K11n185

K11n184

K12a1

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n185 at Knotilus!

Knot K11n185.

A graph, K11n185.

A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation	X6271 X18,4,19,3 X16,5,17,6 X14,8,15,7 X9,21,10,20 X4,12,5,11 X2,13,3,14 X22,16,1,15 X12,18,13,17 X19,9,20,8 X21,11,22,10
Gauss code	1, -7, 2, -6, 3, -1, 4, 10, -5, 11, 6, -9, 7, -4, 8, -3, 9, -2, -10, 5, -11, -8
Dowker-Thistlethwaite code	6 18 16 14 -20 4 2 22 12 -8 -10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n185/ThurstonBennequinNumber Hyperbolic Volume 17.1 A-Polynomial See Data:K11n185/A-polynomial

[edit Notes for K11n185's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n185's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+11t^{2}-24t+31-24t^{-1}+11t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{3,t^{2}+1\right\}$
Determinant and Signature	{ 105, 4 }
Jones polynomial	$-q^{11}+5q^{10}-10q^{9}+14q^{8}-18q^{7}+18q^{6}-16q^{5}+13q^{4}-7q^{3}+3q^{2}$
HOMFLY-PT polynomial (db, data sources)	$-2z^{6}a^{-6}+3z^{4}a^{-4}-7z^{4}a^{-6}+3z^{4}a^{-8}+7z^{2}a^{-4}-9z^{2}a^{-6}+5z^{2}a^{-8}-z^{2}a^{-10}+4a^{-4}-4a^{-6}+a^{-8}$
Kauffman polynomial (db, data sources)	$3z^{9}a^{-7}+3z^{9}a^{-9}+6z^{8}a^{-6}+15z^{8}a^{-8}+9z^{8}a^{-10}+3z^{7}a^{-5}+7z^{7}a^{-7}+14z^{7}a^{-9}+10z^{7}a^{-11}-10z^{6}a^{-6}-24z^{6}a^{-8}-9z^{6}a^{-10}+5z^{6}a^{-12}-18z^{5}a^{-7}-35z^{5}a^{-9}-16z^{5}a^{-11}+z^{5}a^{-13}+6z^{4}a^{-4}+16z^{4}a^{-6}+10z^{4}a^{-8}-5z^{4}a^{-10}-5z^{4}a^{-12}+z^{3}a^{-5}+12z^{3}a^{-7}+16z^{3}a^{-9}+5z^{3}a^{-11}-9z^{2}a^{-4}-14z^{2}a^{-6}-3z^{2}a^{-8}+2z^{2}a^{-10}-3za^{-5}-3za^{-7}+za^{-9}+za^{-11}+4a^{-4}+4a^{-6}+a^{-8}$
The A2 invariant	$3q^{-6}-2q^{-8}+4q^{-10}+q^{-12}-2q^{-14}+4q^{-16}-4q^{-18}+2q^{-20}-3q^{-22}-2q^{-24}+2q^{-26}-3q^{-28}+3q^{-30}+q^{-32}-q^{-34}$
The G2 invariant	Data:K11n185/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n185 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["K11n185"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+11t^{2}-24t+31-24t^{-1}+11t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

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

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 105, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{11}+5q^{10}-10q^{9}+14q^{8}-18q^{7}+18q^{6}-16q^{5}+13q^{4}-7q^{3}+3q^{2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-2z^{6}a^{-6}+3z^{4}a^{-4}-7z^{4}a^{-6}+3z^{4}a^{-8}+7z^{2}a^{-4}-9z^{2}a^{-6}+5z^{2}a^{-8}-z^{2}a^{-10}+4a^{-4}-4a^{-6}+a^{-8}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3z^{9}a^{-7}+3z^{9}a^{-9}+6z^{8}a^{-6}+15z^{8}a^{-8}+9z^{8}a^{-10}+3z^{7}a^{-5}+7z^{7}a^{-7}+14z^{7}a^{-9}+10z^{7}a^{-11}-10z^{6}a^{-6}-24z^{6}a^{-8}-9z^{6}a^{-10}+5z^{6}a^{-12}-18z^{5}a^{-7}-35z^{5}a^{-9}-16z^{5}a^{-11}+z^{5}a^{-13}+6z^{4}a^{-4}+16z^{4}a^{-6}+10z^{4}a^{-8}-5z^{4}a^{-10}-5z^{4}a^{-12}+z^{3}a^{-5}+12z^{3}a^{-7}+16z^{3}a^{-9}+5z^{3}a^{-11}-9z^{2}a^{-4}-14z^{2}a^{-6}-3z^{2}a^{-8}+2z^{2}a^{-10}-3za^{-5}-3za^{-7}+za^{-9}+za^{-11}+4a^{-4}+4a^{-6}+a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_122,}

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["K11n185"];

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]=

{ $-2t^{3}+11t^{2}-24t+31-24t^{-1}+11t^{-2}-2t^{-3}$, $-q^{11}+5q^{10}-10q^{9}+14q^{8}-18q^{7}+18q^{6}-16q^{5}+13q^{4}-7q^{3}+3q^{2}$ }

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]=

{10_122,}

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:

(2, 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 $8$ $16$ $32$ ${\frac {172}{3}}$ ${\frac {68}{3}}$ $128$ ${\frac {736}{3}}$ ${\frac {160}{3}}$ $112$ ${\frac {256}{3}}$ $128$ ${\frac {1376}{3}}$ ${\frac {544}{3}}$ ${\frac {14791}{15}}$ $-{\frac {2148}{5}}$ ${\frac {51244}{45}}$ ${\frac {761}{9}}$ ${\frac {2311}{15}}$

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 K11n185. 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 χ 23                   1 -1 21                 4   4 19               6 1   -5 17             8 4     4 15           10 6       -4 13         8 8         0 11       8 10           2 9     5 8             -3 7   2 8               6 5 1 5                 -4 3 3                   3

Integral Khovanov Homology (db, data source)

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

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.

K11n184 K12a1