K11a67

K11a66

K11a68

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	K11a67 Quick Notes

K11a67 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X8394 X16,5,17,6 X12,8,13,7 X2,9,3,10 X18,12,19,11 X22,13,1,14 X20,15,21,16 X10,18,11,17 X6,19,7,20 X14,21,15,22
Gauss code	1, -5, 2, -1, 3, -10, 4, -2, 5, -9, 6, -4, 7, -11, 8, -3, 9, -6, 10, -8, 11, -7
Dowker-Thistlethwaite code	4 8 16 12 2 18 22 20 10 6 14
Conway Notation	[.3.21.2]

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a67/ThurstonBennequinNumber Hyperbolic Volume 16.0998 A-Polynomial See Data:K11a67/A-polynomial

[edit Notes for K11a67's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $[1,3]$ Rasmussen s-Invariant 0

[edit Notes for K11a67's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+12t^{2}-29t+39-29t^{-1}+12t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 125, 0 }
Jones polynomial	$q^{4}-4q^{3}+9q^{2}-14q+18-20q^{-1}+20q^{-2}-16q^{-3}+12q^{-4}-7q^{-5}+3q^{-6}-q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{6}-a^{6}+2z^{4}a^{4}+3z^{2}a^{4}+a^{4}-z^{6}a^{2}-z^{4}a^{2}+z^{2}a^{2}+2a^{2}-z^{6}-2z^{4}-3z^{2}-2+z^{4}a^{-2}+z^{2}a^{-2}+a^{-2}$
Kauffman polynomial (db, data sources)	$2a^{4}z^{10}+2a^{2}z^{10}+4a^{5}z^{9}+11a^{3}z^{9}+7az^{9}+3a^{6}z^{8}+4a^{4}z^{8}+13a^{2}z^{8}+12z^{8}+a^{7}z^{7}-11a^{5}z^{7}-25a^{3}z^{7}+13z^{7}a^{-1}-11a^{6}z^{6}-27a^{4}z^{6}-40a^{2}z^{6}+9z^{6}a^{-2}-15z^{6}-4a^{7}z^{5}+5a^{5}z^{5}+7a^{3}z^{5}-24az^{5}-18z^{5}a^{-1}+4z^{5}a^{-3}+13a^{6}z^{4}+29a^{4}z^{4}+24a^{2}z^{4}-8z^{4}a^{-2}+z^{4}a^{-4}-z^{4}+5a^{7}z^{3}+5a^{5}z^{3}+8a^{3}z^{3}+18az^{3}+9z^{3}a^{-1}-z^{3}a^{-3}-6a^{6}z^{2}-9a^{4}z^{2}+3z^{2}a^{-2}+6z^{2}-2a^{7}z-3a^{5}z-3a^{3}z-4az-2za^{-1}+a^{6}+a^{4}-2a^{2}-a^{-2}-2$
The A2 invariant	Data:K11a67/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a67/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a67 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["K11a67"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+12t^{2}-29t+39-29t^{-1}+12t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 125, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{4}-4q^{3}+9q^{2}-14q+18-20q^{-1}+20q^{-2}-16q^{-3}+12q^{-4}-7q^{-5}+3q^{-6}-q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{4}z^{10}+2a^{2}z^{10}+4a^{5}z^{9}+11a^{3}z^{9}+7az^{9}+3a^{6}z^{8}+4a^{4}z^{8}+13a^{2}z^{8}+12z^{8}+a^{7}z^{7}-11a^{5}z^{7}-25a^{3}z^{7}+13z^{7}a^{-1}-11a^{6}z^{6}-27a^{4}z^{6}-40a^{2}z^{6}+9z^{6}a^{-2}-15z^{6}-4a^{7}z^{5}+5a^{5}z^{5}+7a^{3}z^{5}-24az^{5}-18z^{5}a^{-1}+4z^{5}a^{-3}+13a^{6}z^{4}+29a^{4}z^{4}+24a^{2}z^{4}-8z^{4}a^{-2}+z^{4}a^{-4}-z^{4}+5a^{7}z^{3}+5a^{5}z^{3}+8a^{3}z^{3}+18az^{3}+9z^{3}a^{-1}-z^{3}a^{-3}-6a^{6}z^{2}-9a^{4}z^{2}+3z^{2}a^{-2}+6z^{2}-2a^{7}z-3a^{5}z-3a^{3}z-4az-2za^{-1}+a^{6}+a^{4}-2a^{2}-a^{-2}-2$

Vassiliev invariants

V2 and V3:

(1, -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 $4$ $-24$ $8$ ${\frac {158}{3}}$ ${\frac {10}{3}}$ $-96$ $-208$ $32$ $-88$ ${\frac {32}{3}}$ $288$ ${\frac {632}{3}}$ ${\frac {40}{3}}$ ${\frac {26671}{30}}$ $-{\frac {2662}{15}}$ ${\frac {18902}{45}}$ ${\frac {2033}{18}}$ ${\frac {1711}{30}}$

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 K11a67. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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, 67]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 67]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[16, 5, 17, 6], X[12, 8, 13, 7], X[2, 9, 3, 10], X[18, 12, 19, 11], X[22, 13, 1, 14], X[20, 15, 21, 16], X[10, 18, 11, 17], X[6, 19, 7, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 67]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -10, 4, -2, 5, -9, 6, -4, 7, -11, 8, -3, 9, -6, 10, -8, 11, -7]
In[5]:=	BR[Knot[11, Alternating, 67]]
Out[5]=	BR[Knot[11, Alternating, 67]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 67]][t]
Out[6]=	2 12 29 2 3 39 - -- + -- - -- - 29 t + 12 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 67]][z]
Out[7]=	2 6 1 + z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 67], Knot[11, Alternating, 104], Knot[11, Alternating, 168]}
In[9]:=	{KnotDet[Knot[11, Alternating, 67]], KnotSignature[Knot[11, Alternating, 67]]}
Out[9]=	{125, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 67]][q]
Out[10]=	-7 3 7 12 16 20 20 2 3 4 18 - q + -- - -- + -- - -- + -- - -- - 14 q + 9 q - 4 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, 67], Knot[11, Alternating, 317]}
In[12]:=	A2Invariant[Knot[11, Alternating, 67]][q]
Out[12]=	-22 -18 3 2 -12 2 5 -6 2 2 -3 - q + q - --- + --- + q - --- + -- - q + -- + 3 q - 16 14 10 8 4 q q q q q 4 6 8 10 12 4 q + 2 q + 2 q - 2 q + q
In[13]:=	Kauffman[Knot[11, Alternating, 67]][a, z]
Out[13]=	-2 2 4 6 2 z 3 5 7 -2 - a - 2 a + a + a - --- - 4 a z - 3 a z - 3 a z - 2 a z + a 2 3 3 2 3 z 4 2 6 2 z 9 z 3 3 3 6 z + ---- - 9 a z - 6 a z - -- + ---- + 18 a z + 8 a z + 2 3 a a a 4 4 5 3 7 3 4 z 8 z 2 4 4 4 6 4 5 a z + 5 a z - z + -- - ---- + 24 a z + 29 a z + 13 a z + 4 2 a a 5 5 6 4 z 18 z 5 3 5 5 5 7 5 6 9 z ---- - ----- - 24 a z + 7 a z + 5 a z - 4 a z - 15 z + ---- - 3 a 2 a a 7 2 6 4 6 6 6 13 z 3 7 5 7 40 a z - 27 a z - 11 a z + ----- - 25 a z - 11 a z + a 7 7 8 2 8 4 8 6 8 9 3 9 a z + 12 z + 13 a z + 4 a z + 3 a z + 7 a z + 11 a z + 5 9 2 10 4 10 4 a z + 2 a z + 2 a z
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 67]], Vassiliev[3][Knot[11, Alternating, 67]]}
Out[14]=	{0, -3}
In[15]:=	Kh[Knot[11, Alternating, 67]][q, t]
Out[15]=	9 1 2 1 5 2 7 5 - + 10 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 q t q t q t q t q t q t q t 9 7 11 9 9 11 3 ----- + ----- + ----- + ----- + ---- + --- + 6 q t + 8 q t + 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t 3 2 5 2 5 3 7 3 9 4 3 q t + 6 q t + q t + 3 q t + q t