9 45

9_44

9_46

Visit 9 45's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 45's page at Knotilus! Visit 9 45's page at the original Knot Atlas! 9 45 Quick Notes

9 45 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,6,11,5 X8394 X2,9,3,10 X7,14,8,15 X18,15,1,16 X16,11,17,12 X12,17,13,18 X13,6,14,7
Gauss code	1, -4, 3, -1, 2, 9, -5, -3, 4, -2, 7, -8, -9, 5, 6, -7, 8, -6
Dowker-Thistlethwaite code	4 8 10 -14 2 16 -6 18 12
Conway Notation	[211,21,2-]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index $\{4,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-10][1] Hyperbolic Volume 8.60203 A-Polynomial See Data:9 45/A-polynomial

[edit Notes for 9 45's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 45's four dimensional invariants]

Polynomial invariants

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

Further Quantum Invariants

Further quantum knot invariants for 9_45.

A1 Invariants.

Weight	Invariant
1	$-q^{17}+q^{15}-q^{13}+q^{11}+q^{5}-q^{3}+2q$
2	$q^{48}-q^{46}-2q^{44}+3q^{42}+q^{40}-4q^{38}+2q^{36}+3q^{34}-4q^{32}-q^{30}+3q^{28}-2q^{26}-2q^{24}+2q^{22}+q^{20}-2q^{18}-q^{16}+5q^{14}-q^{12}-3q^{10}+5q^{8}-3q^{4}+2q^{2}+1$
3	$-q^{93}+q^{91}+2q^{89}-4q^{85}-3q^{83}+5q^{81}+6q^{79}-3q^{77}-9q^{75}-q^{73}+11q^{71}+6q^{69}-9q^{67}-10q^{65}+5q^{63}+13q^{61}-3q^{59}-13q^{57}-q^{55}+13q^{53}+4q^{51}-11q^{49}-4q^{47}+9q^{45}+4q^{43}-7q^{41}-7q^{39}+2q^{37}+7q^{35}+q^{33}-9q^{31}-6q^{29}+9q^{27}+12q^{25}-6q^{23}-14q^{21}+4q^{19}+15q^{17}-10q^{13}-3q^{11}+8q^{9}+4q^{7}-4q^{5}-q^{3}+2q^{-1}$
4	$q^{152}-q^{150}-2q^{148}+q^{144}+6q^{142}+q^{140}-5q^{138}-7q^{136}-7q^{134}+11q^{132}+13q^{130}+5q^{128}-9q^{126}-25q^{124}-5q^{122}+14q^{120}+28q^{118}+16q^{116}-26q^{114}-32q^{112}-15q^{110}+27q^{108}+45q^{106}+4q^{104}-33q^{102}-46q^{100}+49q^{96}+34q^{94}-12q^{92}-51q^{90}-23q^{88}+33q^{86}+42q^{84}+4q^{82}-40q^{80}-26q^{78}+18q^{76}+34q^{74}+7q^{72}-27q^{70}-23q^{68}+6q^{66}+27q^{64}+11q^{62}-13q^{60}-23q^{58}-12q^{56}+17q^{54}+25q^{52}+15q^{50}-23q^{48}-42q^{46}-5q^{44}+33q^{42}+48q^{40}-4q^{38}-57q^{36}-35q^{34}+17q^{32}+61q^{30}+25q^{28}-37q^{26}-40q^{24}-11q^{22}+36q^{20}+31q^{18}-6q^{16}-19q^{14}-16q^{12}+7q^{10}+12q^{8}+3q^{6}-q^{4}-5q^{2}-1+2q^{-2}+q^{-4}$
5	$-q^{225}+q^{223}+2q^{221}-q^{217}-3q^{215}-4q^{213}-q^{211}+7q^{209}+9q^{207}+5q^{205}-3q^{203}-14q^{201}-18q^{199}-7q^{197}+16q^{195}+28q^{193}+24q^{191}+q^{189}-32q^{187}-48q^{185}-30q^{183}+17q^{181}+59q^{179}+67q^{177}+25q^{175}-47q^{173}-97q^{171}-77q^{169}+8q^{167}+96q^{165}+125q^{163}+56q^{161}-67q^{159}-151q^{157}-122q^{155}+12q^{153}+144q^{151}+170q^{149}+58q^{147}-109q^{145}-196q^{143}-116q^{141}+61q^{139}+187q^{137}+159q^{135}-9q^{133}-165q^{131}-175q^{129}-31q^{127}+133q^{125}+168q^{123}+56q^{121}-102q^{119}-153q^{117}-61q^{115}+76q^{113}+131q^{111}+57q^{109}-59q^{107}-108q^{105}-49q^{103}+50q^{101}+92q^{99}+44q^{97}-39q^{95}-78q^{93}-47q^{91}+22q^{89}+72q^{87}+63q^{85}+4q^{83}-63q^{81}-87q^{79}-47q^{77}+41q^{75}+112q^{73}+98q^{71}-9q^{69}-128q^{67}-155q^{65}-44q^{63}+121q^{61}+202q^{59}+108q^{57}-94q^{55}-222q^{53}-164q^{51}+41q^{49}+209q^{47}+200q^{45}+22q^{43}-170q^{41}-199q^{39}-67q^{37}+101q^{35}+170q^{33}+94q^{31}-42q^{29}-119q^{27}-85q^{25}+q^{23}+64q^{21}+64q^{19}+21q^{17}-25q^{15}-38q^{13}-13q^{11}+4q^{9}+13q^{7}+13q^{5}-6q-3q^{-1}+2q^{-7}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{26}-q^{24}+q^{22}+q^{18}+q^{16}-q^{14}-q^{10}+q^{8}+q^{6}+2q^{2}$
1,1	$q^{68}-2q^{66}+6q^{64}-12q^{62}+17q^{60}-24q^{58}+30q^{56}-30q^{54}+25q^{52}-18q^{50}+6q^{48}+12q^{46}-27q^{44}+38q^{42}-48q^{40}+52q^{38}-54q^{36}+44q^{34}-36q^{32}+22q^{30}-6q^{28}-6q^{26}+20q^{24}-26q^{22}+31q^{20}-28q^{18}+22q^{16}-16q^{14}+13q^{12}-4q^{10}+2q^{8}+2q^{4}+2q^{2}$
2,0	$q^{66}+q^{64}-3q^{60}-2q^{58}+q^{56}+2q^{54}+3q^{48}+2q^{46}-2q^{44}-4q^{42}-q^{40}-2q^{38}-2q^{36}-q^{34}+2q^{32}+2q^{30}+q^{28}+2q^{26}-q^{24}-q^{22}+2q^{20}+2q^{18}-2q^{16}-q^{14}+3q^{12}+2q^{10}-q^{8}-q^{6}+3q^{4}+q^{2}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{54}-q^{52}+q^{50}+q^{48}-3q^{46}+q^{44}-q^{42}-4q^{40}+2q^{38}+q^{36}-2q^{34}+2q^{32}+2q^{30}+q^{22}-2q^{20}-2q^{18}+3q^{16}-2q^{14}+5q^{10}+3q^{4}$
1,0,0	$-q^{35}-q^{33}-q^{31}+q^{29}+2q^{25}+q^{23}+q^{21}-q^{19}-q^{17}-q^{15}-q^{13}+q^{11}+q^{9}+2q^{7}+2q^{3}$

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

$q^{128}-q^{126}+3q^{124}-4q^{122}+2q^{120}-5q^{116}+9q^{114}-10q^{112}+8q^{110}-3q^{108}-7q^{106}+10q^{104}-12q^{102}+8q^{100}-3q^{98}-6q^{96}+9q^{94}-7q^{92}+2q^{90}+6q^{88}-9q^{86}+10q^{84}-4q^{82}-2q^{80}+9q^{78}-12q^{76}+15q^{74}-9q^{72}+3q^{70}+7q^{68}-12q^{66}+14q^{64}-12q^{62}+5q^{60}+2q^{58}-9q^{56}+9q^{54}-8q^{52}+q^{50}+6q^{48}-10q^{46}+6q^{44}-q^{42}-7q^{40}+11q^{38}-11q^{36}+8q^{34}-5q^{30}+9q^{28}-8q^{26}+8q^{24}-q^{22}-q^{20}+2q^{18}-2q^{16}+3q^{14}-q^{12}+2q^{10}+q^{8}$

.

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 45"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{2}+6t-9+6t^{-1}-t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 23, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

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 $8$ $-32$ $32$ ${\frac {412}{3}}$ ${\frac {68}{3}}$ $-256$ $-{\frac {1856}{3}}$ $-{\frac {320}{3}}$ $-96$ ${\frac {256}{3}}$ $512$ ${\frac {3296}{3}}$ ${\frac {544}{3}}$ ${\frac {42751}{15}}$ $-{\frac {284}{15}}$ ${\frac {56284}{45}}$ ${\frac {257}{9}}$ ${\frac {2431}{15}}$

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

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

Integral Khovanov Homology (db, data source)

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

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, 45]]
Out[2]=	9
In[3]:=	PD[Knot[9, 45]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], X[7, 14, 8, 15], X[18, 15, 1, 16], X[16, 11, 17, 12], X[12, 17, 13, 18], X[13, 6, 14, 7]]
In[4]:=	GaussCode[Knot[9, 45]]
Out[4]=	GaussCode[1, -4, 3, -1, 2, 9, -5, -3, 4, -2, 7, -8, -9, 5, 6, -7, 8, -6]
In[5]:=	BR[Knot[9, 45]]
Out[5]=	BR[4, {-1, -1, -2, 1, -2, -1, -3, 2, -3}]
In[6]:=	alex = Alexander[Knot[9, 45]][t]
Out[6]=	-2 6 2 -9 - t + - + 6 t - t t
In[7]:=	Conway[Knot[9, 45]][z]
Out[7]=	2 4 1 + 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 45]}
In[9]:=	{KnotDet[Knot[9, 45]], KnotSignature[Knot[9, 45]]}
Out[9]=	{23, -2}
In[10]:=	J=Jones[Knot[9, 45]][q]
Out[10]=	-8 2 3 4 4 4 3 2 -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[9, 45]}
In[12]:=	A2Invariant[Knot[9, 45]][q]
Out[12]=	-26 -24 -22 -18 -16 -14 -10 -8 -6 2 -q - q + q + q + q - q - q + q + q + -- 2 q
In[13]:=	Kauffman[Knot[9, 45]][a, z]
Out[13]=	2 4 6 8 7 9 2 2 4 2 -2 a - 2 a - 2 a - a + 2 a z + 2 a z + 3 a z + 6 a z + 6 2 8 2 3 3 5 3 7 3 9 3 4 4 7 a z + 4 a z + a z - a z - 5 a z - 3 a z - 4 a z - 6 4 8 4 3 5 9 5 4 6 6 6 8 6 10 a z - 6 a z + a z + a z + 2 a z + 4 a z + 2 a z + 5 7 7 7 a z + a z
In[14]:=	{Vassiliev[2][Knot[9, 45]], Vassiliev[3][Knot[9, 45]]}
Out[14]=	{0, -4}
In[15]:=	Kh[Knot[9, 45]][q, t]
Out[15]=	-3 2 1 1 1 2 1 2 2 q + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 q t q t q t q t q t q t q t 2 2 2 2 1 2 ----- + ----- + ----- + ----- + ---- + ---- 9 3 7 3 7 2 5 2 5 3 q t q t q t q t q t q t