K11a85

K11a84

K11a86

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	K11a85 Quick Notes

K11a85 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 X20,14,21,13 X18,16,19,15 X8,18,9,17 X14,20,15,19 X6,21,7,22
Gauss code	1, -5, 2, -1, 3, -11, 4, -9, 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 8 14 6
Conway Notation	[2311112]

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:K11a85/ThurstonBennequinNumber Hyperbolic Volume 14.0551 A-Polynomial See Data:K11a85/A-polynomial

[edit Notes for K11a85'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 K11a85's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-10t^{2}+25t-33+25t^{-1}-10t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+2z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 107, 2 }
Jones polynomial	$q^{9}-4q^{8}+7q^{7}-11q^{6}+15q^{5}-17q^{4}+17q^{3}-14q^{2}+11q-6+3q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}a^{-4}+3z^{4}a^{-2}+2z^{4}a^{-4}-2z^{4}a^{-6}-z^{4}+5z^{2}a^{-2}+2z^{2}a^{-4}-3z^{2}a^{-6}+z^{2}a^{-8}-2z^{2}+3a^{-2}-a^{-6}-1$
Kauffman polynomial (db, data sources)	$z^{10}a^{-4}+z^{10}a^{-6}+3z^{9}a^{-3}+7z^{9}a^{-5}+4z^{9}a^{-7}+4z^{8}a^{-2}+8z^{8}a^{-4}+10z^{8}a^{-6}+6z^{8}a^{-8}+4z^{7}a^{-1}+z^{7}a^{-3}-11z^{7}a^{-5}-4z^{7}a^{-7}+4z^{7}a^{-9}-z^{6}a^{-2}-19z^{6}a^{-4}-33z^{6}a^{-6}-17z^{6}a^{-8}+z^{6}a^{-10}+3z^{6}+az^{5}-4z^{5}a^{-1}-4z^{5}a^{-3}+2z^{5}a^{-5}-10z^{5}a^{-7}-11z^{5}a^{-9}-7z^{4}a^{-2}+17z^{4}a^{-4}+33z^{4}a^{-6}+13z^{4}a^{-8}-2z^{4}a^{-10}-6z^{4}-2az^{3}-2z^{3}a^{-1}+6z^{3}a^{-5}+12z^{3}a^{-7}+6z^{3}a^{-9}+7z^{2}a^{-2}-5z^{2}a^{-4}-11z^{2}a^{-6}-3z^{2}a^{-8}+4z^{2}+az+2za^{-1}+za^{-3}-3za^{-5}-3za^{-7}-3a^{-2}+a^{-6}-1$
The A2 invariant	Data:K11a85/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a85/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a85 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["K11a85"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-10t^{2}+25t-33+25t^{-1}-10t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 107, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{9}-4q^{8}+7q^{7}-11q^{6}+15q^{5}-17q^{4}+17q^{3}-14q^{2}+11q-6+3q^{-1}-q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-2}+z^{6}a^{-4}+3z^{4}a^{-2}+2z^{4}a^{-4}-2z^{4}a^{-6}-z^{4}+5z^{2}a^{-2}+2z^{2}a^{-4}-3z^{2}a^{-6}+z^{2}a^{-8}-2z^{2}+3a^{-2}-a^{-6}-1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-4}+z^{10}a^{-6}+3z^{9}a^{-3}+7z^{9}a^{-5}+4z^{9}a^{-7}+4z^{8}a^{-2}+8z^{8}a^{-4}+10z^{8}a^{-6}+6z^{8}a^{-8}+4z^{7}a^{-1}+z^{7}a^{-3}-11z^{7}a^{-5}-4z^{7}a^{-7}+4z^{7}a^{-9}-z^{6}a^{-2}-19z^{6}a^{-4}-33z^{6}a^{-6}-17z^{6}a^{-8}+z^{6}a^{-10}+3z^{6}+az^{5}-4z^{5}a^{-1}-4z^{5}a^{-3}+2z^{5}a^{-5}-10z^{5}a^{-7}-11z^{5}a^{-9}-7z^{4}a^{-2}+17z^{4}a^{-4}+33z^{4}a^{-6}+13z^{4}a^{-8}-2z^{4}a^{-10}-6z^{4}-2az^{3}-2z^{3}a^{-1}+6z^{3}a^{-5}+12z^{3}a^{-7}+6z^{3}a^{-9}+7z^{2}a^{-2}-5z^{2}a^{-4}-11z^{2}a^{-6}-3z^{2}a^{-8}+4z^{2}+az+2za^{-1}+za^{-3}-3za^{-5}-3za^{-7}-3a^{-2}+a^{-6}-1$

Vassiliev invariants

V2 and V3:

(3, 4)

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 $12$ $32$ $72$ $142$ $18$ $384$ ${\frac {1952}{3}}$ ${\frac {416}{3}}$ $64$ $288$ $512$ $1704$ $216$ ${\frac {30591}{10}}$ ${\frac {3014}{15}}$ ${\frac {16222}{15}}$ ${\frac {1}{6}}$ ${\frac {1471}{10}}$

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=$2 is the signature of K11a85. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 19                       1 1 17                     3   -3 15                   4 1   3 13                 7 3     -4 11               8 4       4 9             9 7         -2 7           8 8           0 5         6 9             3 3       5 8               -3 1     2 7                 5 -1   1 4                   -3 -3   2                     2 -5 1                       -1

Integral Khovanov Homology (db, data source)

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

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, 85]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 85]]
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, 14, 21, 13], X[18, 16, 19, 15], X[8, 18, 9, 17], X[14, 20, 15, 19], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 85]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -11, 4, -9, 5, -2, 6, -3, 7, -10, 8, -4, 9, -8, 10, -7, 11, -6]
In[5]:=	BR[Knot[11, Alternating, 85]]
Out[5]=	BR[Knot[11, Alternating, 85]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 85]][t]
Out[6]=	2 10 25 2 3 -33 + -- - -- + -- + 25 t - 10 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 85]][z]
Out[7]=	2 4 6 1 + 3 z + 2 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 85]}
In[9]:=	{KnotDet[Knot[11, Alternating, 85]], KnotSignature[Knot[11, Alternating, 85]]}
Out[9]=	{107, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 85]][q]
Out[10]=	-2 3 2 3 4 5 6 7 -6 - q + - + 11 q - 14 q + 17 q - 17 q + 15 q - 11 q + 7 q - q 8 9 4 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 85]}
In[12]:=	A2Invariant[Knot[11, Alternating, 85]][q]
Out[12]=	-6 -4 -2 2 4 6 8 10 12 -1 - q + q - q + 4 q - 2 q + 3 q + 2 q - q + 2 q - 14 16 20 22 24 26 28 3 q + 2 q - 2 q + 2 q - 2 q - q + q
In[13]:=	Kauffman[Knot[11, Alternating, 85]][a, z]
Out[13]=	2 2 -6 3 3 z 3 z z 2 z 2 3 z 11 z -1 + a - -- - --- - --- + -- + --- + a z + 4 z - ---- - ----- - 2 7 5 3 a 8 6 a a a a a a 2 2 3 3 3 3 4 5 z 7 z 6 z 12 z 6 z 2 z 3 4 2 z ---- + ---- + ---- + ----- + ---- - ---- - 2 a z - 6 z - ---- + 4 2 9 7 5 a 10 a a a a a a 4 4 4 4 5 5 5 5 5 13 z 33 z 17 z 7 z 11 z 10 z 2 z 4 z 4 z ----- + ----- + ----- - ---- - ----- - ----- + ---- - ---- - ---- + 8 6 4 2 9 7 5 3 a a a a a a a a a 6 6 6 6 6 7 7 5 6 z 17 z 33 z 19 z z 4 z 4 z a z + 3 z + --- - ----- - ----- - ----- - -- + ---- - ---- - 10 8 6 4 2 9 7 a a a a a a a 7 7 7 8 8 8 8 9 9 9 11 z z 4 z 6 z 10 z 8 z 4 z 4 z 7 z 3 z ----- + -- + ---- + ---- + ----- + ---- + ---- + ---- + ---- + ---- + 5 3 a 8 6 4 2 7 5 3 a a a a a a a a a 10 10 z z --- + --- 6 4 a a
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 85]], Vassiliev[3][Knot[11, Alternating, 85]]}
Out[14]=	{0, 4}
In[15]:=	Kh[Knot[11, Alternating, 85]][q, t]
Out[15]=	3 1 2 1 4 2 q 3 5 7 q + 5 q + ----- + ----- + ---- + --- + --- + 8 q t + 6 q t + 5 3 3 2 2 q t t q t q t q t 5 2 7 2 7 3 9 3 9 4 11 4 9 q t + 8 q t + 8 q t + 9 q t + 7 q t + 8 q t + 11 5 13 5 13 6 15 6 15 7 17 7 19 8 4 q t + 7 q t + 3 q t + 4 q t + q t + 3 q t + q t