K11n183

K11n182

K11n184

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n183 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X3,15,4,14 X10,6,11,5 X7,19,8,18 X2,10,3,9 X22,11,1,12 X20,14,21,13 X15,5,16,4 X12,18,13,17 X19,9,20,8 X16,22,17,21
Gauss code	1, -5, -2, 8, 3, -1, -4, 10, 5, -3, 6, -9, 7, 2, -8, -11, 9, 4, -10, -7, 11, -6
Dowker-Thistlethwaite code	6 -14 10 -18 2 22 20 -4 12 -8 16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n183/ThurstonBennequinNumber Hyperbolic Volume 11.4109 A-Polynomial See Data:K11n183/A-polynomial

[edit Notes for K11n183's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -6

[edit Notes for K11n183's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}+t^{2}-6t+9-6t^{-1}+t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+7z^{4}+7z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{4,t^{2}+t+1\right\}$
Determinant and Signature	{ 21, 4 }
Jones polynomial	$q^{12}-3q^{11}+3q^{10}-4q^{9}+4q^{8}-3q^{7}+3q^{6}-q^{5}+q^{3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-6}+6z^{4}a^{-6}+z^{4}a^{-8}+8z^{2}a^{-6}+2z^{2}a^{-8}-3z^{2}a^{-10}+2a^{-6}+2a^{-8}-4a^{-10}+a^{-12}$
Kauffman polynomial (db, data sources)	$2z^{8}a^{-10}+2z^{8}a^{-12}+2z^{7}a^{-9}+5z^{7}a^{-11}+3z^{7}a^{-13}+z^{6}a^{-6}-z^{6}a^{-8}-10z^{6}a^{-10}-7z^{6}a^{-12}+z^{6}a^{-14}-z^{5}a^{-7}-11z^{5}a^{-9}-22z^{5}a^{-11}-12z^{5}a^{-13}-6z^{4}a^{-6}+4z^{4}a^{-8}+17z^{4}a^{-10}+4z^{4}a^{-12}-3z^{4}a^{-14}+2z^{3}a^{-7}+18z^{3}a^{-9}+26z^{3}a^{-11}+10z^{3}a^{-13}+8z^{2}a^{-6}-4z^{2}a^{-8}-14z^{2}a^{-10}-z^{2}a^{-12}+z^{2}a^{-14}-8za^{-9}-9za^{-11}-za^{-13}-2a^{-6}+2a^{-8}+4a^{-10}+a^{-12}$
The A2 invariant	$q^{-10}+q^{-12}+q^{-16}+q^{-18}+3q^{-22}+q^{-24}+q^{-26}-2q^{-28}-3q^{-30}-q^{-32}-2q^{-34}+q^{-36}+q^{-38}$
The G2 invariant	Data:K11n183/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n183 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["K11n183"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}+t^{2}-6t+9-6t^{-1}+t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}+7z^{4}+7z^{2}+1$

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]=

$\left\{4,t^{2}+t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 21, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{8}a^{-10}+2z^{8}a^{-12}+2z^{7}a^{-9}+5z^{7}a^{-11}+3z^{7}a^{-13}+z^{6}a^{-6}-z^{6}a^{-8}-10z^{6}a^{-10}-7z^{6}a^{-12}+z^{6}a^{-14}-z^{5}a^{-7}-11z^{5}a^{-9}-22z^{5}a^{-11}-12z^{5}a^{-13}-6z^{4}a^{-6}+4z^{4}a^{-8}+17z^{4}a^{-10}+4z^{4}a^{-12}-3z^{4}a^{-14}+2z^{3}a^{-7}+18z^{3}a^{-9}+26z^{3}a^{-11}+10z^{3}a^{-13}+8z^{2}a^{-6}-4z^{2}a^{-8}-14z^{2}a^{-10}-z^{2}a^{-12}+z^{2}a^{-14}-8za^{-9}-9za^{-11}-za^{-13}-2a^{-6}+2a^{-8}+4a^{-10}+a^{-12}$

"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["K11n183"];

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]=

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

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:

(7, 17)

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 $28$ $136$ $392$ ${\frac {2642}{3}}$ ${\frac {406}{3}}$ $3808$ ${\frac {18928}{3}}$ ${\frac {3424}{3}}$ $776$ ${\frac {10976}{3}}$ $9248$ ${\frac {73976}{3}}$ ${\frac {11368}{3}}$ ${\frac {1387657}{30}}$ ${\frac {31526}{15}}$ ${\frac {764234}{45}}$ ${\frac {2903}{18}}$ ${\frac {65737}{30}}$

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 K11n183. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 10 χ 25                     1 1 23                   2   -2 21                 1 1   0 19               3 2     -1 17           1 2 1       0 15           2 3         1 13       1 3 2           0 11         2             2 9     1 2               -1 7 1                     1 5 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $i=7$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=1$ $r=2$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=9$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=10$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11n182 K11n184