9 35

9_34

9_36

Visit 9 35's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 35's page at Knotilus! Visit 9 35's page at the original Knot Atlas! 9_35 is also known as the pretzel knot P(3,3,3).

Three-fold symmetric decorative knot

Another three-fold symmetric decorative form

Knot presentations

Planar diagram presentation	X1829 X7,14,8,15 X5,16,6,17 X9,18,10,1 X15,6,16,7 X17,10,18,11 X13,2,14,3 X3,12,4,13 X11,4,12,5
Gauss code	-1, 7, -8, 9, -3, 5, -2, 1, -4, 6, -9, 8, -7, 2, -5, 3, -6, 4
Dowker-Thistlethwaite code	8 12 16 14 18 4 2 6 10
Conway Notation	[3,3,3]

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 1 Bridge index 3 Super bridge index $\{4,6\}$ Nakanishi index 2 Maximal Thurston-Bennequin number [-12][1] Hyperbolic Volume 7.94058 A-Polynomial See Data:9 35/A-polynomial

[edit Notes for 9 35'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 35's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$7t-13+7t^{-1}$
Conway polynomial	$7z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{3,t+1\}$
Determinant and Signature	{ 27, -2 }
Jones polynomial	$q^{-1}-2q^{-2}+3q^{-3}-4q^{-4}+5q^{-5}-3q^{-6}+4q^{-7}-3q^{-8}+q^{-9}-q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$-a^{10}+z^{2}a^{8}-a^{8}+3z^{2}a^{6}+3a^{6}+2z^{2}a^{4}+z^{2}a^{2}$
Kauffman polynomial (db, data sources)	$z^{7}a^{11}-6z^{5}a^{11}+12z^{3}a^{11}-8za^{11}+z^{8}a^{10}-4z^{6}a^{10}+3z^{4}a^{10}+z^{2}a^{10}+a^{10}+4z^{7}a^{9}-18z^{5}a^{9}+23z^{3}a^{9}-9za^{9}+z^{8}a^{8}+z^{6}a^{8}-15z^{4}a^{8}+16z^{2}a^{8}-a^{8}+3z^{7}a^{7}-8z^{5}a^{7}+3z^{3}a^{7}-za^{7}+5z^{6}a^{6}-15z^{4}a^{6}+12z^{2}a^{6}-3a^{6}+4z^{5}a^{5}-6z^{3}a^{5}+3z^{4}a^{4}-2z^{2}a^{4}+2z^{3}a^{3}+z^{2}a^{2}$
The A2 invariant	$-q^{32}-q^{30}-2q^{26}-q^{24}+q^{22}+q^{20}+3q^{18}+2q^{16}+q^{14}-q^{10}+q^{8}-q^{4}+q^{2}$
The G2 invariant	$q^{156}+3q^{152}-3q^{150}+2q^{148}-q^{146}-2q^{144}+7q^{142}-9q^{140}+6q^{138}-2q^{136}-2q^{134}+8q^{132}-12q^{130}+5q^{128}-2q^{126}-3q^{124}+3q^{122}-10q^{120}-2q^{118}+4q^{116}-2q^{114}+q^{112}-8q^{110}-2q^{108}+6q^{106}-6q^{104}+5q^{102}-11q^{100}+6q^{98}+8q^{96}-3q^{94}+8q^{92}-10q^{90}+12q^{88}+4q^{86}-5q^{84}+7q^{82}-5q^{80}+5q^{78}+7q^{76}-3q^{74}+2q^{72}+q^{70}-2q^{68}+4q^{66}-6q^{64}+3q^{62}-2q^{60}-2q^{58}+4q^{56}-4q^{54}+3q^{52}-2q^{50}+q^{48}-q^{46}-q^{44}+2q^{42}-3q^{40}+3q^{38}+q^{36}+q^{34}-q^{30}+2q^{28}-2q^{26}+2q^{24}-q^{22}-q^{16}+q^{14}-q^{12}+q^{10}$

Further Quantum Invariants

Further quantum knot invariants for 9_35.

A1 Invariants.

Weight	Invariant
1	$-q^{21}-2q^{17}+q^{15}+q^{13}+2q^{11}+q^{9}-q^{7}+q^{5}-q^{3}+q$
2	$q^{60}-q^{56}+2q^{54}+2q^{52}-3q^{50}+q^{46}-4q^{44}-3q^{42}+q^{40}-q^{38}-2q^{36}+3q^{34}+3q^{32}+3q^{26}-q^{24}-2q^{22}+q^{20}+q^{18}-2q^{16}+q^{14}+3q^{12}-q^{10}+q^{8}-q^{4}+q^{2}$
3	$-q^{117}+q^{113}+q^{111}-2q^{109}-3q^{107}+5q^{103}+2q^{101}-4q^{99}-4q^{97}+4q^{95}+7q^{93}+3q^{91}-4q^{89}-3q^{87}+4q^{85}+4q^{83}-q^{81}-8q^{79}-3q^{77}+2q^{75}+q^{73}-7q^{71}-4q^{69}+q^{67}+6q^{65}-2q^{63}-4q^{61}+4q^{59}+9q^{57}-q^{55}-8q^{53}+6q^{49}+2q^{47}-6q^{45}-4q^{43}-q^{41}+6q^{39}+4q^{37}-3q^{35}-6q^{33}+4q^{31}+8q^{29}-5q^{25}+3q^{21}-q^{19}-q^{17}+2q^{13}+q^{11}-q^{5}+q^{3}$
4	$q^{192}-q^{188}-q^{186}-q^{184}+3q^{182}+3q^{180}+q^{178}-2q^{176}-8q^{174}-2q^{172}+4q^{170}+9q^{168}+6q^{166}-8q^{164}-11q^{162}-10q^{160}+3q^{158}+15q^{156}+8q^{154}-q^{152}-17q^{150}-15q^{148}+4q^{146}+15q^{144}+22q^{142}+2q^{140}-17q^{138}-15q^{136}-2q^{134}+23q^{132}+23q^{130}-q^{128}-17q^{126}-21q^{124}+q^{122}+19q^{120}+14q^{118}-7q^{116}-25q^{114}-15q^{112}+9q^{110}+17q^{108}+3q^{106}-15q^{104}-12q^{102}+8q^{100}+17q^{98}+6q^{96}-14q^{94}-12q^{92}+10q^{90}+18q^{88}+2q^{86}-18q^{84}-21q^{82}+6q^{80}+20q^{78}+13q^{76}-6q^{74}-25q^{72}-15q^{70}+6q^{68}+21q^{66}+23q^{64}-4q^{62}-27q^{60}-20q^{58}+6q^{56}+35q^{54}+21q^{52}-14q^{50}-31q^{48}-15q^{46}+20q^{44}+22q^{42}-15q^{38}-11q^{36}+8q^{34}+10q^{32}+2q^{30}-3q^{28}-5q^{26}+5q^{24}+q^{22}-2q^{20}-q^{18}-q^{16}+4q^{14}-q^{6}+q^{4}$
5	$-q^{285}+q^{281}+q^{279}+q^{277}-3q^{273}-4q^{271}-q^{269}+2q^{267}+5q^{265}+8q^{263}+3q^{261}-6q^{259}-12q^{257}-11q^{255}-2q^{253}+12q^{251}+20q^{249}+16q^{247}-18q^{243}-27q^{241}-18q^{239}+4q^{237}+27q^{235}+37q^{233}+20q^{231}-11q^{229}-38q^{227}-44q^{225}-23q^{223}+20q^{221}+50q^{219}+46q^{217}+10q^{215}-36q^{213}-66q^{211}-51q^{209}+3q^{207}+57q^{205}+69q^{203}+39q^{201}-26q^{199}-74q^{197}-67q^{195}-10q^{193}+55q^{191}+87q^{189}+58q^{187}-16q^{185}-76q^{183}-76q^{181}-19q^{179}+57q^{177}+92q^{175}+49q^{173}-28q^{171}-83q^{169}-76q^{167}-9q^{165}+65q^{163}+76q^{161}+27q^{159}-46q^{157}-76q^{155}-44q^{153}+29q^{151}+66q^{149}+43q^{147}-11q^{145}-49q^{143}-37q^{141}+15q^{139}+42q^{137}+23q^{135}-19q^{133}-40q^{131}-15q^{129}+30q^{127}+48q^{125}+13q^{123}-44q^{121}-64q^{119}-24q^{117}+36q^{115}+73q^{113}+47q^{111}-23q^{109}-76q^{107}-69q^{105}-16q^{103}+53q^{101}+85q^{99}+61q^{97}-7q^{95}-75q^{93}-95q^{91}-44q^{89}+46q^{87}+106q^{85}+93q^{83}-3q^{81}-100q^{79}-114q^{77}-40q^{75}+66q^{73}+115q^{71}+63q^{69}-34q^{67}-94q^{65}-66q^{63}+11q^{61}+68q^{59}+60q^{57}+4q^{55}-44q^{53}-42q^{51}-3q^{49}+24q^{47}+25q^{45}+3q^{43}-15q^{41}-14q^{39}+2q^{37}+9q^{35}+5q^{33}-q^{31}-q^{29}+q^{25}+q^{23}-2q^{21}-2q^{19}+q^{17}+3q^{15}-q^{7}+q^{5}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{32}-q^{30}-2q^{26}-q^{24}+q^{22}+q^{20}+3q^{18}+2q^{16}+q^{14}-q^{10}+q^{8}-q^{4}+q^{2}$
1,1	$q^{84}+6q^{80}-6q^{78}+12q^{76}-18q^{74}+18q^{72}-22q^{70}+14q^{68}-16q^{66}+4q^{64}+8q^{62}-7q^{60}+20q^{58}-24q^{56}+30q^{54}-35q^{52}+22q^{50}-32q^{48}+16q^{46}-14q^{44}+2q^{42}+8q^{40}-2q^{38}+15q^{36}-4q^{34}+12q^{32}-10q^{30}+7q^{28}-4q^{26}+4q^{24}-6q^{22}+5q^{20}+2q^{18}+4q^{16}-4q^{14}+3q^{12}-2q^{10}+2q^{8}-2q^{6}+q^{4}$
2,0	$q^{82}+q^{80}+q^{78}-q^{76}+q^{74}+3q^{72}+3q^{70}-q^{68}-3q^{66}-q^{64}-q^{62}-5q^{60}-7q^{58}-4q^{56}-2q^{54}-2q^{52}-3q^{50}+2q^{48}+5q^{46}+6q^{44}+4q^{42}+3q^{40}+2q^{38}+q^{36}-q^{34}-q^{32}-q^{30}+q^{28}+3q^{26}-q^{24}-3q^{22}+2q^{20}+4q^{18}+q^{16}-2q^{14}+2q^{10}-q^{8}-q^{6}+q^{4}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{66}+3q^{62}+2q^{60}+q^{58}+q^{56}-3q^{54}-7q^{52}-6q^{50}-7q^{48}-5q^{46}+2q^{44}+2q^{42}+5q^{40}+6q^{38}+4q^{36}+q^{28}+3q^{24}+3q^{22}-q^{20}+q^{18}+q^{16}-2q^{14}+2q^{10}-q^{8}-q^{6}+q^{4}$
1,0,0	$-q^{43}-q^{41}-q^{39}-2q^{35}-q^{33}-q^{31}+q^{29}+q^{27}+3q^{25}+3q^{23}+2q^{21}+q^{19}-q^{13}+q^{11}-q^{5}+q^{3}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{66}-3q^{62}+2q^{60}-3q^{58}+3q^{56}-3q^{54}+3q^{52}-2q^{50}+q^{48}+q^{46}-2q^{44}+4q^{42}-5q^{40}+6q^{38}-6q^{36}+6q^{34}-4q^{32}+4q^{30}-q^{28}+2q^{26}+q^{24}-q^{22}+3q^{20}-3q^{18}+3q^{16}-2q^{14}+2q^{12}-2q^{10}+q^{8}-q^{6}+q^{4}$
1,0	$q^{108}+3q^{100}+2q^{98}-q^{96}-q^{94}+2q^{92}+2q^{90}-q^{88}-5q^{86}-4q^{84}-q^{82}-q^{80}-5q^{78}-7q^{76}-2q^{74}+2q^{72}+2q^{70}-q^{68}+q^{66}+4q^{64}+5q^{62}+2q^{60}+q^{58}+2q^{56}+3q^{54}-q^{52}-3q^{50}-q^{48}+2q^{46}+q^{44}-q^{42}-2q^{40}+2q^{38}+4q^{36}+q^{34}-2q^{32}+2q^{28}+2q^{26}-q^{24}-2q^{22}-q^{20}+q^{18}+2q^{16}-q^{12}-q^{10}+q^{6}$

G2 Invariants.

Weight

Invariant

1,0

$q^{156}+3q^{152}-3q^{150}+2q^{148}-q^{146}-2q^{144}+7q^{142}-9q^{140}+6q^{138}-2q^{136}-2q^{134}+8q^{132}-12q^{130}+5q^{128}-2q^{126}-3q^{124}+3q^{122}-10q^{120}-2q^{118}+4q^{116}-2q^{114}+q^{112}-8q^{110}-2q^{108}+6q^{106}-6q^{104}+5q^{102}-11q^{100}+6q^{98}+8q^{96}-3q^{94}+8q^{92}-10q^{90}+12q^{88}+4q^{86}-5q^{84}+7q^{82}-5q^{80}+5q^{78}+7q^{76}-3q^{74}+2q^{72}+q^{70}-2q^{68}+4q^{66}-6q^{64}+3q^{62}-2q^{60}-2q^{58}+4q^{56}-4q^{54}+3q^{52}-2q^{50}+q^{48}-q^{46}-q^{44}+2q^{42}-3q^{40}+3q^{38}+q^{36}+q^{34}-q^{30}+2q^{28}-2q^{26}+2q^{24}-q^{22}-q^{16}+q^{14}-q^{12}+q^{10}$

.

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 35"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$7t-13+7t^{-1}$

In[5]:=

Conway[K][z]

Out[5]=

$7z^{2}+1$

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]=

$\{3,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 27, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{7}a^{11}-6z^{5}a^{11}+12z^{3}a^{11}-8za^{11}+z^{8}a^{10}-4z^{6}a^{10}+3z^{4}a^{10}+z^{2}a^{10}+a^{10}+4z^{7}a^{9}-18z^{5}a^{9}+23z^{3}a^{9}-9za^{9}+z^{8}a^{8}+z^{6}a^{8}-15z^{4}a^{8}+16z^{2}a^{8}-a^{8}+3z^{7}a^{7}-8z^{5}a^{7}+3z^{3}a^{7}-za^{7}+5z^{6}a^{6}-15z^{4}a^{6}+12z^{2}a^{6}-3a^{6}+4z^{5}a^{5}-6z^{3}a^{5}+3z^{4}a^{4}-2z^{2}a^{4}+2z^{3}a^{3}+z^{2}a^{2}$

Vassiliev invariants

V2 and V3:

(7, -18)

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 $28$ $-144$ $392$ ${\frac {3026}{3}}$ ${\frac {574}{3}}$ $-4032$ $-7584$ $-1344$ $-1296$ ${\frac {10976}{3}}$ $10368$ ${\frac {84728}{3}}$ ${\frac {16072}{3}}$ ${\frac {1720297}{30}}$ $-{\frac {25154}{15}}$ ${\frac {1232834}{45}}$ ${\frac {8471}{18}}$ ${\frac {119977}{30}}$

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 35. 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                 2 1 -1 -5               1     1 -7             3 2     -1 -9           2 1       1 -11         1 3         2 -13       3 2           1 -15       1             1 -17   1 3               -2 -19                     0 -21 1                   -1

Integral Khovanov Homology (db, data source)

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

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[9, 35]]
Out[2]=	9
In[3]:=	PD[Knot[9, 35]]
Out[3]=	PD[X[1, 8, 2, 9], X[7, 14, 8, 15], X[5, 16, 6, 17], X[9, 18, 10, 1], X[15, 6, 16, 7], X[17, 10, 18, 11], X[13, 2, 14, 3], X[3, 12, 4, 13], X[11, 4, 12, 5]]
In[4]:=	GaussCode[Knot[9, 35]]
Out[4]=	GaussCode[-1, 7, -8, 9, -3, 5, -2, 1, -4, 6, -9, 8, -7, 2, -5, 3, -6, 4]
In[5]:=	BR[Knot[9, 35]]
Out[5]=	BR[5, {-1, -1, -2, 1, -2, -2, -3, 2, 2, -4, 3, -2, -4, -3}]
In[6]:=	alex = Alexander[Knot[9, 35]][t]
Out[6]=	7 -13 + - + 7 t t
In[7]:=	Conway[Knot[9, 35]][z]
Out[7]=	2 1 + 7 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 35]}
In[9]:=	{KnotDet[Knot[9, 35]], KnotSignature[Knot[9, 35]]}
Out[9]=	{27, -2}
In[10]:=	J=Jones[Knot[9, 35]][q]
Out[10]=	-10 -9 3 4 3 5 4 3 2 1 -q + q - -- + -- - -- + -- - -- + -- - -- + - 8 7 6 5 4 3 2 q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 35]}
In[12]:=	A2Invariant[Knot[9, 35]][q]
Out[12]=	-32 -30 2 -24 -22 -20 3 2 -14 -10 -q - q - --- - q + q + q + --- + --- + q - q + 26 18 16 q q q -8 -4 -2 q - q + q
In[13]:=	Kauffman[Knot[9, 35]][a, z]
Out[13]=	6 8 10 7 9 11 2 2 4 2 -3 a - a + a - a z - 9 a z - 8 a z + a z - 2 a z + 6 2 8 2 10 2 3 3 5 3 7 3 12 a z + 16 a z + a z + 2 a z - 6 a z + 3 a z + 9 3 11 3 4 4 6 4 8 4 10 4 23 a z + 12 a z + 3 a z - 15 a z - 15 a z + 3 a z + 5 5 7 5 9 5 11 5 6 6 8 6 4 a z - 8 a z - 18 a z - 6 a z + 5 a z + a z - 10 6 7 7 9 7 11 7 8 8 10 8 4 a z + 3 a z + 4 a z + a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[9, 35]], Vassiliev[3][Knot[9, 35]]}
Out[14]=	{0, -18}
In[15]:=	Kh[Knot[9, 35]][q, t]
Out[15]=	-3 1 1 1 3 1 3 2 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 3 2 1 3 2 1 2 ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- 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