K11n165

K11n164

K11n166

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n165 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X3,11,4,10 X14,5,15,6 X16,8,17,7 X20,9,21,10 X11,5,12,4 X18,13,19,14 X2,15,3,16 X22,18,1,17 X12,19,13,20 X8,21,9,22
Gauss code	1, -8, -2, 6, 3, -1, 4, -11, 5, 2, -6, -10, 7, -3, 8, -4, 9, -7, 10, -5, 11, -9
Dowker-Thistlethwaite code	6 -10 14 16 20 -4 18 2 22 12 8

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n165/ThurstonBennequinNumber Hyperbolic Volume 15.8169 A-Polynomial See Data:K11n165/A-polynomial

[edit Notes for K11n165's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for K11n165's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+7t^{2}-20t+29-20t^{-1}+7t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^{2}+t+1\right\}$
Determinant and Signature	{ 85, 0 }
Jones polynomial	$-q^{5}+4q^{4}-8q^{3}+12q^{2}-14q+15-13q^{-1}+10q^{-2}-6q^{-3}+2q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+a^{2}z^{4}+2z^{4}a^{-2}-2z^{4}-a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}-z^{2}+a^{4}-2a^{2}+2$
Kauffman polynomial (db, data sources)	$2az^{9}+2z^{9}a^{-1}+3a^{2}z^{8}+6z^{8}a^{-2}+9z^{8}+a^{3}z^{7}+2az^{7}+8z^{7}a^{-1}+7z^{7}a^{-3}-2a^{2}z^{6}-6z^{6}a^{-2}+4z^{6}a^{-4}-12z^{6}+5a^{3}z^{5}+az^{5}-17z^{5}a^{-1}-12z^{5}a^{-3}+z^{5}a^{-5}+3a^{4}z^{4}+5a^{2}z^{4}-3z^{4}a^{-2}-6z^{4}a^{-4}+5z^{4}-7a^{3}z^{3}-8az^{3}+5z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}-3a^{4}z^{2}-7a^{2}z^{2}+2z^{2}a^{-2}+2z^{2}a^{-4}-4z^{2}+2a^{3}z+3az+za^{-1}+a^{4}+2a^{2}+2$
The A2 invariant	Data:K11n165/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n165/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n165 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["K11n165"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+7t^{2}-20t+29-20t^{-1}+7t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

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

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 85, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+4q^{4}-8q^{3}+12q^{2}-14q+15-13q^{-1}+10q^{-2}-6q^{-3}+2q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_60,}

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["K11n165"];

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]=

{ $-t^{3}+7t^{2}-20t+29-20t^{-1}+7t^{-2}-t^{-3}$, $-q^{5}+4q^{4}-8q^{3}+12q^{2}-14q+15-13q^{-1}+10q^{-2}-6q^{-3}+2q^{-4}$ }

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]=

{10_60,}

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:

(-1, 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 $-4$ $8$ $8$ $-{\frac {14}{3}}$ $-{\frac {10}{3}}$ $-32$ $-{\frac {112}{3}}$ $-{\frac {160}{3}}$ $40$ $-{\frac {32}{3}}$ $32$ ${\frac {56}{3}}$ ${\frac {40}{3}}$ ${\frac {2849}{30}}$ ${\frac {742}{15}}$ ${\frac {538}{45}}$ $-{\frac {161}{18}}$ $-{\frac {511}{30}}$

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 K11n165. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 11                   1 -1 9                 3   3 7               5 1   -4 5             7 3     4 3           7 5       -2 1         8 7         1 -1       6 8           2 -3     4 7             -3 -5   2 6               4 -7   4                 -4 -9 2                   2

Integral Khovanov Homology (db, data source)

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

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.

K11n164 K11n166