K11a87

K11a86

K11a88

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	K11a87 Quick Notes

K11a87 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X16,8,17,7 X18,10,19,9 X2,11,3,12 X20,13,21,14 X8,16,9,15 X6,18,7,17 X22,19,1,20 X14,21,15,22
Gauss code	1, -6, 2, -1, 3, -9, 4, -8, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, -7, 11, -10
Dowker-Thistlethwaite code	4 10 12 16 18 2 20 8 6 22 14
Conway Notation	[31,211,21]

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a87/ThurstonBennequinNumber Hyperbolic Volume 15.1381 A-Polynomial See Data:K11a87/A-polynomial

[edit Notes for K11a87's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a87's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+11t^{2}-28t+39-28t^{-1}+11t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^{2}+t+1\right\}$
Determinant and Signature	{ 121, 0 }
Jones polynomial	$q^{6}-3q^{5}+7q^{4}-12q^{3}+16q^{2}-19q+20-17q^{-1}+13q^{-2}-8q^{-3}+4q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-z^{6}+2a^{2}z^{4}-3z^{4}a^{-2}+z^{4}a^{-4}-z^{4}-a^{4}z^{2}+2a^{2}z^{2}-6z^{2}a^{-2}+2z^{2}a^{-4}+z^{2}-4a^{-2}+2a^{-4}+3$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}+4az^{9}+8z^{9}a^{-1}+4z^{9}a^{-3}+7a^{2}z^{8}+12z^{8}a^{-2}+5z^{8}a^{-4}+14z^{8}+7a^{3}z^{7}+6az^{7}-8z^{7}a^{-1}-4z^{7}a^{-3}+3z^{7}a^{-5}+4a^{4}z^{6}-7a^{2}z^{6}-37z^{6}a^{-2}-13z^{6}a^{-4}+z^{6}a^{-6}-34z^{6}+a^{5}z^{5}-11a^{3}z^{5}-22az^{5}-8z^{5}a^{-1}-6z^{5}a^{-3}-8z^{5}a^{-5}-6a^{4}z^{4}-a^{2}z^{4}+41z^{4}a^{-2}+12z^{4}a^{-4}-3z^{4}a^{-6}+31z^{4}-a^{5}z^{3}+5a^{3}z^{3}+16az^{3}+12z^{3}a^{-1}+8z^{3}a^{-3}+6z^{3}a^{-5}+2a^{4}z^{2}+a^{2}z^{2}-21z^{2}a^{-2}-7z^{2}a^{-4}+2z^{2}a^{-6}-13z^{2}-a^{3}z-3az-3za^{-1}-3za^{-3}-2za^{-5}+4a^{-2}+2a^{-4}+3$
The A2 invariant	Data:K11a87/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a87/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a87 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["K11a87"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+11t^{2}-28t+39-28t^{-1}+11t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\left\{2,t^{2}+t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 121, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-3q^{5}+7q^{4}-12q^{3}+16q^{2}-19q+20-17q^{-1}+13q^{-2}-8q^{-3}+4q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}+4az^{9}+8z^{9}a^{-1}+4z^{9}a^{-3}+7a^{2}z^{8}+12z^{8}a^{-2}+5z^{8}a^{-4}+14z^{8}+7a^{3}z^{7}+6az^{7}-8z^{7}a^{-1}-4z^{7}a^{-3}+3z^{7}a^{-5}+4a^{4}z^{6}-7a^{2}z^{6}-37z^{6}a^{-2}-13z^{6}a^{-4}+z^{6}a^{-6}-34z^{6}+a^{5}z^{5}-11a^{3}z^{5}-22az^{5}-8z^{5}a^{-1}-6z^{5}a^{-3}-8z^{5}a^{-5}-6a^{4}z^{4}-a^{2}z^{4}+41z^{4}a^{-2}+12z^{4}a^{-4}-3z^{4}a^{-6}+31z^{4}-a^{5}z^{3}+5a^{3}z^{3}+16az^{3}+12z^{3}a^{-1}+8z^{3}a^{-3}+6z^{3}a^{-5}+2a^{4}z^{2}+a^{2}z^{2}-21z^{2}a^{-2}-7z^{2}a^{-4}+2z^{2}a^{-6}-13z^{2}-a^{3}z-3az-3za^{-1}-3za^{-3}-2za^{-5}+4a^{-2}+2a^{-4}+3$

Vassiliev invariants

V2 and V3:

(-2, -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 $-8$ $-16$ $32$ ${\frac {116}{3}}$ ${\frac {76}{3}}$ $128$ ${\frac {800}{3}}$ ${\frac {224}{3}}$ $80$ $-{\frac {256}{3}}$ $128$ $-{\frac {928}{3}}$ $-{\frac {608}{3}}$ ${\frac {1769}{15}}$ ${\frac {628}{5}}$ $-{\frac {10204}{45}}$ ${\frac {1111}{9}}$ $-{\frac {1591}{15}}$

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=$0 is the signature of K11a87. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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, 87]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 87]]
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, 10, 19, 9], X[2, 11, 3, 12], X[20, 13, 21, 14], X[8, 16, 9, 15], X[6, 18, 7, 17], X[22, 19, 1, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 87]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -9, 4, -8, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, -7, 11, -10]
In[5]:=	BR[Knot[11, Alternating, 87]]
Out[5]=	BR[Knot[11, Alternating, 87]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 87]][t]
Out[6]=	2 11 28 2 3 39 - -- + -- - -- - 28 t + 11 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 87]][z]
Out[7]=	2 4 6 1 - 2 z - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 87]}
In[9]:=	{KnotDet[Knot[11, Alternating, 87]], KnotSignature[Knot[11, Alternating, 87]]}
Out[9]=	{121, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 87]][q]
Out[10]=	-5 4 8 13 17 2 3 4 5 6 20 - q + -- - -- + -- - -- - 19 q + 16 q - 12 q + 7 q - 3 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 28], Knot[11, Alternating, 87], Knot[11, Alternating, 96]}
In[12]:=	A2Invariant[Knot[11, Alternating, 87]][q]
Out[12]=	-16 -14 2 3 2 2 5 2 4 6 -1 - q + q + --- - --- + -- - -- + -- + 3 q - 2 q - 3 q + 12 10 8 4 2 q q q q q 8 10 12 14 16 18 2 q - 4 q + 2 q + 2 q - q + q
In[13]:=	Kauffman[Knot[11, Alternating, 87]][a, z]
Out[13]=	2 2 2 4 2 z 3 z 3 z 3 2 2 z 7 z 3 + -- + -- - --- - --- - --- - 3 a z - a z - 13 z + ---- - ---- - 4 2 5 3 a 6 4 a a a a a a 2 3 3 3 21 z 2 2 4 2 6 z 8 z 12 z 3 3 3 ----- + a z + 2 a z + ---- + ---- + ----- + 16 a z + 5 a z - 2 5 3 a a a a 4 4 4 5 5 3 4 3 z 12 z 41 z 2 4 4 4 8 z a z + 31 z - ---- + ----- + ----- - a z - 6 a z - ---- - 6 4 2 5 a a a a 5 5 6 6 6 z 8 z 5 3 5 5 5 6 z 13 z ---- - ---- - 22 a z - 11 a z + a z - 34 z + -- - ----- - 3 a 6 4 a a a 6 7 7 7 37 z 2 6 4 6 3 z 4 z 8 z 7 3 7 ----- - 7 a z + 4 a z + ---- - ---- - ---- + 6 a z + 7 a z + 2 5 3 a a a a 8 8 9 9 10 8 5 z 12 z 2 8 4 z 8 z 9 10 z 14 z + ---- + ----- + 7 a z + ---- + ---- + 4 a z + z + --- 4 2 3 a 2 a a a a
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 87]], Vassiliev[3][Knot[11, Alternating, 87]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[11, Alternating, 87]][q, t]
Out[15]=	11 1 3 1 5 3 8 5 -- + 10 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 9 8 3 3 2 5 2 5 3 ---- + --- + 9 q t + 10 q t + 7 q t + 9 q t + 5 q t + 3 q t q t 7 3 7 4 9 4 9 5 11 5 13 6 7 q t + 2 q t + 5 q t + q t + 2 q t + q t