9 7

9_6

9_8

Visit 9 7's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 7's page at Knotilus! Visit 9 7's page at the original Knot Atlas!

9 7 Quick Notes

9 7 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,12,4,13 X5,16,6,17 X7,18,8,1 X17,6,18,7 X9,14,10,15 X13,10,14,11 X15,8,16,9 X11,2,12,3
Gauss code	-1, 9, -2, 1, -3, 5, -4, 8, -6, 7, -9, 2, -7, 6, -8, 3, -5, 4
Dowker-Thistlethwaite code	4 12 16 18 14 2 10 8 6
Conway Notation	[342]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-14][3] Hyperbolic Volume 8.01486 A-Polynomial See Data:9 7/A-polynomial

[edit Notes for 9 7's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 7's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{2}-7t+9-7t^{-1}+3t^{-2}$
Conway polynomial	$3z^{4}+5z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 29, -4 }
Jones polynomial	$q^{-2}-q^{-3}+3q^{-4}-4q^{-5}+5q^{-6}-5q^{-7}+4q^{-8}-3q^{-9}+2q^{-10}-q^{-11}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{10}-a^{10}+z^{4}a^{8}+2z^{2}a^{8}+a^{8}+z^{4}a^{6}+z^{2}a^{6}-a^{6}+z^{4}a^{4}+3z^{2}a^{4}+2a^{4}$
Kauffman polynomial (db, data sources)	$z^{5}a^{13}-3z^{3}a^{13}+za^{13}+2z^{6}a^{12}-6z^{4}a^{12}+3z^{2}a^{12}+2z^{7}a^{11}-6z^{5}a^{11}+5z^{3}a^{11}-2za^{11}+z^{8}a^{10}-2z^{6}a^{10}+2z^{4}a^{10}-2z^{2}a^{10}+a^{10}+3z^{7}a^{9}-9z^{5}a^{9}+11z^{3}a^{9}-3za^{9}+z^{8}a^{8}-3z^{6}a^{8}+7z^{4}a^{8}-4z^{2}a^{8}+a^{8}+z^{7}a^{7}-z^{5}a^{7}+2z^{3}a^{7}-za^{7}+z^{6}a^{6}-2z^{2}a^{6}+a^{6}+z^{5}a^{5}-z^{3}a^{5}-za^{5}+z^{4}a^{4}-3z^{2}a^{4}+2a^{4}$
The A2 invariant	$-q^{34}-q^{28}+q^{26}-q^{18}+q^{16}+q^{12}+2q^{10}+q^{6}$
The G2 invariant	$q^{176}-q^{174}+2q^{172}-3q^{170}+2q^{168}-q^{166}-2q^{164}+7q^{162}-8q^{160}+9q^{158}-7q^{156}+6q^{152}-12q^{150}+13q^{148}-11q^{146}+4q^{144}+4q^{142}-10q^{140}+10q^{138}-6q^{136}+5q^{132}-9q^{130}+5q^{128}-7q^{124}+11q^{122}-12q^{120}+8q^{118}-q^{116}-8q^{114}+13q^{112}-16q^{110}+15q^{108}-7q^{106}-q^{104}+9q^{102}-13q^{100}+14q^{98}-7q^{96}+q^{94}+5q^{92}-7q^{90}+5q^{88}+q^{86}-6q^{84}+8q^{82}-7q^{80}+4q^{76}-9q^{74}+10q^{72}-8q^{70}+4q^{68}-q^{66}-4q^{64}+6q^{62}-6q^{60}+7q^{58}-3q^{56}+3q^{54}+q^{52}-q^{50}+4q^{48}-3q^{46}+4q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+2q^{34}+q^{30}$

Further Quantum Invariants

Further quantum knot invariants for 9_7.

A1 Invariants.

Weight	Invariant
1	$-q^{23}+q^{21}-q^{19}+q^{17}-q^{15}+q^{11}-q^{9}+2q^{7}+q^{3}$
2	$q^{64}-q^{62}-q^{60}+3q^{58}-q^{56}-4q^{54}+3q^{52}+2q^{50}-4q^{48}+2q^{46}+3q^{44}-4q^{42}+2q^{38}-q^{36}-2q^{34}+3q^{30}-4q^{28}-2q^{26}+5q^{24}-2q^{22}-2q^{20}+4q^{18}+2q^{12}+q^{6}$
3	$-q^{123}+q^{121}+q^{119}-q^{117}-2q^{115}+q^{113}+5q^{111}-7q^{107}-3q^{105}+6q^{103}+7q^{101}-4q^{99}-10q^{97}+q^{95}+10q^{93}+4q^{91}-10q^{89}-7q^{87}+9q^{85}+9q^{83}-6q^{81}-9q^{79}+5q^{77}+8q^{75}-3q^{73}-8q^{71}+6q^{67}+2q^{65}-3q^{63}-6q^{61}+3q^{59}+9q^{57}-12q^{53}-4q^{51}+10q^{49}+6q^{47}-11q^{45}-7q^{43}+5q^{41}+7q^{39}-2q^{37}-5q^{35}+q^{33}+2q^{31}+q^{29}+q^{25}+q^{21}+q^{19}+2q^{17}+q^{9}$
4	$q^{200}-q^{198}-q^{196}+q^{194}+2q^{190}-3q^{188}-3q^{186}+2q^{184}+2q^{182}+9q^{180}-3q^{178}-11q^{176}-5q^{174}-q^{172}+18q^{170}+9q^{168}-5q^{166}-13q^{164}-20q^{162}+9q^{160}+18q^{158}+16q^{156}-q^{154}-31q^{152}-16q^{150}+5q^{148}+32q^{146}+26q^{144}-21q^{142}-32q^{140}-18q^{138}+26q^{136}+40q^{134}-3q^{132}-30q^{130}-30q^{128}+13q^{126}+36q^{124}+6q^{122}-19q^{120}-25q^{118}+4q^{116}+24q^{114}+10q^{112}-10q^{110}-17q^{108}-3q^{106}+11q^{104}+16q^{102}+q^{100}-11q^{98}-18q^{96}-4q^{94}+27q^{92}+18q^{90}-4q^{88}-33q^{86}-27q^{84}+25q^{82}+35q^{80}+17q^{78}-31q^{76}-44q^{74}+6q^{72}+28q^{70}+31q^{68}-8q^{66}-35q^{64}-12q^{62}+5q^{60}+24q^{58}+9q^{56}-13q^{54}-9q^{52}-8q^{50}+8q^{48}+9q^{46}-q^{42}-7q^{40}+q^{38}+4q^{36}+2q^{34}+3q^{32}-3q^{30}+q^{26}+q^{24}+3q^{22}+q^{12}$
5	$-q^{295}+q^{293}+q^{291}-q^{289}+q^{281}+2q^{279}-2q^{277}-5q^{275}-2q^{273}+q^{271}+6q^{269}+9q^{267}+5q^{265}-9q^{263}-17q^{261}-11q^{259}+q^{257}+19q^{255}+24q^{253}+13q^{251}-12q^{249}-30q^{247}-27q^{245}-9q^{243}+20q^{241}+40q^{239}+35q^{237}+2q^{235}-35q^{233}-54q^{231}-39q^{229}+11q^{227}+61q^{225}+75q^{223}+27q^{221}-51q^{219}-96q^{217}-73q^{215}+18q^{213}+105q^{211}+112q^{209}+21q^{207}-92q^{205}-132q^{203}-62q^{201}+67q^{199}+140q^{197}+92q^{195}-40q^{193}-132q^{191}-105q^{189}+11q^{187}+112q^{185}+107q^{183}+6q^{181}-91q^{179}-95q^{177}-15q^{175}+67q^{173}+81q^{171}+19q^{169}-50q^{167}-63q^{165}-18q^{163}+35q^{161}+48q^{159}+20q^{157}-20q^{155}-44q^{153}-26q^{151}+8q^{149}+35q^{147}+43q^{145}+12q^{143}-36q^{141}-62q^{139}-36q^{137}+31q^{135}+86q^{133}+72q^{131}-15q^{129}-102q^{127}-111q^{125}-14q^{123}+111q^{121}+142q^{119}+50q^{117}-93q^{115}-160q^{113}-92q^{111}+65q^{109}+157q^{107}+110q^{105}-18q^{103}-127q^{101}-124q^{99}-19q^{97}+87q^{95}+109q^{93}+40q^{91}-44q^{89}-84q^{87}-53q^{85}+11q^{83}+55q^{81}+45q^{79}+5q^{77}-27q^{75}-35q^{73}-14q^{71}+13q^{69}+22q^{67}+13q^{65}-3q^{63}-15q^{61}-11q^{59}+q^{57}+7q^{55}+8q^{53}+4q^{51}-6q^{49}-5q^{47}+2q^{43}+4q^{41}+5q^{39}-q^{37}-2q^{35}+q^{29}+3q^{27}+q^{25}+q^{15}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{34}-q^{28}+q^{26}-q^{18}+q^{16}+q^{12}+2q^{10}+q^{6}$
1,1	$q^{92}-2q^{90}+4q^{88}-8q^{86}+15q^{84}-20q^{82}+26q^{80}-34q^{78}+35q^{76}-34q^{74}+28q^{72}-16q^{70}+3q^{68}+14q^{66}-30q^{64}+42q^{62}-53q^{60}+58q^{58}-58q^{56}+56q^{54}-46q^{52}+36q^{50}-22q^{48}+6q^{46}+q^{44}-18q^{42}+18q^{40}-24q^{38}+21q^{36}-20q^{34}+20q^{32}-14q^{30}+16q^{28}-10q^{26}+12q^{24}-6q^{22}+8q^{20}-2q^{18}+4q^{16}+q^{12}$
2,0	$q^{86}+q^{78}-3q^{74}-q^{72}+q^{70}+q^{68}-q^{66}+3q^{62}+q^{60}-2q^{58}-q^{56}+q^{54}-q^{52}-q^{50}-q^{46}-3q^{44}-2q^{42}-3q^{38}-q^{36}+4q^{34}+2q^{32}-q^{30}+2q^{28}+4q^{26}+2q^{24}-q^{22}+2q^{20}+2q^{18}+q^{12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{74}-q^{72}+q^{68}-2q^{66}+2q^{64}+2q^{62}-3q^{60}+2q^{58}+q^{56}-5q^{54}-q^{52}+q^{50}-3q^{48}-q^{46}+q^{44}+q^{42}+4q^{36}-2q^{34}-3q^{32}+3q^{30}-2q^{28}-3q^{26}+4q^{24}+2q^{22}+q^{20}+3q^{18}+2q^{16}+q^{12}$
1,0,0	$-q^{45}-q^{41}-q^{37}+q^{35}+q^{31}-q^{25}-q^{23}+q^{21}+2q^{17}+q^{15}+2q^{13}+q^{9}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{96}-q^{92}+q^{90}+q^{88}-2q^{86}+4q^{82}+2q^{80}-2q^{78}+2q^{74}-3q^{72}-6q^{70}-q^{68}-2q^{66}-5q^{64}-q^{62}+q^{60}-q^{58}+4q^{54}+3q^{52}+q^{48}+3q^{46}-2q^{44}-5q^{42}-q^{40}-q^{36}+3q^{32}+5q^{30}+3q^{28}+3q^{26}+3q^{24}+2q^{22}+q^{18}$
1,0,0,0	$-q^{56}-q^{52}-q^{50}-q^{46}+q^{44}+q^{40}+q^{38}-q^{32}-q^{30}-q^{28}+q^{26}+2q^{22}+2q^{20}+q^{18}+2q^{16}+q^{12}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{74}+q^{72}-2q^{70}+3q^{68}-4q^{66}+4q^{64}-4q^{62}+3q^{60}-2q^{58}+q^{56}+q^{54}-3q^{52}+5q^{50}-7q^{48}+7q^{46}-7q^{44}+7q^{42}-6q^{40}+4q^{38}-2q^{36}+q^{32}-3q^{30}+4q^{28}-3q^{26}+4q^{24}-2q^{22}+3q^{20}-q^{18}+2q^{16}+q^{12}$
1,0	$q^{120}-q^{116}-q^{114}+q^{112}+2q^{110}-q^{108}-3q^{106}+4q^{102}+3q^{100}-3q^{98}-4q^{96}+q^{94}+4q^{92}+q^{90}-4q^{88}-3q^{86}+q^{84}+2q^{82}-q^{80}-3q^{78}+2q^{74}-3q^{70}-q^{68}+3q^{66}+2q^{64}-2q^{62}-2q^{60}+2q^{58}+3q^{56}-q^{54}-4q^{52}-q^{50}+4q^{48}+2q^{46}-3q^{44}-3q^{42}+4q^{38}+2q^{36}-q^{32}+q^{30}+2q^{28}+2q^{26}+q^{18}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{102}-q^{100}+q^{98}-2q^{96}+3q^{94}-3q^{92}+3q^{90}-3q^{88}+4q^{86}-2q^{84}+2q^{82}-q^{80}-q^{76}-4q^{74}+q^{72}-5q^{70}+3q^{68}-7q^{66}+5q^{64}-5q^{62}+7q^{60}-4q^{58}+5q^{56}-2q^{54}+4q^{52}-q^{46}-3q^{44}+2q^{42}-4q^{40}+q^{38}-3q^{36}+4q^{34}+4q^{30}+q^{28}+4q^{26}+q^{24}+2q^{22}+q^{18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{176}-q^{174}+2q^{172}-3q^{170}+2q^{168}-q^{166}-2q^{164}+7q^{162}-8q^{160}+9q^{158}-7q^{156}+6q^{152}-12q^{150}+13q^{148}-11q^{146}+4q^{144}+4q^{142}-10q^{140}+10q^{138}-6q^{136}+5q^{132}-9q^{130}+5q^{128}-7q^{124}+11q^{122}-12q^{120}+8q^{118}-q^{116}-8q^{114}+13q^{112}-16q^{110}+15q^{108}-7q^{106}-q^{104}+9q^{102}-13q^{100}+14q^{98}-7q^{96}+q^{94}+5q^{92}-7q^{90}+5q^{88}+q^{86}-6q^{84}+8q^{82}-7q^{80}+4q^{76}-9q^{74}+10q^{72}-8q^{70}+4q^{68}-q^{66}-4q^{64}+6q^{62}-6q^{60}+7q^{58}-3q^{56}+3q^{54}+q^{52}-q^{50}+4q^{48}-3q^{46}+4q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+2q^{34}+q^{30}$

.

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 7"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{2}-7t+9-7t^{-1}+3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$3z^{4}+5z^{2}+1$

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]=

{ 29, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

Vassiliev invariants

V2 and V3:

(5, -12)

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 $20$ $-96$ $200$ ${\frac {1654}{3}}$ ${\frac {218}{3}}$ $-1920$ $-3552$ $-576$ $-448$ ${\frac {4000}{3}}$ $4608$ ${\frac {33080}{3}}$ ${\frac {4360}{3}}$ ${\frac {140719}{6}}$ ${\frac {1954}{3}}$ ${\frac {77486}{9}}$ ${\frac {4789}{18}}$ ${\frac {6415}{6}}$

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 9 7. 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
-3										1	1
-5									1	1	0
-7								2			2
-9							2	1			-1
-11						3	2				1
-13					2	2					0
-15				2	3						-1
-17			1	2							1
-19		1	2								-1
-21		1									1
-23	1										-1

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[9, 7]]
Out[2]=	9
In[3]:=	PD[Knot[9, 7]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[5, 16, 6, 17], X[7, 18, 8, 1], X[17, 6, 18, 7], X[9, 14, 10, 15], X[13, 10, 14, 11], X[15, 8, 16, 9], X[11, 2, 12, 3]]
In[4]:=	GaussCode[Knot[9, 7]]
Out[4]=	GaussCode[-1, 9, -2, 1, -3, 5, -4, 8, -6, 7, -9, 2, -7, 6, -8, 3, -5, 4]
In[5]:=	BR[Knot[9, 7]]
Out[5]=	BR[4, {-1, -1, -1, -1, -2, 1, -2, -3, 2, -3, -3}]
In[6]:=	alex = Alexander[Knot[9, 7]][t]
Out[6]=	3 7 2 9 + -- - - - 7 t + 3 t 2 t t
In[7]:=	Conway[Knot[9, 7]][z]
Out[7]=	2 4 1 + 5 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 7]}
In[9]:=	{KnotDet[Knot[9, 7]], KnotSignature[Knot[9, 7]]}
Out[9]=	{29, -4}
In[10]:=	J=Jones[Knot[9, 7]][q]
Out[10]=	-11 2 3 4 5 5 4 3 -3 -2 -q + --- - -- + -- - -- + -- - -- + -- - q + q 10 9 8 7 6 5 4 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 7]}
In[12]:=	A2Invariant[Knot[9, 7]][q]
Out[12]=	-34 -28 -26 -18 -16 -12 2 -6 -q - q + q - q + q + q + --- + q 10 q
In[13]:=	Kauffman[Knot[9, 7]][a, z]
Out[13]=	4 6 8 10 5 7 9 11 13 2 a + a + a + a - a z - a z - 3 a z - 2 a z + a z - 4 2 6 2 8 2 10 2 12 2 5 3 7 3 3 a z - 2 a z - 4 a z - 2 a z + 3 a z - a z + 2 a z + 9 3 11 3 13 3 4 4 8 4 10 4 11 a z + 5 a z - 3 a z + a z + 7 a z + 2 a z - 12 4 5 5 7 5 9 5 11 5 13 5 6 6 6 a z + a z - a z - 9 a z - 6 a z + a z + a z - 8 6 10 6 12 6 7 7 9 7 11 7 8 8 3 a z - 2 a z + 2 a z + a z + 3 a z + 2 a z + a z + 10 8 a z
In[14]:=	{Vassiliev[2][Knot[9, 7]], Vassiliev[3][Knot[9, 7]]}
Out[14]=	{0, -12}
In[15]:=	Kh[Knot[9, 7]][q, t]
Out[15]=	-5 -3 1 1 1 2 1 2 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 23 9 21 8 19 8 19 7 17 7 17 6 q t q t q t q t q t q t 2 3 2 2 3 2 2 1 ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 15 6 15 5 13 5 13 4 11 4 11 3 9 3 9 2 q t q t q t q t q t q t q t q t 2 1 ----- + ---- 7 2 5 q t q t