K11n109

K11n108

K11n110

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n109 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X5,14,6,15 X16,7,17,8 X18,9,19,10 X2,11,3,12 X20,13,21,14 X15,22,16,1 X8,17,9,18 X12,19,13,20 X21,7,22,6
Gauss code	1, -6, 2, -1, -3, 11, 4, -9, 5, -2, 6, -10, 7, 3, -8, -4, 9, -5, 10, -7, -11, 8
Dowker-Thistlethwaite code	4 10 -14 16 18 2 20 -22 8 12 -6

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:K11n109/ThurstonBennequinNumber Hyperbolic Volume 13.0869 A-Polynomial See Data:K11n109/A-polynomial

[edit Notes for K11n109'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 K11n109's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+7t^{2}-13t+15-13t^{-1}+7t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+z^{4}+6z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 57, -4 }
Jones polynomial	$2q^{-2}-4q^{-3}+7q^{-4}-9q^{-5}+10q^{-6}-9q^{-7}+8q^{-8}-5q^{-9}+2q^{-10}-q^{-11}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{10}-2a^{10}+2z^{4}a^{8}+5z^{2}a^{8}+3a^{8}-z^{6}a^{6}-3z^{4}a^{6}-3z^{2}a^{6}-2a^{6}+2z^{4}a^{4}+5z^{2}a^{4}+2a^{4}$
Kauffman polynomial (db, data sources)	$z^{5}a^{13}-3z^{3}a^{13}+2za^{13}+2z^{6}a^{12}-4z^{4}a^{12}+z^{2}a^{12}+3z^{7}a^{11}-6z^{5}a^{11}+4z^{3}a^{11}-2za^{11}+3z^{8}a^{10}-7z^{6}a^{10}+11z^{4}a^{10}-10z^{2}a^{10}+2a^{10}+z^{9}a^{9}+3z^{7}a^{9}-10z^{5}a^{9}+12z^{3}a^{9}-4za^{9}+5z^{8}a^{8}-13z^{6}a^{8}+20z^{4}a^{8}-13z^{2}a^{8}+3a^{8}+z^{9}a^{7}+z^{7}a^{7}-2z^{5}a^{7}+2z^{3}a^{7}+2z^{8}a^{6}-4z^{6}a^{6}+8z^{4}a^{6}-8z^{2}a^{6}+2a^{6}+z^{7}a^{5}+z^{5}a^{5}-3z^{3}a^{5}+3z^{4}a^{4}-6z^{2}a^{4}+2a^{4}$
The A2 invariant	$-q^{34}-q^{32}-2q^{28}+2q^{26}+q^{24}+2q^{20}-2q^{18}+2q^{16}-q^{14}+2q^{10}-q^{8}+2q^{6}$
The G2 invariant	Data:K11n109/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n109 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["K11n109"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+7t^{2}-13t+15-13t^{-1}+7t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 57, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{2}a^{10}-2a^{10}+2z^{4}a^{8}+5z^{2}a^{8}+3a^{8}-z^{6}a^{6}-3z^{4}a^{6}-3z^{2}a^{6}-2a^{6}+2z^{4}a^{4}+5z^{2}a^{4}+2a^{4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n137,}

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["K11n109"];

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]=

{ $-t^{3}+7t^{2}-13t+15-13t^{-1}+7t^{-2}-t^{-3}$, $2q^{-2}-4q^{-3}+7q^{-4}-9q^{-5}+10q^{-6}-9q^{-7}+8q^{-8}-5q^{-9}+2q^{-10}-q^{-11}$ }

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]=

{K11n137,}

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, -16)

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 $24$ $-128$ $288$ $828$ $132$ $-3072$ $-{\frac {17792}{3}}$ $-{\frac {3104}{3}}$ $-864$ $2304$ $8192$ $19872$ $3168$ ${\frac {215311}{5}}$ ${\frac {1476}{5}}$ ${\frac {269564}{15}}$ ${\frac {1201}{3}}$ ${\frac {11951}{5}}$

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 K11n109. 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                 3 1 -2 -7               4 1   3 -9             5 3     -2 -11           5 4       1 -13         4 5         1 -15       4 5           -1 -17     1 4             3 -19   1 4               -3 -21   1                 1 -23 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

K11n108 K11n110