K11a90

K11a89

K11a91

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	K11a90 Quick Notes

K11a90 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X16,7,17,8 X18,9,19,10 X2,11,3,12 X22,14,1,13 X20,16,21,15 X8,17,9,18 X6,19,7,20 X14,22,15,21
Gauss code	1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, -8, 11, -7
Dowker-Thistlethwaite code	4 10 12 16 18 2 22 20 8 6 14
Conway Notation	[313112]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a90/ThurstonBennequinNumber Hyperbolic Volume 12.5188 A-Polynomial See Data:K11a90/A-polynomial

[edit Notes for K11a90's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for K11a90's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-10t^{2}+20t-23+20t^{-1}-10t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+2z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 87, -2 }
Jones polynomial	$-q^{4}+3q^{3}-5q^{2}+9q-11+13q^{-1}-14q^{-2}+12q^{-3}-9q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{6}+a^{6}-2z^{4}a^{4}-4z^{2}a^{4}-a^{4}+z^{6}a^{2}+2z^{4}a^{2}-a^{2}+z^{6}+3z^{4}+3z^{2}+2-z^{4}a^{-2}-2z^{2}a^{-2}$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+6az^{9}+3z^{9}a^{-1}+5a^{4}z^{8}+6a^{2}z^{8}+3z^{8}a^{-2}+4z^{8}+6a^{5}z^{7}+a^{3}z^{7}-16az^{7}-10z^{7}a^{-1}+z^{7}a^{-3}+5a^{6}z^{6}-5a^{4}z^{6}-21a^{2}z^{6}-13z^{6}a^{-2}-24z^{6}+3a^{7}z^{5}-9a^{5}z^{5}-14a^{3}z^{5}+10az^{5}+8z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-6a^{6}z^{4}-2a^{4}z^{4}+18a^{2}z^{4}+17z^{4}a^{-2}+30z^{4}-3a^{7}z^{3}+8a^{5}z^{3}+14a^{3}z^{3}-3az^{3}-2z^{3}a^{-1}+4z^{3}a^{-3}-a^{8}z^{2}+3a^{6}z^{2}+5a^{4}z^{2}-7a^{2}z^{2}-7z^{2}a^{-2}-15z^{2}-3a^{5}z-3a^{3}z+az+za^{-1}-a^{6}-a^{4}+a^{2}+2$
The A2 invariant	Data:K11a90/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a90/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a90 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["K11a90"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-10t^{2}+20t-23+20t^{-1}-10t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 87, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{4}+3q^{3}-5q^{2}+9q-11+13q^{-1}-14q^{-2}+12q^{-3}-9q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+6az^{9}+3z^{9}a^{-1}+5a^{4}z^{8}+6a^{2}z^{8}+3z^{8}a^{-2}+4z^{8}+6a^{5}z^{7}+a^{3}z^{7}-16az^{7}-10z^{7}a^{-1}+z^{7}a^{-3}+5a^{6}z^{6}-5a^{4}z^{6}-21a^{2}z^{6}-13z^{6}a^{-2}-24z^{6}+3a^{7}z^{5}-9a^{5}z^{5}-14a^{3}z^{5}+10az^{5}+8z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-6a^{6}z^{4}-2a^{4}z^{4}+18a^{2}z^{4}+17z^{4}a^{-2}+30z^{4}-3a^{7}z^{3}+8a^{5}z^{3}+14a^{3}z^{3}-3az^{3}-2z^{3}a^{-1}+4z^{3}a^{-3}-a^{8}z^{2}+3a^{6}z^{2}+5a^{4}z^{2}-7a^{2}z^{2}-7z^{2}a^{-2}-15z^{2}-3a^{5}z-3a^{3}z+az+za^{-1}-a^{6}-a^{4}+a^{2}+2$

Vassiliev invariants

V2 and V3:

(-2, 3)

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 $-8$ $24$ $32$ $-{\frac {28}{3}}$ ${\frac {4}{3}}$ $-192$ $-240$ $-128$ $24$ $-{\frac {256}{3}}$ $288$ ${\frac {224}{3}}$ $-{\frac {32}{3}}$ ${\frac {11729}{15}}$ ${\frac {268}{5}}$ ${\frac {16796}{45}}$ $-{\frac {401}{9}}$ ${\frac {929}{15}}$

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 $s=$-2 is the signature of K11a90. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[11, Alternating, 90]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 90]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[16, 7, 17, 8], X[18, 9, 19, 10], X[2, 11, 3, 12], X[22, 14, 1, 13], X[20, 16, 21, 15], X[8, 17, 9, 18], X[6, 19, 7, 20], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 90]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, -8, 11, -7]
In[5]:=	BR[Knot[11, Alternating, 90]]
Out[5]=	BR[Knot[11, Alternating, 90]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 90]][t]
Out[6]=	2 10 20 2 3 -23 + -- - -- + -- + 20 t - 10 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 90]][z]
Out[7]=	2 4 6 1 - 2 z + 2 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 90], Knot[11, Alternating, 118]}
In[9]:=	{KnotDet[Knot[11, Alternating, 90]], KnotSignature[Knot[11, Alternating, 90]]}
Out[9]=	{87, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 90]][q]
Out[10]=	-7 3 6 9 12 14 13 2 3 4 -11 + q - -- + -- - -- + -- - -- + -- + 9 q - 5 q + 3 q - q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 90]}
In[12]:=	A2Invariant[Knot[11, Alternating, 90]][q]
Out[12]=	-22 -18 2 -14 -10 3 -6 2 -2 2 4 2 + q - q + --- - q + q - -- + q - -- + q - q + 3 q + 16 8 4 q q q 10 12 q - q
In[13]:=	Kauffman[Knot[11, Alternating, 90]][a, z]
Out[13]=	2 2 4 6 z 3 5 2 7 z 2 2 2 + a - a - a + - + a z - 3 a z - 3 a z - 15 z - ---- - 7 a z + a 2 a 3 3 4 2 6 2 8 2 4 z 2 z 3 3 3 5 a z + 3 a z - a z + ---- - ---- - 3 a z + 14 a z + 3 a a 4 5 3 7 3 4 17 z 2 4 4 4 6 4 8 a z - 3 a z + 30 z + ----- + 18 a z - 2 a z - 6 a z + 2 a 5 5 8 4 4 z 8 z 5 3 5 5 5 7 5 a z - ---- + ---- + 10 a z - 14 a z - 9 a z + 3 a z - 3 a a 6 7 7 6 13 z 2 6 4 6 6 6 z 10 z 7 24 z - ----- - 21 a z - 5 a z + 5 a z + -- - ----- - 16 a z + 2 3 a a a 8 9 3 7 5 7 8 3 z 2 8 4 8 3 z 9 a z + 6 a z + 4 z + ---- + 6 a z + 5 a z + ---- + 6 a z + 2 a a 3 9 10 2 10 3 a z + z + a z
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 90]], Vassiliev[3][Knot[11, Alternating, 90]]}
Out[14]=	{0, 3}
In[15]:=	Kh[Knot[11, Alternating, 90]][q, t]
Out[15]=	6 8 1 2 1 4 2 5 4 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 q q t q t q t q t q t q t q t 7 5 7 7 6 t 2 3 2 ----- + ----- + ---- + ---- + --- + 5 q t + 3 q t + 6 q t + 7 2 5 2 5 3 q q t q t q t q t 3 3 5 3 5 4 7 4 9 5 2 q t + 3 q t + q t + 2 q t + q t