K11a83

K11a82

K11a84

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a83 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X12,5,13,6 X16,8,17,7 X2,10,3,9 X22,11,1,12 X18,14,19,13 X20,16,21,15 X8,18,9,17 X14,20,15,19 X6,21,7,22
Gauss code	1, -5, 2, -1, 3, -11, 4, -9, 5, -2, 6, -3, 7, -10, 8, -4, 9, -7, 10, -8, 11, -6
Dowker-Thistlethwaite code	4 10 12 16 2 22 18 20 8 14 6

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{2,3\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a83/ThurstonBennequinNumber Hyperbolic Volume 14.7703 A-Polynomial See Data:K11a83/A-polynomial

[edit Notes for K11a83's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a83's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+14t^{2}-23t+27-23t^{-1}+14t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+4z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 113, 4 }
Jones polynomial	$q^{10}-4q^{9}+8q^{8}-13q^{7}+16q^{6}-18q^{5}+18q^{4}-14q^{3}+11q^{2}-6q+3-q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-4}-z^{6}a^{-2}+6z^{6}a^{-4}-2z^{6}a^{-6}-4z^{4}a^{-2}+15z^{4}a^{-4}-8z^{4}a^{-6}+z^{4}a^{-8}-5z^{2}a^{-2}+18z^{2}a^{-4}-11z^{2}a^{-6}+2z^{2}a^{-8}-2a^{-2}+8a^{-4}-6a^{-6}+a^{-8}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-4}+z^{10}a^{-6}+3z^{9}a^{-3}+8z^{9}a^{-5}+5z^{9}a^{-7}+3z^{8}a^{-2}+9z^{8}a^{-4}+16z^{8}a^{-6}+10z^{8}a^{-8}+z^{7}a^{-1}-6z^{7}a^{-3}-13z^{7}a^{-5}+5z^{7}a^{-7}+11z^{7}a^{-9}-12z^{6}a^{-2}-41z^{6}a^{-4}-51z^{6}a^{-6}-14z^{6}a^{-8}+8z^{6}a^{-10}-4z^{5}a^{-1}-6z^{5}a^{-3}-15z^{5}a^{-5}-32z^{5}a^{-7}-15z^{5}a^{-9}+4z^{5}a^{-11}+16z^{4}a^{-2}+52z^{4}a^{-4}+50z^{4}a^{-6}+6z^{4}a^{-8}-7z^{4}a^{-10}+z^{4}a^{-12}+5z^{3}a^{-1}+16z^{3}a^{-3}+31z^{3}a^{-5}+29z^{3}a^{-7}+7z^{3}a^{-9}-2z^{3}a^{-11}-9z^{2}a^{-2}-29z^{2}a^{-4}-24z^{2}a^{-6}-3z^{2}a^{-8}+z^{2}a^{-10}-2za^{-1}-7za^{-3}-13za^{-5}-9za^{-7}-za^{-9}+2a^{-2}+8a^{-4}+6a^{-6}+a^{-8}$
The A2 invariant	$-q^{2}+1-2q^{-2}+q^{-4}+2q^{-6}-q^{-8}+6q^{-10}-q^{-12}+3q^{-14}-3q^{-18}+q^{-20}-4q^{-22}+q^{-24}-q^{-28}+q^{-30}$
The G2 invariant	$q^{12}-2q^{10}+6q^{8}-11q^{6}+14q^{4}-16q^{2}+5+17q^{-2}-48q^{-4}+80q^{-6}-97q^{-8}+78q^{-10}-21q^{-12}-76q^{-14}+181q^{-16}-251q^{-18}+249q^{-20}-156q^{-22}-17q^{-24}+210q^{-26}-357q^{-28}+399q^{-30}-297q^{-32}+94q^{-34}+147q^{-36}-324q^{-38}+373q^{-40}-270q^{-42}+74q^{-44}+147q^{-46}-278q^{-48}+274q^{-50}-123q^{-52}-99q^{-54}+309q^{-56}-392q^{-58}+321q^{-60}-100q^{-62}-190q^{-64}+437q^{-66}-552q^{-68}+482q^{-70}-243q^{-72}-79q^{-74}+363q^{-76}-518q^{-78}+481q^{-80}-289q^{-82}+13q^{-84}+217q^{-86}-334q^{-88}+287q^{-90}-121q^{-92}-84q^{-94}+234q^{-96}-259q^{-98}+151q^{-100}+31q^{-102}-219q^{-104}+327q^{-106}-315q^{-108}+199q^{-110}-15q^{-112}-164q^{-114}+283q^{-116}-310q^{-118}+249q^{-120}-130q^{-122}-q^{-124}+106q^{-126}-168q^{-128}+173q^{-130}-136q^{-132}+81q^{-134}-17q^{-136}-29q^{-138}+53q^{-140}-62q^{-142}+51q^{-144}-32q^{-146}+15q^{-148}+q^{-150}-8q^{-152}+10q^{-154}-10q^{-156}+6q^{-158}-3q^{-160}+q^{-162}$

Further Quantum Invariants

K11a83 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["K11a83"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+14t^{2}-23t+27-23t^{-1}+14t^{-2}-5t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 113, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-4}-z^{6}a^{-2}+6z^{6}a^{-4}-2z^{6}a^{-6}-4z^{4}a^{-2}+15z^{4}a^{-4}-8z^{4}a^{-6}+z^{4}a^{-8}-5z^{2}a^{-2}+18z^{2}a^{-4}-11z^{2}a^{-6}+2z^{2}a^{-8}-2a^{-2}+8a^{-4}-6a^{-6}+a^{-8}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-4}+z^{10}a^{-6}+3z^{9}a^{-3}+8z^{9}a^{-5}+5z^{9}a^{-7}+3z^{8}a^{-2}+9z^{8}a^{-4}+16z^{8}a^{-6}+10z^{8}a^{-8}+z^{7}a^{-1}-6z^{7}a^{-3}-13z^{7}a^{-5}+5z^{7}a^{-7}+11z^{7}a^{-9}-12z^{6}a^{-2}-41z^{6}a^{-4}-51z^{6}a^{-6}-14z^{6}a^{-8}+8z^{6}a^{-10}-4z^{5}a^{-1}-6z^{5}a^{-3}-15z^{5}a^{-5}-32z^{5}a^{-7}-15z^{5}a^{-9}+4z^{5}a^{-11}+16z^{4}a^{-2}+52z^{4}a^{-4}+50z^{4}a^{-6}+6z^{4}a^{-8}-7z^{4}a^{-10}+z^{4}a^{-12}+5z^{3}a^{-1}+16z^{3}a^{-3}+31z^{3}a^{-5}+29z^{3}a^{-7}+7z^{3}a^{-9}-2z^{3}a^{-11}-9z^{2}a^{-2}-29z^{2}a^{-4}-24z^{2}a^{-6}-3z^{2}a^{-8}+z^{2}a^{-10}-2za^{-1}-7za^{-3}-13za^{-5}-9za^{-7}-za^{-9}+2a^{-2}+8a^{-4}+6a^{-6}+a^{-8}$

"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["K11a83"];

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]=

{ $t^{4}-5t^{3}+14t^{2}-23t+27-23t^{-1}+14t^{-2}-5t^{-3}+t^{-4}$, $q^{10}-4q^{9}+8q^{8}-13q^{7}+16q^{6}-18q^{5}+18q^{4}-14q^{3}+11q^{2}-6q+3-q^{-1}$ }

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:

(4, 6)

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 $16$ $48$ $128$ ${\frac {680}{3}}$ ${\frac {88}{3}}$ $768$ $1152$ $224$ $112$ ${\frac {2048}{3}}$ $1152$ ${\frac {10880}{3}}$ ${\frac {1408}{3}}$ ${\frac {90422}{15}}$ ${\frac {2344}{5}}$ ${\frac {87608}{45}}$ ${\frac {58}{9}}$ ${\frac {3782}{15}}$

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 K11a83. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a82 K11a84