K11n155

K11n154

K11n156

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n155 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X8394 X12,6,13,5 X20,8,21,7 X18,9,19,10 X16,11,17,12 X13,1,14,22 X4,16,5,15 X10,17,11,18 X2,19,3,20 X21,15,22,14
Gauss code	1, -10, 2, -8, 3, -1, 4, -2, 5, -9, 6, -3, -7, 11, 8, -6, 9, -5, 10, -4, -11, 7
Dowker-Thistlethwaite code	6 8 12 20 18 16 -22 4 10 2 -14

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:K11n155/ThurstonBennequinNumber Hyperbolic Volume 12.3848 A-Polynomial See Data:K11n155/A-polynomial

[edit Notes for K11n155'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 K11n155's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-8t^{2}+11t-9+11t^{-1}-8t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+4z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 51, 2 }
Jones polynomial	$2q^{5}-4q^{4}+6q^{3}-8q^{2}+8q-8+7q^{-1}-4q^{-2}+3q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}-a^{2}z^{4}+3z^{4}a^{-2}-z^{4}a^{-4}+3z^{4}-2a^{2}z^{2}+z^{2}a^{-2}-3z^{2}a^{-4}+z^{2}+a^{2}-a^{-4}+a^{-6}$
Kauffman polynomial (db, data sources)	$2az^{9}+2z^{9}a^{-1}+3a^{2}z^{8}+4z^{8}a^{-2}+7z^{8}+a^{3}z^{7}-6az^{7}-3z^{7}a^{-1}+4z^{7}a^{-3}-14a^{2}z^{6}-10z^{6}a^{-2}+3z^{6}a^{-4}-27z^{6}-4a^{3}z^{5}+2az^{5}-z^{5}a^{-1}-6z^{5}a^{-3}+z^{5}a^{-5}+19a^{2}z^{4}+8z^{4}a^{-2}-3z^{4}a^{-4}+30z^{4}+4a^{3}z^{3}+az^{3}-7z^{3}a^{-1}-z^{3}a^{-3}+3z^{3}a^{-5}-6a^{2}z^{2}-6z^{2}a^{-2}+3z^{2}a^{-4}+3z^{2}a^{-6}-12z^{2}+2az+6za^{-1}+3za^{-3}-za^{-5}-a^{2}-a^{-4}-a^{-6}$
The A2 invariant	Data:K11n155/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n155/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n155 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["K11n155"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-8t^{2}+11t-9+11t^{-1}-8t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 51, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2az^{9}+2z^{9}a^{-1}+3a^{2}z^{8}+4z^{8}a^{-2}+7z^{8}+a^{3}z^{7}-6az^{7}-3z^{7}a^{-1}+4z^{7}a^{-3}-14a^{2}z^{6}-10z^{6}a^{-2}+3z^{6}a^{-4}-27z^{6}-4a^{3}z^{5}+2az^{5}-z^{5}a^{-1}-6z^{5}a^{-3}+z^{5}a^{-5}+19a^{2}z^{4}+8z^{4}a^{-2}-3z^{4}a^{-4}+30z^{4}+4a^{3}z^{3}+az^{3}-7z^{3}a^{-1}-z^{3}a^{-3}+3z^{3}a^{-5}-6a^{2}z^{2}-6z^{2}a^{-2}+3z^{2}a^{-4}+3z^{2}a^{-6}-12z^{2}+2az+6za^{-1}+3za^{-3}-za^{-5}-a^{2}-a^{-4}-a^{-6}$

"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["K11n155"];

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]=

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

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]=

{}

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:

(-3, -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 $-12$ $-24$ $72$ $34$ $6$ $288$ $368$ $96$ $40$ $-288$ $288$ $-408$ $-72$ ${\frac {4049}{10}}$ $-{\frac {2494}{15}}$ ${\frac {6938}{15}}$ $-{\frac {433}{6}}$ ${\frac {849}{10}}$

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=$2 is the signature of K11n155. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n154 K11n156