K11n169

K11n168

K11n170

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n169 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X3,11,4,10 X5,17,6,16 X12,8,13,7 X9,19,10,18 X11,3,12,2 X20,14,21,13 X22,16,1,15 X17,5,18,4 X19,9,20,8 X14,22,15,21
Gauss code	1, 6, -2, 9, -3, -1, 4, 10, -5, 2, -6, -4, 7, -11, 8, 3, -9, 5, -10, -7, 11, -8
Dowker-Thistlethwaite code	6 -10 -16 12 -18 -2 20 22 -4 -8 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{3,4\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n169/ThurstonBennequinNumber Hyperbolic Volume 10.8173 A-Polynomial See Data:K11n169/A-polynomial

[edit Notes for K11n169's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -6

[edit Notes for K11n169's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{3}-6t^{2}+6t-5+6t^{-1}-6t^{-2}+3t^{-3}$
Conway polynomial	$3z^{6}+12z^{4}+9z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 35, 6 }
Jones polynomial	$-q^{12}+2q^{11}-4q^{10}+5q^{9}-6q^{8}+6q^{7}-4q^{6}+4q^{5}-2q^{4}+q^{3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-6}+2z^{6}a^{-8}+4z^{4}a^{-6}+10z^{4}a^{-8}-2z^{4}a^{-10}+3z^{2}a^{-6}+14z^{2}a^{-8}-8z^{2}a^{-10}+6a^{-8}-6a^{-10}+a^{-12}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-9}+z^{9}a^{-11}+3z^{8}a^{-8}+4z^{8}a^{-10}+z^{8}a^{-12}+2z^{7}a^{-7}-z^{7}a^{-9}-3z^{7}a^{-11}+z^{6}a^{-6}-14z^{6}a^{-8}-18z^{6}a^{-10}-3z^{6}a^{-12}-7z^{5}a^{-7}-7z^{5}a^{-9}+3z^{5}a^{-11}+3z^{5}a^{-13}-4z^{4}a^{-6}+22z^{4}a^{-8}+30z^{4}a^{-10}+6z^{4}a^{-12}+2z^{4}a^{-14}+4z^{3}a^{-7}+11z^{3}a^{-9}+2z^{3}a^{-11}-4z^{3}a^{-13}+z^{3}a^{-15}+3z^{2}a^{-6}-18z^{2}a^{-8}-22z^{2}a^{-10}-3z^{2}a^{-12}-2z^{2}a^{-14}-5za^{-9}-2za^{-11}+2za^{-13}-za^{-15}+6a^{-8}+6a^{-10}+a^{-12}$
The A2 invariant	Data:K11n169/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n169/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n169 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["K11n169"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{3}-6t^{2}+6t-5+6t^{-1}-6t^{-2}+3t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$3z^{6}+12z^{4}+9z^{2}+1$

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]=

{ 35, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-6}+2z^{6}a^{-8}+4z^{4}a^{-6}+10z^{4}a^{-8}-2z^{4}a^{-10}+3z^{2}a^{-6}+14z^{2}a^{-8}-8z^{2}a^{-10}+6a^{-8}-6a^{-10}+a^{-12}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-9}+z^{9}a^{-11}+3z^{8}a^{-8}+4z^{8}a^{-10}+z^{8}a^{-12}+2z^{7}a^{-7}-z^{7}a^{-9}-3z^{7}a^{-11}+z^{6}a^{-6}-14z^{6}a^{-8}-18z^{6}a^{-10}-3z^{6}a^{-12}-7z^{5}a^{-7}-7z^{5}a^{-9}+3z^{5}a^{-11}+3z^{5}a^{-13}-4z^{4}a^{-6}+22z^{4}a^{-8}+30z^{4}a^{-10}+6z^{4}a^{-12}+2z^{4}a^{-14}+4z^{3}a^{-7}+11z^{3}a^{-9}+2z^{3}a^{-11}-4z^{3}a^{-13}+z^{3}a^{-15}+3z^{2}a^{-6}-18z^{2}a^{-8}-22z^{2}a^{-10}-3z^{2}a^{-12}-2z^{2}a^{-14}-5za^{-9}-2za^{-11}+2za^{-13}-za^{-15}+6a^{-8}+6a^{-10}+a^{-12}$

"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["K11n169"];

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]=

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

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:

(9, 25)

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 $36$ $200$ $648$ $1482$ $222$ $7200$ ${\frac {36464}{3}}$ ${\frac {6368}{3}}$ $1544$ $7776$ $20000$ $53352$ $7992$ ${\frac {1019133}{10}}$ ${\frac {17854}{5}}$ ${\frac {190502}{5}}$ ${\frac {1313}{2}}$ ${\frac {49213}{10}}$

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=$6 is the signature of K11n169. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 χ 25                   1 -1 23                 1   1 21               3 1   -2 19             2 1     1 17           4 3       -1 15         2 2         0 13       2 4           2 11     2 2             0 9     2               2 7 1 2                 -1 5 1                   1

Integral Khovanov Homology (db, data source)

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

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.

K11n168 K11n170