K11n159

K11n158

K11n160

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n159 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X10,3,11,4 X12,6,13,5 X7,18,8,19 X9,20,10,21 X16,11,17,12 X22,13,1,14 X4,16,5,15 X2,17,3,18 X19,8,20,9 X14,21,15,22
Gauss code	1, -9, 2, -8, 3, -1, -4, 10, -5, -2, 6, -3, 7, -11, 8, -6, 9, 4, -10, 5, 11, -7
Dowker-Thistlethwaite code	6 10 12 -18 -20 16 22 4 2 -8 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n159/ThurstonBennequinNumber Hyperbolic Volume 15.181 A-Polynomial See Data:K11n159/A-polynomial

[edit Notes for K11n159's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for K11n159's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-6t^{2}+17t-23+17t^{-1}-6t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 71, -2 }
Jones polynomial	$-1+5q^{-1}-8q^{-2}+11q^{-3}-12q^{-4}+12q^{-5}-10q^{-6}+7q^{-7}-4q^{-8}+q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{8}-2z^{4}a^{6}-3z^{2}a^{6}-a^{6}+z^{6}a^{4}+3z^{4}a^{4}+4z^{2}a^{4}+a^{4}-z^{4}a^{2}+a^{2}$
Kauffman polynomial (db, data sources)	$z^{6}a^{10}-2z^{4}a^{10}+z^{2}a^{10}+4z^{7}a^{9}-11z^{5}a^{9}+7z^{3}a^{9}+5z^{8}a^{8}-12z^{6}a^{8}+4z^{4}a^{8}+z^{2}a^{8}+2z^{9}a^{7}+4z^{7}a^{7}-21z^{5}a^{7}+13z^{3}a^{7}-2za^{7}+9z^{8}a^{6}-21z^{6}a^{6}+13z^{4}a^{6}-5z^{2}a^{6}+a^{6}+2z^{9}a^{5}+2z^{7}a^{5}-9z^{5}a^{5}+6z^{3}a^{5}-2za^{5}+4z^{8}a^{4}-8z^{6}a^{4}+12z^{4}a^{4}-6z^{2}a^{4}+a^{4}+2z^{7}a^{3}+z^{5}a^{3}+z^{3}a^{3}+5z^{4}a^{2}-z^{2}a^{2}-a^{2}+z^{3}a$
The A2 invariant	$q^{28}-q^{26}-2q^{24}+2q^{22}-2q^{20}+q^{18}+q^{16}-2q^{14}+2q^{12}-2q^{10}+3q^{8}+q^{6}-q^{4}+3q^{2}-1$
The G2 invariant	Data:K11n159/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n159 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["K11n159"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-6t^{2}+17t-23+17t^{-1}-6t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 71, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{6}a^{10}-2z^{4}a^{10}+z^{2}a^{10}+4z^{7}a^{9}-11z^{5}a^{9}+7z^{3}a^{9}+5z^{8}a^{8}-12z^{6}a^{8}+4z^{4}a^{8}+z^{2}a^{8}+2z^{9}a^{7}+4z^{7}a^{7}-21z^{5}a^{7}+13z^{3}a^{7}-2za^{7}+9z^{8}a^{6}-21z^{6}a^{6}+13z^{4}a^{6}-5z^{2}a^{6}+a^{6}+2z^{9}a^{5}+2z^{7}a^{5}-9z^{5}a^{5}+6z^{3}a^{5}-2za^{5}+4z^{8}a^{4}-8z^{6}a^{4}+12z^{4}a^{4}-6z^{2}a^{4}+a^{4}+2z^{7}a^{3}+z^{5}a^{3}+z^{3}a^{3}+5z^{4}a^{2}-z^{2}a^{2}-a^{2}+z^{3}a$

"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["K11n159"];

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]=

{ $t^{3}-6t^{2}+17t-23+17t^{-1}-6t^{-2}+t^{-3}$, $-1+5q^{-1}-8q^{-2}+11q^{-3}-12q^{-4}+12q^{-5}-10q^{-6}+7q^{-7}-4q^{-8}+q^{-9}$ }

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]=

{}

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:

(2, -3)

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 $8$ $-24$ $32$ ${\frac {268}{3}}$ ${\frac {44}{3}}$ $-192$ $-368$ $-96$ $-24$ ${\frac {256}{3}}$ $288$ ${\frac {2144}{3}}$ ${\frac {352}{3}}$ ${\frac {22951}{15}}$ ${\frac {1252}{5}}$ ${\frac {21124}{45}}$ $-{\frac {439}{9}}$ ${\frac {871}{15}}$

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 $s=$-2 is the signature of K11n159. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n158 K11n160