K11n157

K11n156

K11n158

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n157 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X3,11,4,10 X5,12,6,13 X14,7,15,8 X9,16,10,17 X11,19,12,18 X22,13,1,14 X20,16,21,15 X17,4,18,5 X19,3,20,2 X8,21,9,22
Gauss code	1, 10, -2, 9, -3, -1, 4, -11, -5, 2, -6, 3, 7, -4, 8, 5, -9, 6, -10, -8, 11, -7
Dowker-Thistlethwaite code	6 -10 -12 14 -16 -18 22 20 -4 -2 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:K11n157/ThurstonBennequinNumber Hyperbolic Volume 14.7471 A-Polynomial See Data:K11n157/A-polynomial

[edit Notes for K11n157'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 K11n157's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+6t^{2}-15t+21-15t^{-1}+6t^{-2}-t^{-3}$
Conway polynomial	$1-z^{6}$
2nd Alexander ideal (db, data sources)	$\left\{3,t^{2}+1\right\}$
Determinant and Signature	{ 65, 0 }
Jones polynomial	$-q^{3}+4q^{2}-7q+10-11q^{-1}+11q^{-2}-9q^{-3}+7q^{-4}-4q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{6}+a^{4}z^{4}-3a^{2}z^{4}+2z^{4}+a^{4}z^{2}-3a^{2}z^{2}-z^{2}a^{-2}+3z^{2}+1$
Kauffman polynomial (db, data sources)	$2a^{3}z^{9}+2az^{9}+5a^{4}z^{8}+8a^{2}z^{8}+3z^{8}+4a^{5}z^{7}+2a^{3}z^{7}-az^{7}+z^{7}a^{-1}+a^{6}z^{6}-13a^{4}z^{6}-20a^{2}z^{6}-6z^{6}-11a^{5}z^{5}-16a^{3}z^{5}-2az^{5}+3z^{5}a^{-1}-2a^{6}z^{4}+5a^{4}z^{4}+13a^{2}z^{4}+4z^{4}a^{-2}+10z^{4}+6a^{5}z^{3}+9a^{3}z^{3}+az^{3}-z^{3}a^{-1}+z^{3}a^{-3}+a^{6}z^{2}+a^{4}z^{2}-3a^{2}z^{2}-2z^{2}a^{-2}-5z^{2}+1$
The A2 invariant	$q^{18}-2q^{16}+q^{14}-2q^{10}+3q^{8}-q^{6}+2q^{4}-1+2q^{-2}-2q^{-4}+2q^{-6}+q^{-8}-q^{-10}$
The G2 invariant	Data:K11n157/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n157 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["K11n157"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+6t^{2}-15t+21-15t^{-1}+6t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$1-z^{6}$

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]=

$\left\{3,t^{2}+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 65, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{6}+a^{4}z^{4}-3a^{2}z^{4}+2z^{4}+a^{4}z^{2}-3a^{2}z^{2}-z^{2}a^{-2}+3z^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{3}z^{9}+2az^{9}+5a^{4}z^{8}+8a^{2}z^{8}+3z^{8}+4a^{5}z^{7}+2a^{3}z^{7}-az^{7}+z^{7}a^{-1}+a^{6}z^{6}-13a^{4}z^{6}-20a^{2}z^{6}-6z^{6}-11a^{5}z^{5}-16a^{3}z^{5}-2az^{5}+3z^{5}a^{-1}-2a^{6}z^{4}+5a^{4}z^{4}+13a^{2}z^{4}+4z^{4}a^{-2}+10z^{4}+6a^{5}z^{3}+9a^{3}z^{3}+az^{3}-z^{3}a^{-1}+z^{3}a^{-3}+a^{6}z^{2}+a^{4}z^{2}-3a^{2}z^{2}-2z^{2}a^{-2}-5z^{2}+1$

"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["K11n157"];

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]=

{ $-t^{3}+6t^{2}-15t+21-15t^{-1}+6t^{-2}-t^{-3}$, $-q^{3}+4q^{2}-7q+10-11q^{-1}+11q^{-2}-9q^{-3}+7q^{-4}-4q^{-5}+q^{-6}$ }

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:

(0, 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 $0$ $0$ $0$ $0$ $0$ $0$ $0$ $-32$ $32$ $0$ $0$ $0$ $0$ $0$ $128$ $-128$ $0$ $-32$

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 K11n157. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n156 K11n158