K11n158

K11n157

K11n159

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n158 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X10,3,11,4 X12,6,13,5 X18,8,19,7 X20,10,21,9 X16,11,17,12 X13,1,14,22 X4,16,5,15 X2,17,3,18 X8,20,9,19 X21,15,22,14
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 Reversible Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n158/ThurstonBennequinNumber Hyperbolic Volume 12.7254 A-Polynomial See Data:K11n158/A-polynomial

[edit Notes for K11n158's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n158's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-4t^{3}+7t^{2}-7t+7-7t^{-1}+7t^{-2}-4t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+4z^{6}+3z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 45, 4 }
Jones polynomial	$q^{8}-3q^{7}+5q^{6}-7q^{5}+7q^{4}-7q^{3}+7q^{2}-4q+3-q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-4}-z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-4z^{4}a^{-2}+12z^{4}a^{-4}-5z^{4}a^{-6}-3z^{2}a^{-2}+10z^{2}a^{-4}-7z^{2}a^{-6}+z^{2}a^{-8}+a^{-2}+2a^{-4}-3a^{-6}+a^{-8}$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-3}+2z^{9}a^{-5}+3z^{8}a^{-2}+7z^{8}a^{-4}+4z^{8}a^{-6}+z^{7}a^{-1}-6z^{7}a^{-3}-4z^{7}a^{-5}+3z^{7}a^{-7}-14z^{6}a^{-2}-29z^{6}a^{-4}-14z^{6}a^{-6}+z^{6}a^{-8}-4z^{5}a^{-1}+z^{5}a^{-3}-2z^{5}a^{-5}-7z^{5}a^{-7}+19z^{4}a^{-2}+35z^{4}a^{-4}+17z^{4}a^{-6}+z^{4}a^{-8}+4z^{3}a^{-1}+2z^{3}a^{-3}+5z^{3}a^{-7}+3z^{3}a^{-9}-7z^{2}a^{-2}-17z^{2}a^{-4}-12z^{2}a^{-6}-z^{2}a^{-8}+z^{2}a^{-10}+2za^{-3}+2za^{-5}-za^{-7}-za^{-9}-a^{-2}+2a^{-4}+3a^{-6}+a^{-8}$
The A2 invariant	$-q^{2}+1+2q^{-4}+2q^{-6}+q^{-8}+2q^{-10}-3q^{-12}+q^{-14}-2q^{-16}-q^{-22}+q^{-24}-q^{-26}+q^{-28}$
The G2 invariant	Data:K11n158/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n158 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["K11n158"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-4t^{3}+7t^{2}-7t+7-7t^{-1}+7t^{-2}-4t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+4z^{6}+3z^{4}+z^{2}+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 45, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{8}-3q^{7}+5q^{6}-7q^{5}+7q^{4}-7q^{3}+7q^{2}-4q+3-q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-4}-z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-4z^{4}a^{-2}+12z^{4}a^{-4}-5z^{4}a^{-6}-3z^{2}a^{-2}+10z^{2}a^{-4}-7z^{2}a^{-6}+z^{2}a^{-8}+a^{-2}+2a^{-4}-3a^{-6}+a^{-8}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{9}a^{-3}+2z^{9}a^{-5}+3z^{8}a^{-2}+7z^{8}a^{-4}+4z^{8}a^{-6}+z^{7}a^{-1}-6z^{7}a^{-3}-4z^{7}a^{-5}+3z^{7}a^{-7}-14z^{6}a^{-2}-29z^{6}a^{-4}-14z^{6}a^{-6}+z^{6}a^{-8}-4z^{5}a^{-1}+z^{5}a^{-3}-2z^{5}a^{-5}-7z^{5}a^{-7}+19z^{4}a^{-2}+35z^{4}a^{-4}+17z^{4}a^{-6}+z^{4}a^{-8}+4z^{3}a^{-1}+2z^{3}a^{-3}+5z^{3}a^{-7}+3z^{3}a^{-9}-7z^{2}a^{-2}-17z^{2}a^{-4}-12z^{2}a^{-6}-z^{2}a^{-8}+z^{2}a^{-10}+2za^{-3}+2za^{-5}-za^{-7}-za^{-9}-a^{-2}+2a^{-4}+3a^{-6}+a^{-8}$

"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["K11n158"];

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]=

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

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:

(1, 0)

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 $4$ $0$ $8$ $-{\frac {130}{3}}$ $-{\frac {62}{3}}$ $0$ $-224$ $-32$ $-96$ ${\frac {32}{3}}$ $0$ $-{\frac {520}{3}}$ $-{\frac {248}{3}}$ $-{\frac {23249}{30}}$ ${\frac {2978}{15}}$ $-{\frac {25378}{45}}$ $-{\frac {2287}{18}}$ $-{\frac {2129}{30}}$

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=$4 is the signature of K11n158. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n157 K11n159