K11a98

K11a97

K11a99

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	K11a98 Quick Notes

K11a98 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X18,7,19,8 X16,9,17,10 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, -5, 9, -4, 10, -8, 11, -7
Dowker-Thistlethwaite code	4 10 12 18 16 2 22 20 8 6 14
Conway Notation	[43112]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a98/ThurstonBennequinNumber Hyperbolic Volume 12.0217 A-Polynomial See Data:K11a98/A-polynomial

[edit Notes for K11a98's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for K11a98's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$4t^{2}-19t+31-19t^{-1}+4t^{-2}$
Conway polynomial	$4z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 77, 0 }
Jones polynomial	$-q^{5}+3q^{4}-5q^{3}+9q^{2}-11q+12-12q^{-1}+10q^{-2}-7q^{-3}+4q^{-4}-2q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$a^{6}-2z^{2}a^{4}-a^{4}+z^{4}a^{2}-z^{2}a^{2}+2z^{4}+z^{2}+z^{4}a^{-2}+a^{-2}-z^{2}a^{-4}$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+2a^{3}z^{9}+5az^{9}+3z^{9}a^{-1}+2a^{4}z^{8}+4z^{8}a^{-2}+2z^{8}+2a^{5}z^{7}-5a^{3}z^{7}-18az^{7}-7z^{7}a^{-1}+4z^{7}a^{-3}+a^{6}z^{6}-4a^{4}z^{6}-4a^{2}z^{6}-8z^{6}a^{-2}+3z^{6}a^{-4}-10z^{6}-7a^{5}z^{5}+6a^{3}z^{5}+33az^{5}+11z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}-4a^{6}z^{4}-2a^{4}z^{4}+8a^{2}z^{4}+5z^{4}a^{-2}-7z^{4}a^{-4}+18z^{4}+6a^{5}z^{3}-7a^{3}z^{3}-25az^{3}-7z^{3}a^{-1}+3z^{3}a^{-3}-2z^{3}a^{-5}+4a^{6}z^{2}+4a^{4}z^{2}-6a^{2}z^{2}+z^{2}a^{-2}+3z^{2}a^{-4}-8z^{2}-a^{5}z+3a^{3}z+6az+2za^{-1}-a^{6}-a^{4}-a^{-2}$
The A2 invariant	Data:K11a98/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a98/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a98 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["K11a98"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$4t^{2}-19t+31-19t^{-1}+4t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 77, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+3q^{4}-5q^{3}+9q^{2}-11q+12-12q^{-1}+10q^{-2}-7q^{-3}+4q^{-4}-2q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{10}+z^{10}+2a^{3}z^{9}+5az^{9}+3z^{9}a^{-1}+2a^{4}z^{8}+4z^{8}a^{-2}+2z^{8}+2a^{5}z^{7}-5a^{3}z^{7}-18az^{7}-7z^{7}a^{-1}+4z^{7}a^{-3}+a^{6}z^{6}-4a^{4}z^{6}-4a^{2}z^{6}-8z^{6}a^{-2}+3z^{6}a^{-4}-10z^{6}-7a^{5}z^{5}+6a^{3}z^{5}+33az^{5}+11z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}-4a^{6}z^{4}-2a^{4}z^{4}+8a^{2}z^{4}+5z^{4}a^{-2}-7z^{4}a^{-4}+18z^{4}+6a^{5}z^{3}-7a^{3}z^{3}-25az^{3}-7z^{3}a^{-1}+3z^{3}a^{-3}-2z^{3}a^{-5}+4a^{6}z^{2}+4a^{4}z^{2}-6a^{2}z^{2}+z^{2}a^{-2}+3z^{2}a^{-4}-8z^{2}-a^{5}z+3a^{3}z+6az+2za^{-1}-a^{6}-a^{4}-a^{-2}$

Vassiliev invariants

V2 and V3:

(-3, 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 $-12$ $24$ $72$ $34$ $6$ $-288$ $-368$ $-160$ $24$ $-288$ $288$ $-408$ $-72$ ${\frac {4049}{10}}$ $-{\frac {1054}{15}}$ ${\frac {5018}{15}}$ $-{\frac {241}{6}}$ ${\frac {529}{10}}$

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 K11a98. 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                     2   2 7                   3 1   -2 5                 6 2     4 3               5 3       -2 1             7 6         1 -1           6 6           0 -3         4 6             -2 -5       3 6               3 -7     1 4                 -3 -9   1 3                   2 -11   1                     -1 -13 1                       1

Integral Khovanov Homology (db, data source)

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

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, 98]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 98]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[18, 7, 19, 8], X[16, 9, 17, 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, 98]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -11, 8, -5, 9, -4, 10, -8, 11, -7]
In[5]:=	BR[Knot[11, Alternating, 98]]
Out[5]=	BR[Knot[11, Alternating, 98]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 98]][t]
Out[6]=	4 19 2 31 + -- - -- - 19 t + 4 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 98]][z]
Out[7]=	2 4 1 - 3 z + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 98], Knot[11, Alternating, 119]}
In[9]:=	{KnotDet[Knot[11, Alternating, 98]], KnotSignature[Knot[11, Alternating, 98]]}
Out[9]=	{77, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 98]][q]
Out[10]=	-6 2 4 7 10 12 2 3 4 5 12 + q - -- + -- - -- + -- - -- - 11 q + 9 q - 5 q + 3 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, 98]}
In[12]:=	A2Invariant[Knot[11, Alternating, 98]][q]
Out[12]=	-20 -18 -16 -14 3 2 -4 -2 2 4 -2 + q + q - q + q - --- + -- - q + q + 2 q - q + 10 8 q q 6 8 10 12 14 16 q + 3 q - 2 q + q + q - q
In[13]:=	Kauffman[Knot[11, Alternating, 98]][a, z]
Out[13]=	2 2 -2 4 6 2 z 3 5 2 3 z z -a - a - a + --- + 6 a z + 3 a z - a z - 8 z + ---- + -- - a 4 2 a a 3 3 3 2 2 4 2 6 2 2 z 3 z 7 z 3 6 a z + 4 a z + 4 a z - ---- + ---- - ---- - 25 a z - 5 3 a a a 4 4 3 3 5 3 4 7 z 5 z 2 4 4 4 7 a z + 6 a z + 18 z - ---- + ---- + 8 a z - 2 a z - 4 2 a a 5 5 5 6 4 z 8 z 11 z 5 3 5 5 5 6 4 a z + -- - ---- + ----- + 33 a z + 6 a z - 7 a z - 10 z + 5 3 a a a 6 6 7 7 3 z 8 z 2 6 4 6 6 6 4 z 7 z 7 ---- - ---- - 4 a z - 4 a z + a z + ---- - ---- - 18 a z - 4 2 3 a a a a 8 9 3 7 5 7 8 4 z 4 8 3 z 9 3 9 5 a z + 2 a z + 2 z + ---- + 2 a z + ---- + 5 a z + 2 a z + 2 a a 10 2 10 z + a z
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 98]], Vassiliev[3][Knot[11, Alternating, 98]]}
Out[14]=	{0, 3}
In[15]:=	Kh[Knot[11, Alternating, 98]][q, t]
Out[15]=	6 1 1 1 3 1 4 3 - + 7 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 6 4 6 6 3 3 2 5 2 ----- + ----- + ---- + --- + 6 q t + 5 q t + 3 q t + 6 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 2 q t + 3 q t + q t + 2 q t + q t