K11a95

K11a94

K11a96

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	K11a95 Quick Notes

K11a95 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X12,6,13,5 X18,8,19,7 X2,10,3,9 X8,12,9,11 X22,14,1,13 X20,16,21,15 X6,18,7,17 X16,20,17,19 X14,22,15,21
Gauss code	1, -5, 2, -1, 3, -9, 4, -6, 5, -2, 6, -3, 7, -11, 8, -10, 9, -4, 10, -8, 11, -7
Dowker-Thistlethwaite code	4 10 12 18 2 8 22 20 6 16 14
Conway Notation	[24122]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a95/ThurstonBennequinNumber Hyperbolic Volume 11.7607 A-Polynomial See Data:K11a95/A-polynomial

[edit Notes for K11a95's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a95's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$6t^{2}-18t+25-18t^{-1}+6t^{-2}$
Conway polynomial	$6z^{4}+6z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 73, 4 }
Jones polynomial	$-q^{13}+3q^{12}-5q^{11}+7q^{10}-10q^{9}+11q^{8}-11q^{7}+10q^{6}-7q^{5}+5q^{4}-2q^{3}+q^{2}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-4}+2z^{4}a^{-6}+2z^{4}a^{-8}+z^{4}a^{-10}+2z^{2}a^{-4}+3z^{2}a^{-6}+2z^{2}a^{-8}-z^{2}a^{-12}+a^{-4}+a^{-6}-a^{-10}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-10}+z^{10}a^{-12}+2z^{9}a^{-9}+5z^{9}a^{-11}+3z^{9}a^{-13}+3z^{8}a^{-8}+2z^{8}a^{-10}+2z^{8}a^{-12}+3z^{8}a^{-14}+3z^{7}a^{-7}-15z^{7}a^{-11}-11z^{7}a^{-13}+z^{7}a^{-15}+3z^{6}a^{-6}-3z^{6}a^{-8}-10z^{6}a^{-10}-17z^{6}a^{-12}-13z^{6}a^{-14}+2z^{5}a^{-5}-z^{5}a^{-7}-6z^{5}a^{-9}+10z^{5}a^{-11}+9z^{5}a^{-13}-4z^{5}a^{-15}+z^{4}a^{-4}-3z^{4}a^{-6}+2z^{4}a^{-8}+9z^{4}a^{-10}+18z^{4}a^{-12}+15z^{4}a^{-14}-2z^{3}a^{-5}-2z^{3}a^{-7}+5z^{3}a^{-9}+z^{3}a^{-11}+4z^{3}a^{-15}-2z^{2}a^{-4}+2z^{2}a^{-6}-z^{2}a^{-8}-5z^{2}a^{-10}-4z^{2}a^{-12}-4z^{2}a^{-14}+2za^{-7}-za^{-9}-3za^{-11}-za^{-13}-za^{-15}+a^{-4}-a^{-6}+a^{-10}$
The A2 invariant	Data:K11a95/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a95/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a95 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["K11a95"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$6t^{2}-18t+25-18t^{-1}+6t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 73, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{13}+3q^{12}-5q^{11}+7q^{10}-10q^{9}+11q^{8}-11q^{7}+10q^{6}-7q^{5}+5q^{4}-2q^{3}+q^{2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{4}a^{-4}+2z^{4}a^{-6}+2z^{4}a^{-8}+z^{4}a^{-10}+2z^{2}a^{-4}+3z^{2}a^{-6}+2z^{2}a^{-8}-z^{2}a^{-12}+a^{-4}+a^{-6}-a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-10}+z^{10}a^{-12}+2z^{9}a^{-9}+5z^{9}a^{-11}+3z^{9}a^{-13}+3z^{8}a^{-8}+2z^{8}a^{-10}+2z^{8}a^{-12}+3z^{8}a^{-14}+3z^{7}a^{-7}-15z^{7}a^{-11}-11z^{7}a^{-13}+z^{7}a^{-15}+3z^{6}a^{-6}-3z^{6}a^{-8}-10z^{6}a^{-10}-17z^{6}a^{-12}-13z^{6}a^{-14}+2z^{5}a^{-5}-z^{5}a^{-7}-6z^{5}a^{-9}+10z^{5}a^{-11}+9z^{5}a^{-13}-4z^{5}a^{-15}+z^{4}a^{-4}-3z^{4}a^{-6}+2z^{4}a^{-8}+9z^{4}a^{-10}+18z^{4}a^{-12}+15z^{4}a^{-14}-2z^{3}a^{-5}-2z^{3}a^{-7}+5z^{3}a^{-9}+z^{3}a^{-11}+4z^{3}a^{-15}-2z^{2}a^{-4}+2z^{2}a^{-6}-z^{2}a^{-8}-5z^{2}a^{-10}-4z^{2}a^{-12}-4z^{2}a^{-14}+2za^{-7}-za^{-9}-3za^{-11}-za^{-13}-za^{-15}+a^{-4}-a^{-6}+a^{-10}$

Vassiliev invariants

V2 and V3:

(6, 15)

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 $24$ $120$ $288$ $732$ $92$ $2880$ $5008$ $800$ $568$ $2304$ $7200$ $17568$ $2208$ ${\frac {176751}{5}}$ ${\frac {25828}{15}}$ ${\frac {176164}{15}}$ ${\frac {929}{3}}$ ${\frac {6991}{5}}$

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=$4 is the signature of K11a95. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 10 11 χ 27                       1 -1 25                     2   2 23                   3 1   -2 21                 4 2     2 19               6 3       -3 17             5 4         1 15           6 6           0 13         4 5             -1 11       3 6               3 9     2 4                 -2 7     3                   3 5 1 2                     -1 3 1                       1

Integral Khovanov Homology (db, data source)

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

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, 95]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 95]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[12, 6, 13, 5], X[18, 8, 19, 7], X[2, 10, 3, 9], X[8, 12, 9, 11], X[22, 14, 1, 13], X[20, 16, 21, 15], X[6, 18, 7, 17], X[16, 20, 17, 19], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 95]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -6, 5, -2, 6, -3, 7, -11, 8, -10, 9, -4, 10, -8, 11, -7]
In[5]:=	BR[Knot[11, Alternating, 95]]
Out[5]=	BR[Knot[11, Alternating, 95]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 95]][t]
Out[6]=	6 18 2 25 + -- - -- - 18 t + 6 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 95]][z]
Out[7]=	2 4 1 + 6 z + 6 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 53], Knot[11, Alternating, 95]}
In[9]:=	{KnotDet[Knot[11, Alternating, 95]], KnotSignature[Knot[11, Alternating, 95]]}
Out[9]=	{73, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 95]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 q - 2 q + 5 q - 7 q + 10 q - 11 q + 11 q - 10 q + 7 q - 11 12 13 5 q + 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 95]}
In[12]:=	A2Invariant[Knot[11, Alternating, 95]][q]
Out[12]=	6 8 10 12 14 16 22 24 26 28 30 q - q + 2 q + q - q + 3 q + q - q + q - 2 q - q + 32 34 36 38 40 q - 2 q + q + q - q
In[13]:=	Kauffman[Knot[11, Alternating, 95]][a, z]
Out[13]=	2 2 2 -10 -6 -4 z z 3 z z 2 z 4 z 4 z 5 z a - a + a - --- - --- - --- - -- + --- - ---- - ---- - ---- - 15 13 11 9 7 14 12 10 a a a a a a a a 2 2 2 3 3 3 3 3 4 4 z 2 z 2 z 4 z z 5 z 2 z 2 z 15 z 18 z -- + ---- - ---- + ---- + --- + ---- - ---- - ---- + ----- + ----- + 8 6 4 15 11 9 7 5 14 12 a a a a a a a a a a 4 4 4 4 5 5 5 5 5 5 9 z 2 z 3 z z 4 z 9 z 10 z 6 z z 2 z ---- + ---- - ---- + -- - ---- + ---- + ----- - ---- - -- + ---- - 10 8 6 4 15 13 11 9 7 5 a a a a a a a a a a 6 6 6 6 6 7 7 7 7 13 z 17 z 10 z 3 z 3 z z 11 z 15 z 3 z ----- - ----- - ----- - ---- + ---- + --- - ----- - ----- + ---- + 14 12 10 8 6 15 13 11 7 a a a a a a a a a 8 8 8 8 9 9 9 10 10 3 z 2 z 2 z 3 z 3 z 5 z 2 z z z ---- + ---- + ---- + ---- + ---- + ---- + ---- + --- + --- 14 12 10 8 13 11 9 12 10 a a a a a a a a a
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 95]], Vassiliev[3][Knot[11, Alternating, 95]]}
Out[14]=	{0, 15}
In[15]:=	Kh[Knot[11, Alternating, 95]][q, t]
Out[15]=	3 5 5 7 2 9 2 9 3 11 3 11 4 q + q + 2 q t + 3 q t + 2 q t + 4 q t + 3 q t + 6 q t + 13 4 13 5 15 5 15 6 17 6 17 7 4 q t + 5 q t + 6 q t + 6 q t + 5 q t + 4 q t + 19 7 19 8 21 8 21 9 23 9 23 10 6 q t + 3 q t + 4 q t + 2 q t + 3 q t + q t + 25 10 27 11 2 q t + q t