K11n182

K11n181

K11n183

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n182 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X3,12,4,13 X16,5,17,6 X18,8,19,7 X14,10,15,9 X11,4,12,5 X20,13,21,14 X22,16,1,15 X2,17,3,18 X8,20,9,19 X10,21,11,22
Gauss code	1, -9, -2, 6, 3, -1, 4, -10, 5, -11, -6, 2, 7, -5, 8, -3, 9, -4, 10, -7, 11, -8
Dowker-Thistlethwaite code	6 -12 16 18 14 -4 20 22 2 8 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n182/ThurstonBennequinNumber Hyperbolic Volume 16.3897 A-Polynomial See Data:K11n182/A-polynomial

[edit Notes for K11n182's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant 0

[edit Notes for K11n182's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+11t^{2}-18t+23-18t^{-1}+11t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 93, 0 }
Jones polynomial	$-q^{5}+4q^{4}-8q^{3}+13q^{2}-15q+16-15q^{-1}+11q^{-2}-7q^{-3}+3q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+5z^{6}-4a^{2}z^{4}-3z^{4}a^{-2}+8z^{4}+a^{4}z^{2}-6a^{2}z^{2}-2z^{2}a^{-2}+4z^{2}+2a^{4}-3a^{2}+a^{-2}+1$
Kauffman polynomial (db, data sources)	$3az^{9}+3z^{9}a^{-1}+6a^{2}z^{8}+7z^{8}a^{-2}+13z^{8}+3a^{3}z^{7}+2az^{7}+6z^{7}a^{-1}+7z^{7}a^{-3}-14a^{2}z^{6}-9z^{6}a^{-2}+4z^{6}a^{-4}-27z^{6}-7az^{5}-19z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}+6a^{4}z^{4}+23a^{2}z^{4}-6z^{4}a^{-4}+23z^{4}-5a^{3}z^{3}+2az^{3}+12z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-9a^{4}z^{2}-18a^{2}z^{2}+2z^{2}a^{-2}+2z^{2}a^{-4}-9z^{2}+3a^{3}z+2az-2za^{-1}-za^{-3}+2a^{4}+3a^{2}-a^{-2}+1$
The A2 invariant	$q^{16}+3q^{12}-2q^{10}+q^{8}-q^{6}-4q^{4}+3q^{2}-4+4q^{-2}-q^{-4}+q^{-6}+3q^{-8}-2q^{-10}+2q^{-12}-q^{-14}$
The G2 invariant	Data:K11n182/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n182 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["K11n182"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+11t^{2}-18t+23-18t^{-1}+11t^{-2}-5t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 93, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+4q^{4}-8q^{3}+13q^{2}-15q+16-15q^{-1}+11q^{-2}-7q^{-3}+3q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+5z^{6}-4a^{2}z^{4}-3z^{4}a^{-2}+8z^{4}+a^{4}z^{2}-6a^{2}z^{2}-2z^{2}a^{-2}+4z^{2}+2a^{4}-3a^{2}+a^{-2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3az^{9}+3z^{9}a^{-1}+6a^{2}z^{8}+7z^{8}a^{-2}+13z^{8}+3a^{3}z^{7}+2az^{7}+6z^{7}a^{-1}+7z^{7}a^{-3}-14a^{2}z^{6}-9z^{6}a^{-2}+4z^{6}a^{-4}-27z^{6}-7az^{5}-19z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}+6a^{4}z^{4}+23a^{2}z^{4}-6z^{4}a^{-4}+23z^{4}-5a^{3}z^{3}+2az^{3}+12z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-9a^{4}z^{2}-18a^{2}z^{2}+2z^{2}a^{-2}+2z^{2}a^{-4}-9z^{2}+3a^{3}z+2az-2za^{-1}-za^{-3}+2a^{4}+3a^{2}-a^{-2}+1$

"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["K11n182"];

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]=

{ $t^{4}-5t^{3}+11t^{2}-18t+23-18t^{-1}+11t^{-2}-5t^{-3}+t^{-4}$, $-q^{5}+4q^{4}-8q^{3}+13q^{2}-15q+16-15q^{-1}+11q^{-2}-7q^{-3}+3q^{-4}$ }

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, 2)

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 $-12$ $16$ $72$ $82$ $30$ $-192$ $-{\frac {992}{3}}$ $-{\frac {224}{3}}$ $-80$ $-288$ $128$ $-984$ $-360$ $-{\frac {5471}{10}}$ $-{\frac {1174}{15}}$ $-{\frac {4462}{15}}$ ${\frac {287}{6}}$ $-{\frac {671}{10}}$

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=$0 is the signature of K11n182. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 11                   1 -1 9                 3   3 7               5 1   -4 5             8 3     5 3           7 5       -2 1         9 8         1 -1       7 8           1 -3     4 8             -4 -5   3 7               4 -7   4                 -4 -9 3                   3

Integral Khovanov Homology (db, data source)

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

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.

K11n181 K11n183