7}+z^{3}a^{7}+za^{7}+2z^{8}a^{6}+z^{6}a^{6}-3z^{4}a^{6}+2z^{2}a^{6}-a^{6}+z^{7}a^{5}+3z^{5}a^{5}-4z^{3}a^{5}+3z^{4}a^{4}-4z^{2}a^{4}+a^{4}$

"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["K11n108"];

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]=

{ $-t^{3}+8t^{2}-17t+21-17t^{-1}+8t^{-2}-t^{-3}$, $2q^{-2}-5q^{-3}+9q^{-4}-11q^{-5}+13q^{-6}-12q^{-7}+10q^{-8}-7q^{-9}+3q^{-10}-q^{-11}$ }

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:

(6, -15)

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 $24$ $-120$ $288$ $764$ $124$ $-2880$ $-5360$ $-928$ $-792$ $2304$ $7200$ $18336$ $2976$ ${\frac {190431}{5}}$ ${\frac {636}{5}}$ ${\frac {80108}{5}}$ $363$ ${\frac {10831}{5}}$

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 K11n108. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                   2 2 -5                 4 1 -3 -7               5 1   4 -9             6 4     -2 -11           7 5       2 -13         5 6         1 -15       5 7           -2 -17     2 5             3 -19   1 5               -4 -21   2                 2 -23 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$

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.

K11n107 K11n109