K11a99

K11a98

K11a100

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	K11a99 Quick Notes

K11a99 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X20,8,21,7 X16,10,17,9 X2,11,3,12 X18,13,19,14 X8,16,9,15 X22,17,1,18 X6,20,7,19 X14,21,15,22
Gauss code	1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -11, 8, -5, 9, -7, 10, -4, 11, -9
Dowker-Thistlethwaite code	4 10 12 20 16 2 18 8 22 6 14
Conway Notation	[211:02:02]

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a99/ThurstonBennequinNumber Hyperbolic Volume 16.6086 A-Polynomial See Data:K11a99/A-polynomial

[edit Notes for K11a99'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 K11a99's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-17t^{2}+28t-31+28t^{-1}-17t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}-z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 135, 2 }
Jones polynomial	$q^{7}-4q^{6}+9q^{5}-14q^{4}+19q^{3}-22q^{2}+21q-18+14q^{-1}-8q^{-2}+4q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-10z^{4}a^{-2}+3z^{4}a^{-4}+7z^{4}-2a^{2}z^{2}-10z^{2}a^{-2}+3z^{2}a^{-4}+7z^{2}-4a^{-2}+2a^{-4}+3$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}+5az^{9}+13z^{9}a^{-1}+8z^{9}a^{-3}+4a^{2}z^{8}+19z^{8}a^{-2}+13z^{8}a^{-4}+10z^{8}+a^{3}z^{7}-12az^{7}-27z^{7}a^{-1}-z^{7}a^{-3}+13z^{7}a^{-5}-14a^{2}z^{6}-62z^{6}a^{-2}-18z^{6}a^{-4}+9z^{6}a^{-6}-49z^{6}-3a^{3}z^{5}+az^{5}-25z^{5}a^{-3}-17z^{5}a^{-5}+4z^{5}a^{-7}+16a^{2}z^{4}+55z^{4}a^{-2}+7z^{4}a^{-4}-8z^{4}a^{-6}+z^{4}a^{-8}+55z^{4}+3a^{3}z^{3}+10az^{3}+15z^{3}a^{-1}+18z^{3}a^{-3}+9z^{3}a^{-5}-z^{3}a^{-7}-6a^{2}z^{2}-23z^{2}a^{-2}-5z^{2}a^{-4}+3z^{2}a^{-6}-21z^{2}-a^{3}z-3az-3za^{-1}-3za^{-3}-2za^{-5}+4a^{-2}+2a^{-4}+3$
The A2 invariant	$-q^{12}+q^{10}+q^{8}-q^{6}+4q^{4}-2q^{2}+2+q^{-2}-4q^{-4}+3q^{-6}-5q^{-8}+3q^{-10}-q^{-14}+3q^{-16}-2q^{-18}+q^{-20}$
The G2 invariant	$q^{60}-3q^{58}+9q^{56}-19q^{54}+28q^{52}-33q^{50}+17q^{48}+26q^{46}-91q^{44}+165q^{42}-204q^{40}+167q^{38}-37q^{36}-174q^{34}+394q^{32}-522q^{30}+480q^{28}-242q^{26}-135q^{24}+516q^{22}-741q^{20}+713q^{18}-405q^{16}-48q^{14}+467q^{12}-683q^{10}+599q^{8}-269q^{6}-156q^{4}+498q^{2}-586+391q^{-2}+18q^{-4}-460q^{-6}+748q^{-8}-756q^{-10}+454q^{-12}+43q^{-14}-573q^{-16}+935q^{-18}-987q^{-20}+703q^{-22}-172q^{-24}-401q^{-26}+798q^{-28}-894q^{-30}+643q^{-32}-188q^{-34}-279q^{-36}+566q^{-38}-560q^{-40}+292q^{-42}+105q^{-44}-431q^{-46}+544q^{-48}-402q^{-50}+68q^{-52}+293q^{-54}-544q^{-56}+602q^{-58}-444q^{-60}+168q^{-62}+140q^{-64}-372q^{-66}+462q^{-68}-420q^{-70}+276q^{-72}-96q^{-74}-65q^{-76}+179q^{-78}-227q^{-80}+213q^{-82}-152q^{-84}+78q^{-86}-5q^{-88}-46q^{-90}+68q^{-92}-74q^{-94}+57q^{-96}-33q^{-98}+13q^{-100}+4q^{-102}-10q^{-104}+11q^{-106}-10q^{-108}+6q^{-110}-3q^{-112}+q^{-114}$

Further Quantum Invariants

K11a99 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["K11a99"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-17t^{2}+28t-31+28t^{-1}-17t^{-2}+6t^{-3}-t^{-4}$

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]=

{ 135, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{7}-4q^{6}+9q^{5}-14q^{4}+19q^{3}-22q^{2}+21q-18+14q^{-1}-8q^{-2}+4q^{-3}-q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}+5az^{9}+13z^{9}a^{-1}+8z^{9}a^{-3}+4a^{2}z^{8}+19z^{8}a^{-2}+13z^{8}a^{-4}+10z^{8}+a^{3}z^{7}-12az^{7}-27z^{7}a^{-1}-z^{7}a^{-3}+13z^{7}a^{-5}-14a^{2}z^{6}-62z^{6}a^{-2}-18z^{6}a^{-4}+9z^{6}a^{-6}-49z^{6}-3a^{3}z^{5}+az^{5}-25z^{5}a^{-3}-17z^{5}a^{-5}+4z^{5}a^{-7}+16a^{2}z^{4}+55z^{4}a^{-2}+7z^{4}a^{-4}-8z^{4}a^{-6}+z^{4}a^{-8}+55z^{4}+3a^{3}z^{3}+10az^{3}+15z^{3}a^{-1}+18z^{3}a^{-3}+9z^{3}a^{-5}-z^{3}a^{-7}-6a^{2}z^{2}-23z^{2}a^{-2}-5z^{2}a^{-4}+3z^{2}a^{-6}-21z^{2}-a^{3}z-3az-3za^{-1}-3za^{-3}-2za^{-5}+4a^{-2}+2a^{-4}+3$

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 $-8$ $-16$ $32$ ${\frac {116}{3}}$ ${\frac {76}{3}}$ $128$ ${\frac {800}{3}}$ ${\frac {224}{3}}$ $80$ $-{\frac {256}{3}}$ $128$ $-{\frac {928}{3}}$ $-{\frac {608}{3}}$ ${\frac {1769}{15}}$ ${\frac {948}{5}}$ $-{\frac {13084}{45}}$ ${\frac {1111}{9}}$ $-{\frac {1591}{15}}$

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=$2 is the signature of K11a99. 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 15                       1 1 13                     3   -3 11                   6 1   5 9                 8 3     -5 7               11 6       5 5             11 8         -3 3           10 11           -1 1         9 12             3 -1       5 9               -4 -3     3 9                 6 -5   1 5                   -4 -7   3                     3 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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, 99]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 99]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[20, 8, 21, 7], X[16, 10, 17, 9], X[2, 11, 3, 12], X[18, 13, 19, 14], X[8, 16, 9, 15], X[22, 17, 1, 18], X[6, 20, 7, 19], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 99]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -11, 8, -5, 9, -7, 10, -4, 11, -9]
In[5]:=	BR[Knot[11, Alternating, 99]]
Out[5]=	BR[Knot[11, Alternating, 99]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 99]][t]
Out[6]=	-4 6 17 28 2 3 4 -31 - t + -- - -- + -- + 28 t - 17 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 99]][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, 99], Knot[11, Alternating, 277]}
In[9]:=	{KnotDet[Knot[11, Alternating, 99]], KnotSignature[Knot[11, Alternating, 99]]}
Out[9]=	{135, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 99]][q]
Out[10]=	-4 4 8 14 2 3 4 5 6 -18 - q + -- - -- + -- + 21 q - 22 q + 19 q - 14 q + 9 q - 4 q + 3 2 q q q 7 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 99]}
In[12]:=	A2Invariant[Knot[11, Alternating, 99]][q]
Out[12]=	-12 -10 -8 -6 4 2 2 4 6 8 2 - q + q + q - q + -- - -- + q - 4 q + 3 q - 5 q + 4 2 q q 10 14 16 18 20 3 q - q + 3 q - 2 q + q
In[13]:=	Kauffman[Knot[11, Alternating, 99]][a, z]
Out[13]=	2 2 2 4 2 z 3 z 3 z 3 2 3 z 5 z 3 + -- + -- - --- - --- - --- - 3 a z - a z - 21 z + ---- - ---- - 4 2 5 3 a 6 4 a a a a a a 2 3 3 3 3 23 z 2 2 z 9 z 18 z 15 z 3 3 3 ----- - 6 a z - -- + ---- + ----- + ----- + 10 a z + 3 a z + 2 7 5 3 a a a a a 4 4 4 4 5 5 5 4 z 8 z 7 z 55 z 2 4 4 z 17 z 25 z 55 z + -- - ---- + ---- + ----- + 16 a z + ---- - ----- - ----- + 8 6 4 2 7 5 3 a a a a a a a 6 6 6 7 5 3 5 6 9 z 18 z 62 z 2 6 13 z a z - 3 a z - 49 z + ---- - ----- - ----- - 14 a z + ----- - 6 4 2 5 a a a a 7 7 8 8 z 27 z 7 3 7 8 13 z 19 z 2 8 -- - ----- - 12 a z + a z + 10 z + ----- + ----- + 4 a z + 3 a 4 2 a a a 9 9 10 8 z 13 z 9 10 2 z ---- + ----- + 5 a z + 2 z + ----- 3 a 2 a a
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 99]], Vassiliev[3][Knot[11, Alternating, 99]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[11, Alternating, 99]][q, t]
Out[15]=	3 1 3 1 5 3 9 5 12 q + 10 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t 9 9 q 3 5 5 2 7 2 7 3 --- + --- + 11 q t + 11 q t + 8 q t + 11 q t + 6 q t + q t t 9 3 9 4 11 4 11 5 13 5 15 6 8 q t + 3 q t + 6 q t + q t + 3 q t + q t