K11a96

K11a95

K11a97

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	K11a96 Quick Notes

K11a96 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X18,7,19,8 X14,9,15,10 X2,11,3,12 X22,14,1,13 X8,15,9,16 X20,18,21,17 X6,19,7,20 X16,22,17,21
Gauss code	1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -5, 8, -11, 9, -4, 10, -9, 11, -7
Dowker-Thistlethwaite code	4 10 12 18 14 2 22 8 20 6 16
Conway Notation	[2221112]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a96/ThurstonBennequinNumber Hyperbolic Volume 14.8744 A-Polynomial See Data:K11a96/A-polynomial

[edit Notes for K11a96's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $[0,3]$ Rasmussen s-Invariant 0

[edit Notes for K11a96's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+9t^{2}-29t+43-29t^{-1}+9t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+3z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 121, 0 }
Jones polynomial	$-q^{5}+4q^{4}-8q^{3}+13q^{2}-17q+20-19q^{-1}+16q^{-2}-12q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$a^{6}-3z^{2}a^{4}-2a^{4}+3z^{4}a^{2}+4z^{2}a^{2}+2a^{2}-z^{6}-2z^{4}-4z^{2}-1+2z^{4}a^{-2}+2z^{2}a^{-2}+a^{-2}-z^{2}a^{-4}$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+7az^{9}+4z^{9}a^{-1}+4a^{4}z^{8}+9a^{2}z^{8}+7z^{8}a^{-2}+12z^{8}+3a^{5}z^{7}+2a^{3}z^{7}-3az^{7}+5z^{7}a^{-1}+7z^{7}a^{-3}+a^{6}z^{6}-6a^{4}z^{6}-17a^{2}z^{6}-7z^{6}a^{-2}+4z^{6}a^{-4}-21z^{6}-8a^{5}z^{5}-15a^{3}z^{5}-11az^{5}-16z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}-3a^{4}z^{4}+2a^{2}z^{4}-z^{4}a^{-2}-6z^{4}a^{-4}+7z^{4}+7a^{5}z^{3}+11a^{3}z^{3}+6az^{3}+8z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+3a^{6}z^{2}+7a^{4}z^{2}+6a^{2}z^{2}+3z^{2}a^{-2}+2z^{2}a^{-4}+3z^{2}-2a^{5}z-2a^{3}z-a^{6}-2a^{4}-2a^{2}-a^{-2}-1$
The A2 invariant	Data:K11a96/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a96/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a96 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["K11a96"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+9t^{2}-29t+43-29t^{-1}+9t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 121, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+4q^{4}-8q^{3}+13q^{2}-17q+20-19q^{-1}+16q^{-2}-12q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{6}-3z^{2}a^{4}-2a^{4}+3z^{4}a^{2}+4z^{2}a^{2}+2a^{2}-z^{6}-2z^{4}-4z^{2}-1+2z^{4}a^{-2}+2z^{2}a^{-2}+a^{-2}-z^{2}a^{-4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+7az^{9}+4z^{9}a^{-1}+4a^{4}z^{8}+9a^{2}z^{8}+7z^{8}a^{-2}+12z^{8}+3a^{5}z^{7}+2a^{3}z^{7}-3az^{7}+5z^{7}a^{-1}+7z^{7}a^{-3}+a^{6}z^{6}-6a^{4}z^{6}-17a^{2}z^{6}-7z^{6}a^{-2}+4z^{6}a^{-4}-21z^{6}-8a^{5}z^{5}-15a^{3}z^{5}-11az^{5}-16z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}-3a^{4}z^{4}+2a^{2}z^{4}-z^{4}a^{-2}-6z^{4}a^{-4}+7z^{4}+7a^{5}z^{3}+11a^{3}z^{3}+6az^{3}+8z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+3a^{6}z^{2}+7a^{4}z^{2}+6a^{2}z^{2}+3z^{2}a^{-2}+2z^{2}a^{-4}+3z^{2}-2a^{5}z-2a^{3}z-a^{6}-2a^{4}-2a^{2}-a^{-2}-1$

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 $-8$ $16$ $32$ $-{\frac {28}{3}}$ $-{\frac {20}{3}}$ $-128$ $-{\frac {320}{3}}$ $-{\frac {224}{3}}$ $48$ $-{\frac {256}{3}}$ $128$ ${\frac {224}{3}}$ ${\frac {160}{3}}$ ${\frac {5729}{15}}$ ${\frac {524}{15}}$ ${\frac {9716}{45}}$ $-{\frac {449}{9}}$ ${\frac {449}{15}}$

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=$0 is the signature of K11a96. 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 11                       1 -1 9                     3   3 7                   5 1   -4 5                 8 3     5 3               9 5       -4 1             11 8         3 -1           9 10           1 -3         7 10             -3 -5       5 9               4 -7     2 7                 -5 -9   1 5                   4 -11   2                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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, 96]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 96]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[18, 7, 19, 8], X[14, 9, 15, 10], X[2, 11, 3, 12], X[22, 14, 1, 13], X[8, 15, 9, 16], X[20, 18, 21, 17], X[6, 19, 7, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 96]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -5, 8, -11, 9, -4, 10, -9, 11, -7]
In[5]:=	BR[Knot[11, Alternating, 96]]
Out[5]=	BR[Knot[11, Alternating, 96]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 96]][t]
Out[6]=	-3 9 29 2 3 43 - t + -- - -- - 29 t + 9 t - t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 96]][z]
Out[7]=	2 4 6 1 - 2 z + 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 96]}
In[9]:=	{KnotDet[Knot[11, Alternating, 96]], KnotSignature[Knot[11, Alternating, 96]]}
Out[9]=	{121, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 96]][q]
Out[10]=	-6 3 7 12 16 19 2 3 4 5 20 + q - -- + -- - -- + -- - -- - 17 q + 13 q - 8 q + 4 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 28], Knot[11, Alternating, 87], Knot[11, Alternating, 96]}
In[12]:=	A2Invariant[Knot[11, Alternating, 96]][q]
Out[12]=	-20 -18 2 -14 -12 4 3 -6 -4 3 -2 + q + q - --- + q + q - --- + -- - q - q + -- + 16 10 8 2 q q q q 2 4 8 10 12 14 16 4 q - 3 q + 3 q - 3 q + 2 q + q - q
In[13]:=	Kauffman[Knot[11, Alternating, 96]][a, z]
Out[13]=	2 2 -2 2 4 6 3 5 2 2 z 3 z -1 - a - 2 a - 2 a - a - 2 a z - 2 a z + 3 z + ---- + ---- + 4 2 a a 3 3 3 2 2 4 2 6 2 z 5 z 8 z 3 3 3 6 a z + 7 a z + 3 a z - -- + ---- + ---- + 6 a z + 11 a z + 5 3 a a a 4 4 5 5 3 4 6 z z 2 4 4 4 6 4 z 7 a z + 7 z - ---- - -- + 2 a z - 3 a z - 3 a z + -- - 4 2 5 a a a 5 5 6 6 11 z 16 z 5 3 5 5 5 6 4 z 7 z ----- - ----- - 11 a z - 15 a z - 8 a z - 21 z + ---- - ---- - 3 a 4 2 a a a 7 7 2 6 4 6 6 6 7 z 5 z 7 3 7 17 a z - 6 a z + a z + ---- + ---- - 3 a z + 2 a z + 3 a a 8 9 5 7 8 7 z 2 8 4 8 4 z 9 3 a z + 12 z + ---- + 9 a z + 4 a z + ---- + 7 a z + 2 a a 3 9 10 2 10 3 a z + z + a z
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 96]], Vassiliev[3][Knot[11, Alternating, 96]]}
Out[14]=	{0, 2}
In[15]:=	Kh[Knot[11, Alternating, 96]][q, t]
Out[15]=	10 1 2 1 5 2 7 5 -- + 11 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 q t q t q t q t q t q t q t 9 7 10 9 3 3 2 5 2 ----- + ----- + ---- + --- + 8 q t + 9 q t + 5 q t + 8 q t + 5 2 3 2 3 q t q t q t q t 5 3 7 3 7 4 9 4 11 5 3 q t + 5 q t + q t + 3 q t + q t