K11a83

K11a82

K11a84

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	K11a83 Quick Notes

K11a83 Further Notes and Views

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
Conway Notation	[21111,3,2]

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]=

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

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)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[11, Alternating, 83]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 83]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[12, 5, 13, 6], X[16, 8, 17, 7], X[2, 10, 3, 9], X[22, 11, 1, 12], X[18, 14, 19, 13], X[20, 16, 21, 15], X[8, 18, 9, 17], X[14, 20, 15, 19], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 83]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -11, 4, -9, 5, -2, 6, -3, 7, -10, 8, -4, 9, -7, 10, -8, 11, -6]
In[5]:=	BR[Knot[11, Alternating, 83]]
Out[5]=	BR[Knot[11, Alternating, 83]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 83]][t]
Out[6]=	-4 5 14 23 2 3 4 27 + t - -- + -- - -- - 23 t + 14 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 83]][z]
Out[7]=	2 4 6 8 1 + 4 z + 4 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 83]}
In[9]:=	{KnotDet[Knot[11, Alternating, 83]], KnotSignature[Knot[11, Alternating, 83]]}
Out[9]=	{113, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 83]][q]
Out[10]=	1 2 3 4 5 6 7 8 3 - - - 6 q + 11 q - 14 q + 18 q - 18 q + 16 q - 13 q + 8 q - q 9 10 4 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 83]}
In[12]:=	A2Invariant[Knot[11, Alternating, 83]][q]
Out[12]=	-2 2 4 6 8 10 12 14 18 20 1 - q - 2 q + q + 2 q - q + 6 q - q + 3 q - 3 q + q - 22 24 28 30 4 q + q - q + q
In[13]:=	Kauffman[Knot[11, Alternating, 83]][a, z]
Out[13]=	2 2 2 -8 6 8 2 z 9 z 13 z 7 z 2 z z 3 z 24 z a + -- + -- + -- - -- - --- - ---- - --- - --- + --- - ---- - ----- - 6 4 2 9 7 5 3 a 10 8 6 a a a a a a a a a a 2 2 3 3 3 3 3 3 4 29 z 9 z 2 z 7 z 29 z 31 z 16 z 5 z z ----- - ---- - ---- + ---- + ----- + ----- + ----- + ---- + --- - 4 2 11 9 7 5 3 a 12 a a a a a a a a 4 4 4 4 4 5 5 5 5 7 z 6 z 50 z 52 z 16 z 4 z 15 z 32 z 15 z ---- + ---- + ----- + ----- + ----- + ---- - ----- - ----- - ----- - 10 8 6 4 2 11 9 7 5 a a a a a a a a a 5 5 6 6 6 6 6 7 7 6 z 4 z 8 z 14 z 51 z 41 z 12 z 11 z 5 z ---- - ---- + ---- - ----- - ----- - ----- - ----- + ----- + ---- - 3 a 10 8 6 4 2 9 7 a a a a a a a a 7 7 7 8 8 8 8 9 9 13 z 6 z z 10 z 16 z 9 z 3 z 5 z 8 z ----- - ---- + -- + ----- + ----- + ---- + ---- + ---- + ---- + 5 3 a 8 6 4 2 7 5 a a a a a a a a 9 10 10 3 z z z ---- + --- + --- 3 6 4 a a a
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 83]], Vassiliev[3][Knot[11, Alternating, 83]]}
Out[14]=	{0, 6}
In[15]:=	Kh[Knot[11, Alternating, 83]][q, t]
Out[15]=	3 3 5 1 2 q 4 q 2 q 5 7 7 q + 5 q + ----- + ---- + -- + --- + ---- + 8 q t + 6 q t + 3 3 2 2 t t q t q t t 7 2 9 2 9 3 11 3 11 4 13 4 10 q t + 8 q t + 8 q t + 10 q t + 8 q t + 8 q t + 13 5 15 5 15 6 17 6 17 7 19 7 21 8 5 q t + 8 q t + 3 q t + 5 q t + q t + 3 q t + q t