9 2

9_1

9_3

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9 2 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,12,4,13 X5,18,6,1 X7,16,8,17 X9,14,10,15 X13,10,14,11 X15,8,16,9 X17,6,18,7 X11,2,12,3
Gauss code	-1, 9, -2, 1, -3, 8, -4, 7, -5, 6, -9, 2, -6, 5, -7, 4, -8, 3
Dowker-Thistlethwaite code	4 12 18 16 14 2 10 8 6
Conway Notation	[72]

Minimum Braid Representative: Length is 12, width is 5. Braid index is 5.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index $\{4,7\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-12][1] Hyperbolic Volume 3.48666 A-Polynomial See Data:9 2/A-polynomial

[edit Notes for 9 2's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $1$ Rasmussen s-Invariant -2

[edit Notes for 9 2's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$4t-7+4t^{-1}$
Conway polynomial	$4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 15, -2 }
Jones polynomial	$q^{-1}-q^{-2}+2q^{-3}-2q^{-4}+2q^{-5}-2q^{-6}+2q^{-7}-q^{-8}+q^{-9}-q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$-a^{10}+z^{2}a^{8}+a^{8}+z^{2}a^{6}+z^{2}a^{4}+z^{2}a^{2}+a^{2}$
Kauffman polynomial (db, data sources)	$z^{7}a^{11}-6z^{5}a^{11}+10z^{3}a^{11}-4za^{11}+z^{8}a^{10}-6z^{6}a^{10}+11z^{4}a^{10}-7z^{2}a^{10}+a^{10}+2z^{7}a^{9}-10z^{5}a^{9}+13z^{3}a^{9}-4za^{9}+z^{8}a^{8}-5z^{6}a^{8}+8z^{4}a^{8}-6z^{2}a^{8}+a^{8}+z^{7}a^{7}-3z^{5}a^{7}+z^{3}a^{7}+z^{6}a^{6}-2z^{4}a^{6}+z^{5}a^{5}-z^{3}a^{5}+z^{4}a^{4}+z^{3}a^{3}+z^{2}a^{2}-a^{2}$
The A2 invariant	$-q^{32}-q^{30}+q^{24}+q^{22}+q^{8}+q^{6}+q^{2}$
The G2 invariant	$q^{156}+q^{152}-q^{150}+q^{142}-2q^{140}+q^{138}-q^{136}-q^{134}-2q^{130}-q^{128}-q^{126}-q^{124}-q^{118}+q^{112}-q^{108}+q^{106}+q^{104}+2q^{102}+q^{98}+q^{94}+q^{92}-2q^{90}+q^{88}+q^{86}+q^{76}-q^{72}+q^{66}-q^{62}-q^{52}+q^{48}+q^{38}+q^{34}+q^{28}+q^{24}+q^{20}+q^{14}+q^{10}$

Further Quantum Invariants

Further quantum knot invariants for 9_2.

A1 Invariants.

Weight	Invariant
1	$-q^{21}+q^{15}+q^{5}+q$
2	$q^{60}-q^{56}-q^{50}+q^{46}-q^{30}-q^{28}+q^{18}+q^{16}+q^{14}+q^{8}+q^{2}$
3	$-q^{117}+q^{113}+q^{111}-q^{107}+q^{103}-q^{99}-q^{97}+q^{93}+q^{73}+q^{71}-q^{67}-q^{61}-q^{59}-q^{53}+q^{49}+q^{47}-q^{43}-q^{41}-q^{39}+q^{37}-q^{33}-q^{31}+q^{29}+2q^{27}-q^{23}+q^{21}+2q^{19}+q^{17}+q^{11}+q^{3}$
4	$q^{192}-q^{188}-q^{186}-q^{184}+q^{182}+q^{180}+q^{178}-2q^{174}+q^{170}+q^{168}+q^{166}-q^{164}-q^{162}-q^{160}+q^{156}-q^{134}-q^{132}+q^{128}+2q^{126}-q^{122}+q^{118}+2q^{116}-2q^{112}-q^{110}+q^{106}-2q^{102}-q^{100}+q^{96}+q^{94}+q^{90}+q^{88}+q^{86}-q^{82}-q^{80}+q^{76}-q^{72}-q^{70}-q^{68}-q^{66}+q^{62}-q^{60}-2q^{58}-2q^{56}+2q^{52}+q^{50}-2q^{46}+q^{44}+2q^{42}+q^{40}-q^{38}-2q^{36}+q^{34}+2q^{32}+q^{30}-q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{14}+q^{4}$
5	$-q^{285}+q^{281}+q^{279}+q^{277}-q^{273}-2q^{271}-q^{269}+q^{265}+2q^{263}+q^{261}-q^{259}-2q^{257}-q^{255}+q^{251}+2q^{249}+q^{247}-q^{243}-q^{241}-q^{239}+q^{235}+q^{213}+q^{211}-q^{207}-2q^{205}-2q^{203}+2q^{199}+2q^{197}-2q^{193}-2q^{191}-q^{189}+2q^{187}+4q^{185}+2q^{183}-q^{181}-2q^{179}-2q^{177}+2q^{173}+2q^{171}-2q^{167}-2q^{165}-q^{163}+q^{159}+q^{157}-q^{155}-q^{153}-q^{151}+q^{147}+2q^{145}+q^{143}-q^{139}-q^{137}+q^{135}+2q^{133}+q^{131}-q^{127}+q^{123}-q^{119}-2q^{117}-q^{115}+q^{113}+2q^{111}+q^{109}-q^{105}-q^{103}-2q^{101}+q^{97}+2q^{95}+2q^{93}+q^{91}-2q^{89}-3q^{87}-2q^{85}+q^{81}+q^{79}-q^{77}-2q^{75}-3q^{73}-q^{71}+q^{69}+q^{67}-q^{63}+q^{57}-q^{53}+2q^{49}+3q^{47}+q^{45}-q^{43}-q^{41}-q^{39}+q^{37}+2q^{35}+q^{33}+q^{23}+q^{21}+q^{19}+q^{17}+q^{5}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{32}-q^{30}+q^{24}+q^{22}+q^{8}+q^{6}+q^{2}$
1,1	$q^{84}+2q^{80}-2q^{78}+2q^{76}-4q^{74}+2q^{72}-4q^{70}+4q^{62}-q^{60}+4q^{58}-4q^{56}+2q^{54}-4q^{52}+2q^{50}-2q^{48}+2q^{46}-2q^{42}-2q^{40}-2q^{38}-2q^{36}+2q^{30}+2q^{28}+2q^{26}+2q^{24}+q^{20}+2q^{16}+2q^{12}+2q^{8}+q^{4}$
2,0	$q^{82}+q^{80}+q^{78}-q^{76}-q^{74}-q^{72}-q^{70}-q^{68}-q^{66}+q^{64}+q^{62}+q^{60}-q^{44}-2q^{42}-2q^{40}-q^{38}+q^{32}+q^{30}+q^{28}+2q^{26}+q^{24}+q^{20}+q^{18}+q^{16}+q^{12}+q^{10}+q^{4}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{66}+q^{62}-q^{58}-q^{56}-q^{54}-q^{52}-q^{50}-q^{46}-q^{42}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{16}+q^{12}+2q^{10}+q^{8}+q^{4}$
1,0,0	$-q^{43}-q^{41}-q^{39}+q^{33}+q^{31}+q^{29}+q^{11}+q^{9}+q^{7}+q^{3}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{66}-q^{62}-q^{58}+q^{56}-q^{54}+q^{52}+q^{50}+q^{46}+q^{42}-2q^{40}+q^{38}-q^{36}+q^{34}-q^{32}+q^{30}+q^{16}+q^{12}+q^{8}+q^{4}$
1,0	$q^{108}+q^{100}-q^{96}-q^{94}-q^{88}-q^{86}+q^{82}-q^{76}-q^{68}-q^{66}+q^{56}+q^{54}+q^{48}+q^{46}+q^{26}+q^{18}+q^{16}+q^{14}+q^{6}$

G2 Invariants.

Weight	Invariant
1,0	$q^{156}+q^{152}-q^{150}+q^{142}-2q^{140}+q^{138}-q^{136}-q^{134}-2q^{130}-q^{128}-q^{126}-q^{124}-q^{118}+q^{112}-q^{108}+q^{106}+q^{104}+2q^{102}+q^{98}+q^{94}+q^{92}-2q^{90}+q^{88}+q^{86}+q^{76}-q^{72}+q^{66}-q^{62}-q^{52}+q^{48}+q^{38}+q^{34}+q^{28}+q^{24}+q^{20}+q^{14}+q^{10}$

.

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["9 2"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$4t-7+4t^{-1}$

In[5]:=

Conway[K][z]

Out[5]=

$4z^{2}+1$

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]=

{ 15, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {7_4, ...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n13, ...}

Vassiliev invariants

V2 and V3:

(4, -10)

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 $16$ $-80$ $128$ ${\frac {1304}{3}}$ ${\frac {184}{3}}$ $-1280$ $-{\frac {8000}{3}}$ $-{\frac {1280}{3}}$ $-400$ ${\frac {2048}{3}}$ $3200$ ${\frac {20864}{3}}$ ${\frac {2944}{3}}$ ${\frac {249422}{15}}$ $-{\frac {856}{5}}$ ${\frac {315368}{45}}$ ${\frac {2482}{9}}$ ${\frac {13742}{15}}$

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

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{-2}-q^{-3}+2q^{-5}-2q^{-6}+3q^{-8}-2q^{-9}+2q^{-11}-2q^{-12}+2q^{-14}-3q^{-15}+3q^{-17}-3q^{-18}+3q^{-20}-3q^{-21}+3q^{-23}-2q^{-24}-q^{-25}+2q^{-26}-q^{-27}-q^{-28}+q^{-29}$
3	$q^{-3}-q^{-4}+2q^{-7}-2q^{-8}+q^{-10}+3q^{-11}-3q^{-12}-2q^{-13}+2q^{-14}+5q^{-15}-4q^{-16}-4q^{-17}+2q^{-18}+6q^{-19}-3q^{-20}-6q^{-21}+2q^{-22}+6q^{-23}-2q^{-24}-5q^{-25}+2q^{-26}+5q^{-27}-3q^{-28}-4q^{-29}+2q^{-30}+4q^{-31}-3q^{-32}-3q^{-33}+2q^{-34}+3q^{-35}-2q^{-36}-2q^{-37}+2q^{-38}+2q^{-39}-2q^{-40}-2q^{-41}+2q^{-42}+2q^{-43}-2q^{-44}-2q^{-45}+2q^{-46}+2q^{-47}-q^{-48}-3q^{-49}+q^{-50}+2q^{-51}-2q^{-53}+q^{-55}+q^{-56}-q^{-57}$
4	$q^{-4}-q^{-5}+2q^{-9}-2q^{-10}+q^{-11}+2q^{-14}-4q^{-15}+2q^{-16}+q^{-17}+q^{-18}+q^{-19}-7q^{-20}+3q^{-21}+3q^{-22}+2q^{-23}-10q^{-25}+5q^{-26}+4q^{-27}+3q^{-28}-2q^{-29}-12q^{-30}+5q^{-31}+5q^{-32}+5q^{-33}-3q^{-34}-13q^{-35}+5q^{-36}+5q^{-37}+5q^{-38}-2q^{-39}-12q^{-40}+4q^{-41}+4q^{-42}+5q^{-43}-q^{-44}-11q^{-45}+4q^{-46}+4q^{-47}+4q^{-48}-11q^{-50}+3q^{-51}+3q^{-52}+3q^{-53}+2q^{-54}-10q^{-55}+2q^{-56}+2q^{-57}+2q^{-58}+4q^{-59}-8q^{-60}+q^{-61}+q^{-62}+q^{-63}+5q^{-64}-6q^{-65}+5q^{-69}-5q^{-70}+5q^{-74}-5q^{-75}+5q^{-79}-4q^{-80}-q^{-81}-q^{-82}+5q^{-84}-2q^{-85}-q^{-86}-q^{-87}-q^{-88}+3q^{-89}-q^{-92}-q^{-93}+q^{-94}$
5	Not Available
6	Not Available
7	Not Available

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[9, 2]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[5, 18, 6, 1], X[7, 16, 8, 17], X[9, 14, 10, 15], X[13, 10, 14, 11], X[15, 8, 16, 9], X[17, 6, 18, 7], X[11, 2, 12, 3]]
In[3]:=	GaussCode[Knot[9, 2]]
Out[3]=	GaussCode[-1, 9, -2, 1, -3, 8, -4, 7, -5, 6, -9, 2, -6, 5, -7, 4, -8, 3]
In[4]:=	DTCode[Knot[9, 2]]
Out[4]=	DTCode[4, 12, 18, 16, 14, 2, 10, 8, 6]
In[5]:=	br = BR[Knot[9, 2]]
Out[5]=	BR[5, {-1, -1, -1, -2, 1, -2, -3, 2, -3, -4, 3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[9, 2]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[9, 2]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 2]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 1, 2, {4, 7}, 1}
In[10]:=	alex = Alexander[Knot[9, 2]][t]
Out[10]=	4 -7 + - + 4 t t
In[11]:=	Conway[Knot[9, 2]][z]
Out[11]=	2 1 + 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[7, 4], Knot[9, 2]}
In[13]:=	{KnotDet[Knot[9, 2]], KnotSignature[Knot[9, 2]]}
Out[13]=	{15, -2}
In[14]:=	Jones[Knot[9, 2]][q]
Out[14]=	-10 -9 -8 2 2 2 2 2 -2 1 -q + q - q + -- - -- + -- - -- + -- - q + - 7 6 5 4 3 q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 2], Knot[11, NonAlternating, 13]}
In[16]:=	A2Invariant[Knot[9, 2]][q]
Out[16]=	-32 -30 -24 -22 -8 -6 -2 -q - q + q + q + q + q + q
In[17]:=	HOMFLYPT[Knot[9, 2]][a, z]
Out[17]=	2 8 10 2 2 4 2 6 2 8 2 a + a - a + a z + a z + a z + a z
In[18]:=	Kauffman[Knot[9, 2]][a, z]
Out[18]=	2 8 10 9 11 2 2 8 2 10 2 -a + a + a - 4 a z - 4 a z + a z - 6 a z - 7 a z + 3 3 5 3 7 3 9 3 11 3 4 4 6 4 a z - a z + a z + 13 a z + 10 a z + a z - 2 a z + 8 4 10 4 5 5 7 5 9 5 11 5 6 6 8 a z + 11 a z + a z - 3 a z - 10 a z - 6 a z + a z - 8 6 10 6 7 7 9 7 11 7 8 8 10 8 5 a z - 6 a z + a z + 2 a z + a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[9, 2]], Vassiliev[3][Knot[9, 2]]}
Out[19]=	{4, -10}
In[20]:=	Kh[Knot[9, 2]][q, t]
Out[20]=	-3 1 1 1 1 1 1 1 q + - + ------ + ------ + ------ + ------ + ------ + ------ + q 21 9 17 8 17 7 15 6 13 6 13 5 q t q t q t q t q t q t 1 1 1 1 1 1 1 1 ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- 11 5 11 4 9 4 9 3 7 3 7 2 5 2 3 q t q t q t q t q t q t q t q t
In[21]:=	ColouredJones[Knot[9, 2], 2][q]
Out[21]=	-29 -28 -27 2 -25 2 3 3 3 3 3 q - q - q + --- - q - --- + --- - --- + --- - --- + --- - 26 24 23 21 20 18 17 q q q q q q q 3 2 2 2 2 3 2 2 -3 -2 --- + --- - --- + --- - -- + -- - -- + -- - q + q 15 14 12 11 9 8 6 5 q q q q q q q q

See/edit the Rolfsen_Splice_Template. Back to the top.

9_1 9_3