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-]

Minimum Braid Representative: Length is 9, width is 4. Braid index is 4.

A Morse Link Presentation:

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]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

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

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)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{-1}+q^{-2}-5q^{-3}+4q^{-4}+6q^{-5}-13q^{-6}+6q^{-7}+12q^{-8}-19q^{-9}+5q^{-10}+15q^{-11}-18q^{-12}+q^{-13}+15q^{-14}-13q^{-15}-3q^{-16}+12q^{-17}-6q^{-18}-4q^{-19}+6q^{-20}-q^{-21}-2q^{-22}+q^{-23}$
3	$2q^{-1}-2q^{-2}-q^{-3}-3q^{-4}+10q^{-5}+2q^{-6}-12q^{-7}-10q^{-8}+20q^{-9}+17q^{-10}-23q^{-11}-28q^{-12}+28q^{-13}+35q^{-14}-26q^{-15}-43q^{-16}+25q^{-17}+45q^{-18}-20q^{-19}-48q^{-20}+16q^{-21}+45q^{-22}-9q^{-23}-43q^{-24}+3q^{-25}+38q^{-26}+6q^{-27}-34q^{-28}-11q^{-29}+26q^{-30}+16q^{-31}-18q^{-32}-19q^{-33}+11q^{-34}+17q^{-35}-3q^{-36}-14q^{-37}-q^{-38}+9q^{-39}+3q^{-40}-5q^{-41}-2q^{-42}+q^{-43}+2q^{-44}-q^{-45}$
4	$1+q^{-1}-3q^{-2}-4q^{-3}+4q^{-4}+5q^{-5}+10q^{-6}-8q^{-7}-27q^{-8}+q^{-9}+18q^{-10}+47q^{-11}-3q^{-12}-74q^{-13}-28q^{-14}+21q^{-15}+109q^{-16}+33q^{-17}-118q^{-18}-80q^{-19}-q^{-20}+162q^{-21}+85q^{-22}-133q^{-23}-118q^{-24}-38q^{-25}+181q^{-26}+123q^{-27}-123q^{-28}-126q^{-29}-67q^{-30}+170q^{-31}+133q^{-32}-99q^{-33}-110q^{-34}-88q^{-35}+141q^{-36}+129q^{-37}-65q^{-38}-83q^{-39}-104q^{-40}+97q^{-41}+115q^{-42}-21q^{-43}-45q^{-44}-113q^{-45}+41q^{-46}+87q^{-47}+18q^{-48}+q^{-49}-98q^{-50}-8q^{-51}+41q^{-52}+31q^{-53}+38q^{-54}-57q^{-55}-26q^{-56}-q^{-57}+14q^{-58}+44q^{-59}-15q^{-60}-14q^{-61}-15q^{-62}-5q^{-63}+24q^{-64}+q^{-65}-7q^{-67}-7q^{-68}+6q^{-69}+q^{-70}+2q^{-71}-q^{-72}-2q^{-73}+q^{-74}$
5	$2q-2-3q^{-2}-3q^{-3}+6q^{-4}+15q^{-5}-2q^{-6}-9q^{-7}-20q^{-8}-28q^{-9}+19q^{-10}+61q^{-11}+41q^{-12}-9q^{-13}-83q^{-14}-114q^{-15}-15q^{-16}+138q^{-17}+177q^{-18}+67q^{-19}-152q^{-20}-282q^{-21}-147q^{-22}+167q^{-23}+369q^{-24}+245q^{-25}-143q^{-26}-450q^{-27}-352q^{-28}+109q^{-29}+497q^{-30}+447q^{-31}-49q^{-32}-531q^{-33}-517q^{-34}-2q^{-35}+524q^{-36}+566q^{-37}+58q^{-38}-517q^{-39}-588q^{-40}-90q^{-41}+484q^{-42}+590q^{-43}+125q^{-44}-458q^{-45}-579q^{-46}-140q^{-47}+415q^{-48}+559q^{-49}+164q^{-50}-375q^{-51}-531q^{-52}-182q^{-53}+316q^{-54}+500q^{-55}+213q^{-56}-259q^{-57}-457q^{-58}-237q^{-59}+179q^{-60}+408q^{-61}+264q^{-62}-101q^{-63}-345q^{-64}-272q^{-65}+15q^{-66}+264q^{-67}+274q^{-68}+55q^{-69}-177q^{-70}-244q^{-71}-111q^{-72}+87q^{-73}+194q^{-74}+142q^{-75}-10q^{-76}-132q^{-77}-137q^{-78}-45q^{-79}+60q^{-80}+113q^{-81}+74q^{-82}-9q^{-83}-68q^{-84}-74q^{-85}-28q^{-86}+28q^{-87}+54q^{-88}+41q^{-89}+4q^{-90}-32q^{-91}-36q^{-92}-14q^{-93}+7q^{-94}+23q^{-95}+20q^{-96}+q^{-97}-13q^{-98}-10q^{-99}-5q^{-100}+9q^{-102}+5q^{-103}-2q^{-104}-2q^{-105}-q^{-106}-2q^{-107}+q^{-108}+2q^{-109}-q^{-110}$
6	$q^{3}+q^{2}-3q-2+q^{-2}+2q^{-3}+9q^{-4}+12q^{-5}-9q^{-6}-29q^{-7}-21q^{-8}-3q^{-9}+12q^{-10}+64q^{-11}+80q^{-12}+12q^{-13}-96q^{-14}-144q^{-15}-105q^{-16}-30q^{-17}+189q^{-18}+329q^{-19}+209q^{-20}-109q^{-21}-385q^{-22}-456q^{-23}-320q^{-24}+241q^{-25}+743q^{-26}+736q^{-27}+164q^{-28}-552q^{-29}-989q^{-30}-969q^{-31}-14q^{-32}+1061q^{-33}+1437q^{-34}+780q^{-35}-406q^{-36}-1393q^{-37}-1736q^{-38}-569q^{-39}+1049q^{-40}+1957q^{-41}+1449q^{-42}+7q^{-43}-1459q^{-44}-2262q^{-45}-1126q^{-46}+778q^{-47}+2124q^{-48}+1861q^{-49}+417q^{-50}-1283q^{-51}-2432q^{-52}-1447q^{-53}+486q^{-54}+2046q^{-55}+1969q^{-56}+654q^{-57}-1066q^{-58}-2371q^{-59}-1536q^{-60}+291q^{-61}+1886q^{-62}+1902q^{-63}+755q^{-64}-874q^{-65}-2210q^{-66}-1524q^{-67}+129q^{-68}+1679q^{-69}+1773q^{-70}+846q^{-71}-640q^{-72}-1974q^{-73}-1505q^{-74}-108q^{-75}+1368q^{-76}+1599q^{-77}+993q^{-78}-282q^{-79}-1617q^{-80}-1467q^{-81}-445q^{-82}+899q^{-83}+1317q^{-84}+1134q^{-85}+183q^{-86}-1086q^{-87}-1302q^{-88}-766q^{-89}+312q^{-90}+848q^{-91}+1107q^{-92}+600q^{-93}-430q^{-94}-901q^{-95}-866q^{-96}-208q^{-97}+247q^{-98}+788q^{-99}+741q^{-100}+132q^{-101}-340q^{-102}-623q^{-103}-415q^{-104}-240q^{-105}+291q^{-106}+515q^{-107}+348q^{-108}+98q^{-109}-203q^{-110}-256q^{-111}-373q^{-112}-74q^{-113}+144q^{-114}+215q^{-115}+199q^{-116}+72q^{-117}+7q^{-118}-209q^{-119}-135q^{-120}-63q^{-121}+17q^{-122}+73q^{-123}+90q^{-124}+110q^{-125}-36q^{-126}-38q^{-127}-60q^{-128}-42q^{-129}-24q^{-130}+14q^{-131}+68q^{-132}+11q^{-133}+15q^{-134}-9q^{-135}-15q^{-136}-27q^{-137}-13q^{-138}+18q^{-139}+2q^{-140}+11q^{-141}+4q^{-142}+3q^{-143}-9q^{-144}-7q^{-145}+4q^{-146}-2q^{-147}+2q^{-148}+q^{-149}+2q^{-150}-q^{-151}-2q^{-152}+q^{-153}$
7	$2q^{5}-2q^{4}-2q^{2}-3q+1+6q^{-1}+7q^{-2}+6q^{-3}-2q^{-4}-8q^{-5}-18q^{-6}-34q^{-7}-16q^{-8}+29q^{-9}+65q^{-10}+73q^{-11}+46q^{-12}-11q^{-13}-101q^{-14}-202q^{-15}-199q^{-16}-19q^{-17}+190q^{-18}+369q^{-19}+405q^{-20}+224q^{-21}-147q^{-22}-632q^{-23}-869q^{-24}-589q^{-25}+54q^{-26}+846q^{-27}+1385q^{-28}+1241q^{-29}+400q^{-30}-963q^{-31}-2082q^{-32}-2135q^{-33}-1084q^{-34}+838q^{-35}+2644q^{-36}+3207q^{-37}+2149q^{-38}-366q^{-39}-3085q^{-40}-4318q^{-41}-3411q^{-42}-412q^{-43}+3201q^{-44}+5271q^{-45}+4772q^{-46}+1462q^{-47}-2997q^{-48}-5978q^{-49}-6027q^{-50}-2620q^{-51}+2505q^{-52}+6357q^{-53}+7045q^{-54}+3739q^{-55}-1857q^{-56}-6403q^{-57}-7743q^{-58}-4699q^{-59}+1135q^{-60}+6235q^{-61}+8154q^{-62}+5394q^{-63}-510q^{-64}-5911q^{-65}-8262q^{-66}-5857q^{-67}-27q^{-68}+5562q^{-69}+8234q^{-70}+6087q^{-71}+373q^{-72}-5224q^{-73}-8059q^{-74}-6166q^{-75}-628q^{-76}+4927q^{-77}+7873q^{-78}+6152q^{-79}+773q^{-80}-4673q^{-81}-7635q^{-82}-6090q^{-83}-914q^{-84}+4409q^{-85}+7408q^{-86}+6027q^{-87}+1058q^{-88}-4113q^{-89}-7122q^{-90}-5982q^{-91}-1282q^{-92}+3734q^{-93}+6811q^{-94}+5939q^{-95}+1572q^{-96}-3231q^{-97}-6397q^{-98}-5908q^{-99}-1973q^{-100}+2617q^{-101}+5884q^{-102}+5825q^{-103}+2418q^{-104}-1845q^{-105}-5209q^{-106}-5683q^{-107}-2900q^{-108}+974q^{-109}+4399q^{-110}+5383q^{-111}+3312q^{-112}-29q^{-113}-3407q^{-114}-4897q^{-115}-3620q^{-116}-892q^{-117}+2314q^{-118}+4207q^{-119}+3675q^{-120}+1685q^{-121}-1153q^{-122}-3295q^{-123}-3471q^{-124}-2273q^{-125}+84q^{-126}+2259q^{-127}+2971q^{-128}+2515q^{-129}+799q^{-130}-1181q^{-131}-2230q^{-132}-2421q^{-133}-1390q^{-134}+237q^{-135}+1378q^{-136}+2015q^{-137}+1592q^{-138}+460q^{-139}-531q^{-140}-1398q^{-141}-1480q^{-142}-847q^{-143}-114q^{-144}+745q^{-145}+1108q^{-146}+892q^{-147}+516q^{-148}-180q^{-149}-650q^{-150}-709q^{-151}-637q^{-152}-181q^{-153}+233q^{-154}+421q^{-155}+540q^{-156}+318q^{-157}+48q^{-158}-124q^{-159}-348q^{-160}-310q^{-161}-168q^{-162}-49q^{-163}+158q^{-164}+181q^{-165}+159q^{-166}+145q^{-167}-10q^{-168}-86q^{-169}-111q^{-170}-125q^{-171}-33q^{-172}-2q^{-173}+26q^{-174}+92q^{-175}+56q^{-176}+30q^{-177}-3q^{-178}-47q^{-179}-24q^{-180}-26q^{-181}-26q^{-182}+11q^{-183}+18q^{-184}+23q^{-185}+17q^{-186}-9q^{-187}-4q^{-189}-13q^{-190}-4q^{-191}-2q^{-192}+6q^{-193}+7q^{-194}-2q^{-195}+2q^{-197}-2q^{-198}-q^{-199}-2q^{-200}+q^{-201}+2q^{-202}-q^{-203}$

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, 45]]
Out[2]=	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[3]:=	GaussCode[Knot[9, 45]]
Out[3]=	GaussCode[1, -4, 3, -1, 2, 9, -5, -3, 4, -2, 7, -8, -9, 5, 6, -7, 8, -6]
In[4]:=	DTCode[Knot[9, 45]]
Out[4]=	DTCode[4, 8, 10, -14, 2, 16, -6, 18, 12]
In[5]:=	br = BR[Knot[9, 45]]
Out[5]=	BR[4, {-1, -1, -2, 1, -2, -1, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 45]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 45]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 45]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 3, {4, 5}, 1}
In[10]:=	alex = Alexander[Knot[9, 45]][t]
Out[10]=	-2 6 2 -9 - t + - + 6 t - t t
In[11]:=	Conway[Knot[9, 45]][z]
Out[11]=	2 4 1 + 2 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 45]}
In[13]:=	{KnotDet[Knot[9, 45]], KnotSignature[Knot[9, 45]]}
Out[13]=	{23, -2}
In[14]:=	Jones[Knot[9, 45]][q]
Out[14]=	-8 2 3 4 4 4 3 2 -q + -- - -- + -- - -- + -- - -- + - 7 6 5 4 3 2 q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 45]}
In[16]:=	A2Invariant[Knot[9, 45]][q]
Out[16]=	-26 -24 -22 -18 -16 -14 -10 -8 -6 2 -q - q + q + q + q - q - q + q + q + -- 2 q
In[17]:=	HOMFLYPT[Knot[9, 45]][a, z]
Out[17]=	2 4 6 8 2 2 4 2 6 2 4 4 2 a - 2 a + 2 a - a + 2 a z - 2 a z + 2 a z - a z
In[18]:=	Kauffman[Knot[9, 45]][a, z]
Out[18]=	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[19]:=	{Vassiliev[2][Knot[9, 45]], Vassiliev[3][Knot[9, 45]]}
Out[19]=	{2, -4}
In[20]:=	Kh[Knot[9, 45]][q, t]
Out[20]=	-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
In[21]:=	ColouredJones[Knot[9, 45], 2][q]
Out[21]=	-23 2 -21 6 4 6 12 3 13 15 -13 q - --- - q + --- - --- - --- + --- - --- - --- + --- + q - 22 20 19 18 17 16 15 14 q q q q q q q q 18 15 5 19 12 6 13 6 4 5 -2 1 --- + --- + --- - -- + -- + -- - -- + -- + -- - -- + q + - 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q

See/edit the Rolfsen_Splice_Template. Back to the top.

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