K11a89

K11a88

K11a90

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	K11a89 Quick Notes

K11a89 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X16,8,17,7 X18,9,19,10 X2,11,3,12 X22,14,1,13 X6,16,7,15 X20,18,21,17 X8,19,9,20 X14,22,15,21
Gauss code	1, -6, 2, -1, 3, -8, 4, -10, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, -9, 11, -7
Dowker-Thistlethwaite code	4 10 12 16 18 2 22 6 20 8 14
Conway Notation	[2122112]

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:K11a89/ThurstonBennequinNumber Hyperbolic Volume 14.7281 A-Polynomial See Data:K11a89/A-polynomial

[edit Notes for K11a89'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 K11a89's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-12t^{2}+28t-35+28t^{-1}-12t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 119, 2 }
Jones polynomial	$-q^{8}+4q^{7}-8q^{6}+13q^{5}-17q^{4}+19q^{3}-19q^{2}+16q-11+7q^{-1}-3q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}a^{-4}+z^{4}a^{-2}+2z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}-z^{2}a^{-2}+2z^{2}a^{-4}-z^{2}a^{-6}-3z^{2}+a^{2}-a^{-2}+a^{-4}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+7z^{9}a^{-3}+4z^{9}a^{-5}+9z^{8}a^{-2}+12z^{8}a^{-4}+7z^{8}a^{-6}+4z^{8}+3az^{7}+z^{7}a^{-1}-4z^{7}a^{-3}+5z^{7}a^{-5}+7z^{7}a^{-7}+a^{2}z^{6}-21z^{6}a^{-2}-24z^{6}a^{-4}-7z^{6}a^{-6}+4z^{6}a^{-8}-7z^{6}-8az^{5}-13z^{5}a^{-1}-12z^{5}a^{-3}-19z^{5}a^{-5}-11z^{5}a^{-7}+z^{5}a^{-9}-3a^{2}z^{4}+14z^{4}a^{-2}+15z^{4}a^{-4}-2z^{4}a^{-6}-6z^{4}a^{-8}+6az^{3}+10z^{3}a^{-1}+13z^{3}a^{-3}+15z^{3}a^{-5}+5z^{3}a^{-7}-z^{3}a^{-9}+3a^{2}z^{2}-5z^{2}a^{-2}-4z^{2}a^{-4}+3z^{2}a^{-6}+2z^{2}a^{-8}+3z^{2}-az-2za^{-1}-4za^{-3}-4za^{-5}-za^{-7}-a^{2}+a^{-2}+a^{-4}$
The A2 invariant	Data:K11a89/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a89/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a89 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["K11a89"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-12t^{2}+28t-35+28t^{-1}-12t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 119, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+4q^{7}-8q^{6}+13q^{5}-17q^{4}+19q^{3}-19q^{2}+16q-11+7q^{-1}-3q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+7z^{9}a^{-3}+4z^{9}a^{-5}+9z^{8}a^{-2}+12z^{8}a^{-4}+7z^{8}a^{-6}+4z^{8}+3az^{7}+z^{7}a^{-1}-4z^{7}a^{-3}+5z^{7}a^{-5}+7z^{7}a^{-7}+a^{2}z^{6}-21z^{6}a^{-2}-24z^{6}a^{-4}-7z^{6}a^{-6}+4z^{6}a^{-8}-7z^{6}-8az^{5}-13z^{5}a^{-1}-12z^{5}a^{-3}-19z^{5}a^{-5}-11z^{5}a^{-7}+z^{5}a^{-9}-3a^{2}z^{4}+14z^{4}a^{-2}+15z^{4}a^{-4}-2z^{4}a^{-6}-6z^{4}a^{-8}+6az^{3}+10z^{3}a^{-1}+13z^{3}a^{-3}+15z^{3}a^{-5}+5z^{3}a^{-7}-z^{3}a^{-9}+3a^{2}z^{2}-5z^{2}a^{-2}-4z^{2}a^{-4}+3z^{2}a^{-6}+2z^{2}a^{-8}+3z^{2}-az-2za^{-1}-4za^{-3}-4za^{-5}-za^{-7}-a^{2}+a^{-2}+a^{-4}$

Vassiliev invariants

V2 and V3:

(-2, -1)

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 $-8$ $-8$ $32$ ${\frac {116}{3}}$ ${\frac {52}{3}}$ $64$ ${\frac {208}{3}}$ $-{\frac {32}{3}}$ $24$ $-{\frac {256}{3}}$ $32$ $-{\frac {928}{3}}$ $-{\frac {416}{3}}$ $-{\frac {4231}{15}}$ $-{\frac {1756}{15}}$ $-{\frac {5764}{45}}$ ${\frac {199}{9}}$ $-{\frac {151}{15}}$

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 K11a89. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     3   3 13                   5 1   -4 11                 8 3     5 9               9 5       -4 7             10 8         2 5           9 9           0 3         7 10             -3 1       5 10               5 -1     2 6                 -4 -3   1 5                   4 -5   2                     -2 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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, 89]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 89]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[16, 8, 17, 7], X[18, 9, 19, 10], X[2, 11, 3, 12], X[22, 14, 1, 13], X[6, 16, 7, 15], X[20, 18, 21, 17], X[8, 19, 9, 20], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 89]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -8, 4, -10, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, -9, 11, -7]
In[5]:=	BR[Knot[11, Alternating, 89]]
Out[5]=	BR[Knot[11, Alternating, 89]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 89]][t]
Out[6]=	2 12 28 2 3 -35 + -- - -- + -- + 28 t - 12 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 89]][z]
Out[7]=	2 6 1 - 2 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 89]}
In[9]:=	{KnotDet[Knot[11, Alternating, 89]], KnotSignature[Knot[11, Alternating, 89]]}
Out[9]=	{119, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 89]][q]
Out[10]=	-3 3 7 2 3 4 5 6 -11 + q - -- + - + 16 q - 19 q + 19 q - 17 q + 13 q - 8 q + 2 q q 7 8 4 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 34], Knot[11, Alternating, 89]}
In[12]:=	A2Invariant[Knot[11, Alternating, 89]][q]
Out[12]=	-10 -6 3 -2 2 4 6 8 12 14 q - q + -- - q + 3 q - 4 q + 2 q - 2 q + 2 q - 3 q + 4 q 16 18 20 22 24 4 q - q - q + 2 q - q
In[13]:=	Kauffman[Knot[11, Alternating, 89]][a, z]
Out[13]=	2 2 -4 -2 2 z 4 z 4 z 2 z 2 2 z 3 z a + a - a - -- - --- - --- - --- - a z + 3 z + ---- + ---- - 7 5 3 a 8 6 a a a a a 2 2 3 3 3 3 3 4 z 5 z 2 2 z 5 z 15 z 13 z 10 z 3 ---- - ---- + 3 a z - -- + ---- + ----- + ----- + ----- + 6 a z - 4 2 9 7 5 3 a a a a a a a 4 4 4 4 5 5 5 5 6 z 2 z 15 z 14 z 2 4 z 11 z 19 z 12 z ---- - ---- + ----- + ----- - 3 a z + -- - ----- - ----- - ----- - 8 6 4 2 9 7 5 3 a a a a a a a a 5 6 6 6 6 7 13 z 5 6 4 z 7 z 24 z 21 z 2 6 7 z ----- - 8 a z - 7 z + ---- - ---- - ----- - ----- + a z + ---- + a 8 6 4 2 7 a a a a a 7 7 7 8 8 8 9 5 z 4 z z 7 8 7 z 12 z 9 z 4 z ---- - ---- + -- + 3 a z + 4 z + ---- + ----- + ---- + ---- + 5 3 a 6 4 2 5 a a a a a a 9 9 10 10 7 z 3 z z z ---- + ---- + --- + --- 3 a 4 2 a a a
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 89]], Vassiliev[3][Knot[11, Alternating, 89]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[11, Alternating, 89]][q, t]
Out[15]=	3 1 2 1 5 2 6 5 q 10 q + 7 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 10 q t + 9 q t + 9 q t + 10 q t + 8 q t + 9 q t + 5 q t + 11 4 11 5 13 5 13 6 15 6 17 7 8 q t + 3 q t + 5 q t + q t + 3 q t + q t