K11a66

K11a65

K11a67

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	K11a66 Quick Notes

K11a66 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X8394 X16,5,17,6 X12,8,13,7 X2,9,3,10 X18,12,19,11 X20,13,21,14 X22,15,1,16 X10,18,11,17 X6,19,7,20 X14,21,15,22
Gauss code	1, -5, 2, -1, 3, -10, 4, -2, 5, -9, 6, -4, 7, -11, 8, -3, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code	4 8 16 12 2 18 20 22 10 6 14
Conway Notation	[.30.21.2]

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a66/ThurstonBennequinNumber Hyperbolic Volume 15.8011 A-Polynomial See Data:K11a66/A-polynomial

[edit Notes for K11a66's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a66's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-15t^{2}+24t-27+24t^{-1}-15t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}+z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 119, -2 }
Jones polynomial	$q^{3}-4q^{2}+8q-12+17q^{-1}-19q^{-2}+19q^{-3}-16q^{-4}+12q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+2a^{4}z^{6}-5a^{2}z^{6}+z^{6}-a^{6}z^{4}+8a^{4}z^{4}-9a^{2}z^{4}+3z^{4}-3a^{6}z^{2}+10a^{4}z^{2}-7a^{2}z^{2}+2z^{2}-2a^{6}+4a^{4}-2a^{2}+1$
Kauffman polynomial (db, data sources)	$2a^{4}z^{10}+2a^{2}z^{10}+5a^{5}z^{9}+11a^{3}z^{9}+6az^{9}+6a^{6}z^{8}+7a^{4}z^{8}+8a^{2}z^{8}+7z^{8}+5a^{7}z^{7}-6a^{5}z^{7}-28a^{3}z^{7}-13az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-9a^{6}z^{6}-26a^{4}z^{6}-35a^{2}z^{6}+z^{6}a^{-2}-20z^{6}+a^{9}z^{5}-7a^{7}z^{5}+5a^{5}z^{5}+29a^{3}z^{5}+6az^{5}-10z^{5}a^{-1}-5a^{8}z^{4}+8a^{6}z^{4}+36a^{4}z^{4}+41a^{2}z^{4}-2z^{4}a^{-2}+16z^{4}-2a^{9}z^{3}+2a^{7}z^{3}-3a^{5}z^{3}-12a^{3}z^{3}-az^{3}+4z^{3}a^{-1}+2a^{8}z^{2}-7a^{6}z^{2}-21a^{4}z^{2}-17a^{2}z^{2}-5z^{2}+a^{9}z+2a^{3}z+az+2a^{6}+4a^{4}+2a^{2}+1$
The A2 invariant	$-q^{24}-2q^{18}+3q^{16}-2q^{14}+2q^{12}+2q^{10}-2q^{8}+4q^{6}-4q^{4}+3q^{2}-q^{-2}+2q^{-4}-2q^{-6}+q^{-8}$
The G2 invariant	$q^{128}-2q^{126}+5q^{124}-8q^{122}+9q^{120}-8q^{118}+q^{116}+13q^{114}-30q^{112}+48q^{110}-58q^{108}+49q^{106}-22q^{104}-26q^{102}+86q^{100}-138q^{98}+166q^{96}-158q^{94}+93q^{92}+8q^{90}-140q^{88}+268q^{86}-338q^{84}+321q^{82}-195q^{80}-17q^{78}+243q^{76}-408q^{74}+439q^{72}-312q^{70}+66q^{68}+200q^{66}-367q^{64}+359q^{62}-166q^{60}-109q^{58}+341q^{56}-418q^{54}+280q^{52}+13q^{50}-346q^{48}+586q^{46}-611q^{44}+417q^{42}-53q^{40}-338q^{38}+616q^{36}-692q^{34}+539q^{32}-221q^{30}-147q^{28}+437q^{26}-537q^{24}+432q^{22}-158q^{20}-154q^{18}+360q^{16}-388q^{14}+209q^{12}+79q^{10}-342q^{8}+475q^{6}-401q^{4}+155q^{2}+161-417q^{-2}+514q^{-4}-431q^{-6}+212q^{-8}+43q^{-10}-244q^{-12}+335q^{-14}-307q^{-16}+202q^{-18}-63q^{-20}-51q^{-22}+109q^{-24}-122q^{-26}+95q^{-28}-53q^{-30}+19q^{-32}+9q^{-34}-19q^{-36}+18q^{-38}-14q^{-40}+7q^{-42}-3q^{-44}+q^{-46}$

Further Quantum Invariants

K11a66 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["K11a66"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-15t^{2}+24t-27+24t^{-1}-15t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 119, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{4}z^{10}+2a^{2}z^{10}+5a^{5}z^{9}+11a^{3}z^{9}+6az^{9}+6a^{6}z^{8}+7a^{4}z^{8}+8a^{2}z^{8}+7z^{8}+5a^{7}z^{7}-6a^{5}z^{7}-28a^{3}z^{7}-13az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-9a^{6}z^{6}-26a^{4}z^{6}-35a^{2}z^{6}+z^{6}a^{-2}-20z^{6}+a^{9}z^{5}-7a^{7}z^{5}+5a^{5}z^{5}+29a^{3}z^{5}+6az^{5}-10z^{5}a^{-1}-5a^{8}z^{4}+8a^{6}z^{4}+36a^{4}z^{4}+41a^{2}z^{4}-2z^{4}a^{-2}+16z^{4}-2a^{9}z^{3}+2a^{7}z^{3}-3a^{5}z^{3}-12a^{3}z^{3}-az^{3}+4z^{3}a^{-1}+2a^{8}z^{2}-7a^{6}z^{2}-21a^{4}z^{2}-17a^{2}z^{2}-5z^{2}+a^{9}z+2a^{3}z+az+2a^{6}+4a^{4}+2a^{2}+1$

Vassiliev invariants

V2 and V3:

(2, -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 $8$ $-32$ $32$ ${\frac {316}{3}}$ ${\frac {20}{3}}$ $-256$ $-{\frac {1376}{3}}$ $-{\frac {32}{3}}$ $-96$ ${\frac {256}{3}}$ $512$ ${\frac {2528}{3}}$ ${\frac {160}{3}}$ ${\frac {30991}{15}}$ $-{\frac {764}{15}}$ ${\frac {30604}{45}}$ ${\frac {1025}{9}}$ ${\frac {991}{15}}$

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 K11a66. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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, 66]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 66]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[16, 5, 17, 6], X[12, 8, 13, 7], X[2, 9, 3, 10], X[18, 12, 19, 11], X[20, 13, 21, 14], X[22, 15, 1, 16], X[10, 18, 11, 17], X[6, 19, 7, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 66]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -10, 4, -2, 5, -9, 6, -4, 7, -11, 8, -3, 9, -6, 10, -7, 11, -8]
In[5]:=	BR[Knot[11, Alternating, 66]]
Out[5]=	BR[Knot[11, Alternating, 66]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 66]][t]
Out[6]=	-4 6 15 24 2 3 4 -27 - t + -- - -- + -- + 24 t - 15 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 66]][z]
Out[7]=	2 4 6 8 1 + 2 z + z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 66], Knot[11, Alternating, 163]}
In[9]:=	{KnotDet[Knot[11, Alternating, 66]], KnotSignature[Knot[11, Alternating, 66]]}
Out[9]=	{119, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 66]][q]
Out[10]=	-8 3 7 12 16 19 19 17 2 3 -12 - q + -- - -- + -- - -- + -- - -- + -- + 8 q - 4 q + q 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 66], Knot[11, Alternating, 121]}
In[12]:=	A2Invariant[Knot[11, Alternating, 66]][q]
Out[12]=	-24 2 3 2 2 2 2 4 4 3 2 4 -q - --- + --- - --- + --- + --- - -- + -- - -- + -- - q + 2 q - 18 16 14 12 10 8 6 4 2 q q q q q q q q q 6 8 2 q + q
In[13]:=	Kauffman[Knot[11, Alternating, 66]][a, z]
Out[13]=	2 4 6 3 9 2 2 2 1 + 2 a + 4 a + 2 a + a z + 2 a z + a z - 5 z - 17 a z - 3 4 2 6 2 8 2 4 z 3 3 3 5 3 21 a z - 7 a z + 2 a z + ---- - a z - 12 a z - 3 a z + a 4 7 3 9 3 4 2 z 2 4 4 4 6 4 2 a z - 2 a z + 16 z - ---- + 41 a z + 36 a z + 8 a z - 2 a 5 8 4 10 z 5 3 5 5 5 7 5 9 5 5 a z - ----- + 6 a z + 29 a z + 5 a z - 7 a z + a z - a 6 7 6 z 2 6 4 6 6 6 8 6 4 z 20 z + -- - 35 a z - 26 a z - 9 a z + 3 a z + ---- - 2 a a 7 3 7 5 7 7 7 8 2 8 4 8 13 a z - 28 a z - 6 a z + 5 a z + 7 z + 8 a z + 7 a z + 6 8 9 3 9 5 9 2 10 4 10 6 a z + 6 a z + 11 a z + 5 a z + 2 a z + 2 a z
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 66]], Vassiliev[3][Knot[11, Alternating, 66]]}
Out[14]=	{0, -4}
In[15]:=	Kh[Knot[11, Alternating, 66]][q, t]
Out[15]=	8 10 1 2 1 5 2 7 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 q q t q t q t q t q t q t q t 9 7 10 9 9 10 5 t 2 ----- + ----- + ----- + ----- + ---- + ---- + --- + 7 q t + 3 q t + 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t 3 2 3 3 5 3 7 4 5 q t + q t + 3 q t + q t