K11a84

K11a83

K11a85

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	K11a84 Quick Notes

K11a84 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 X20,13,21,14 X18,15,19,16 X6,18,7,17 X14,19,15,20 X8,21,9,22
Gauss code	1, -5, 2, -1, 3, -9, 4, -11, 5, -2, 6, -3, 7, -10, 8, -4, 9, -8, 10, -7, 11, -6
Dowker-Thistlethwaite code	4 10 12 16 2 22 20 18 6 14 8
Conway Notation	[232112]

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:K11a84/ThurstonBennequinNumber Hyperbolic Volume 13.8696 A-Polynomial See Data:K11a84/A-polynomial

[edit Notes for K11a84's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a84's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+10t^{2}-23t+31-23t^{-1}+10t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 101, 0 }
Jones polynomial	$-q^{5}+3q^{4}-6q^{3}+11q^{2}-14q+16-16q^{-1}+14q^{-2}-10q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{6}-z^{6}+a^{4}z^{4}-3a^{2}z^{4}+2z^{4}a^{-2}-2z^{4}+2a^{4}z^{2}-4a^{2}z^{2}+4z^{2}a^{-2}-z^{2}a^{-4}-2z^{2}+a^{4}-a^{2}+3a^{-2}-a^{-4}-1$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+6az^{9}+3z^{9}a^{-1}+4a^{4}z^{8}+6a^{2}z^{8}+4z^{8}a^{-2}+6z^{8}+3a^{5}z^{7}-3a^{3}z^{7}-10az^{7}+4z^{7}a^{-3}+a^{6}z^{6}-10a^{4}z^{6}-18a^{2}z^{6}-z^{6}a^{-2}+3z^{6}a^{-4}-11z^{6}-9a^{5}z^{5}-4a^{3}z^{5}+9az^{5}-z^{5}a^{-1}-4z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}+7a^{4}z^{4}+18a^{2}z^{4}-8z^{4}a^{-2}-6z^{4}a^{-4}+6z^{4}+7a^{5}z^{3}+3a^{3}z^{3}-7az^{3}-3z^{3}a^{-1}-2z^{3}a^{-3}-2z^{3}a^{-5}+2a^{6}z^{2}-3a^{4}z^{2}-9a^{2}z^{2}+9z^{2}a^{-2}+4z^{2}a^{-4}+z^{2}-a^{5}z+2az+2za^{-1}+2za^{-3}+za^{-5}+a^{4}+a^{2}-3a^{-2}-a^{-4}-1$
The A2 invariant	Data:K11a84/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a84/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a84 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["K11a84"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+10t^{2}-23t+31-23t^{-1}+10t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 101, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+3q^{4}-6q^{3}+11q^{2}-14q+16-16q^{-1}+14q^{-2}-10q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

Vassiliev invariants

V2 and V3:

(-1, 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 $-4$ $16$ $8$ ${\frac {82}{3}}$ ${\frac {62}{3}}$ $-64$ $-{\frac {416}{3}}$ $-{\frac {224}{3}}$ $-16$ $-{\frac {32}{3}}$ $128$ $-{\frac {328}{3}}$ $-{\frac {248}{3}}$ ${\frac {4769}{30}}$ ${\frac {3662}{15}}$ $-{\frac {11342}{45}}$ ${\frac {991}{18}}$ $-{\frac {2911}{30}}$

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 K11a84. 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                   4 1   -3 5                 7 2     5 3               7 4       -3 1             9 7         2 -1           8 8           0 -3         6 8             -2 -5       4 8               4 -7     2 6                 -4 -9   1 4                   3 -11   2                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

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

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, 84]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 84]]
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[20, 13, 21, 14], X[18, 15, 19, 16], X[6, 18, 7, 17], X[14, 19, 15, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 84]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -11, 5, -2, 6, -3, 7, -10, 8, -4, 9, -8, 10, -7, 11, -6]
In[5]:=	BR[Knot[11, Alternating, 84]]
Out[5]=	BR[Knot[11, Alternating, 84]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 84]][t]
Out[6]=	2 10 23 2 3 31 - -- + -- - -- - 23 t + 10 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 84]][z]
Out[7]=	2 4 6 1 - z - 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 119], Knot[11, Alternating, 84]}
In[9]:=	{KnotDet[Knot[11, Alternating, 84]], KnotSignature[Knot[11, Alternating, 84]]}
Out[9]=	{101, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 84]][q]
Out[10]=	-6 3 6 10 14 16 2 3 4 5 16 + q - -- + -- - -- + -- - -- - 14 q + 11 q - 6 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, 84], Knot[11, Alternating, 88]}
In[12]:=	A2Invariant[Knot[11, Alternating, 84]][q]
Out[12]=	-18 -16 -14 -12 3 3 -6 -4 -2 2 -3 + q - q + q + q - --- + -- - q - q + q + 3 q - 10 8 q q 4 6 8 10 12 16 q + 2 q + 3 q - 2 q + q - q
In[13]:=	Kauffman[Knot[11, Alternating, 84]][a, z]
Out[13]=	2 -4 3 2 4 z 2 z 2 z 5 2 4 z -1 - a - -- + a + a + -- + --- + --- + 2 a z - a z + z + ---- + 2 5 3 a 4 a a a a 2 3 3 3 9 z 2 2 4 2 6 2 2 z 2 z 3 z 3 ---- - 9 a z - 3 a z + 2 a z - ---- - ---- - ---- - 7 a z + 2 5 3 a a a a 4 4 3 3 5 3 4 6 z 8 z 2 4 4 4 3 a z + 7 a z + 6 z - ---- - ---- + 18 a z + 7 a z - 4 2 a a 5 5 5 6 4 z 4 z z 5 3 5 5 5 6 3 a z + -- - ---- - -- + 9 a z - 4 a z - 9 a z - 11 z + 5 3 a a a 6 6 7 3 z z 2 6 4 6 6 6 4 z 7 3 7 ---- - -- - 18 a z - 10 a z + a z + ---- - 10 a z - 3 a z + 4 2 3 a a a 8 9 5 7 8 4 z 2 8 4 8 3 z 9 3 9 3 a z + 6 z + ---- + 6 a z + 4 a z + ---- + 6 a z + 3 a z + 2 a a 10 2 10 z + a z
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 84]], Vassiliev[3][Knot[11, Alternating, 84]]}
Out[14]=	{0, 2}
In[15]:=	Kh[Knot[11, Alternating, 84]][q, t]
Out[15]=	8 1 2 1 4 2 6 4 - + 9 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 8 6 8 8 3 3 2 5 2 ----- + ----- + ---- + --- + 7 q t + 7 q t + 4 q t + 7 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 + 4 q t + q t + 2 q t + q t