9 29

9_28

9_30

Visit 9 29's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 29's page at Knotilus! Visit 9 29's page at the original Knot Atlas! 9 29 Quick Notes

9 29 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X16,11,17,12 X10,4,11,3 X2,15,3,16 X14,5,15,6 X18,8,1,7 X4,10,5,9 X12,17,13,18 X8,13,9,14
Gauss code	1, -4, 3, -7, 5, -1, 6, -9, 7, -3, 2, -8, 9, -5, 4, -2, 8, -6
Dowker-Thistlethwaite code	6 10 14 18 4 16 8 2 12
Conway Notation	[.2.20.2]

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 2 3-genus 3 Bridge index 3 Super bridge index $\{4,7\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-8][-3] Hyperbolic Volume 12.2059 A-Polynomial See Data:9 29/A-polynomial

[edit Notes for 9 29's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for 9 29's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-5t^{2}+12t-15+12t^{-1}-5t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 51, -2 }
Jones polynomial	$q^{3}-3q^{2}+5q-7+9q^{-1}-8q^{-2}+8q^{-3}-6q^{-4}+3q^{-5}-q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$a^{2}z^{6}-a^{4}z^{4}+4a^{2}z^{4}-2z^{4}-2a^{4}z^{2}+7a^{2}z^{2}+z^{2}a^{-2}-5z^{2}-2a^{4}+5a^{2}+a^{-2}-3$
Kauffman polynomial (db, data sources)	$2a^{2}z^{8}+2z^{8}+6a^{3}z^{7}+9az^{7}+3z^{7}a^{-1}+8a^{4}z^{6}+6a^{2}z^{6}+z^{6}a^{-2}-z^{6}+6a^{5}z^{5}-8a^{3}z^{5}-24az^{5}-10z^{5}a^{-1}+3a^{6}z^{4}-13a^{4}z^{4}-24a^{2}z^{4}-3z^{4}a^{-2}-11z^{4}+a^{7}z^{3}-5a^{5}z^{3}-a^{3}z^{3}+14az^{3}+9z^{3}a^{-1}+8a^{4}z^{2}+17a^{2}z^{2}+3z^{2}a^{-2}+12z^{2}+2a^{5}z+2a^{3}z-az-za^{-1}-2a^{4}-5a^{2}-a^{-2}-3$
The A2 invariant	$-q^{18}+q^{16}-2q^{14}-q^{12}+2q^{10}+4q^{6}+q^{2}-2q^{-2}+q^{-4}-q^{-6}+q^{-10}$
The G2 invariant	$q^{100}-2q^{98}+3q^{96}-4q^{94}+3q^{92}-2q^{90}-q^{88}+8q^{86}-12q^{84}+16q^{82}-19q^{80}+13q^{78}-6q^{76}-9q^{74}+28q^{72}-39q^{70}+44q^{68}-37q^{66}+16q^{64}+13q^{62}-46q^{60}+63q^{58}-61q^{56}+31q^{54}+5q^{52}-41q^{50}+57q^{48}-42q^{46}+9q^{44}+30q^{42}-59q^{40}+56q^{38}-21q^{36}-30q^{34}+78q^{32}-91q^{30}+77q^{28}-24q^{26}-29q^{24}+77q^{22}-96q^{20}+88q^{18}-50q^{16}+47q^{12}-69q^{10}+69q^{8}-36q^{6}-5q^{4}+36q^{2}-55+41q^{-2}-7q^{-4}-39q^{-6}+68q^{-8}-70q^{-10}+39q^{-12}+10q^{-14}-56q^{-16}+77q^{-18}-70q^{-20}+40q^{-22}-3q^{-24}-32q^{-26}+47q^{-28}-40q^{-30}+27q^{-32}-6q^{-34}-7q^{-36}+11q^{-38}-10q^{-40}+6q^{-42}-2q^{-44}+q^{-46}$

Further Quantum Invariants

Further quantum knot invariants for 9_29.

A1 Invariants.

Weight	Invariant
1	$-q^{13}+2q^{11}-3q^{9}+2q^{7}+q^{3}+2q-2q^{-1}+2q^{-3}-2q^{-5}+q^{-7}$
2	$q^{36}-2q^{34}+q^{32}+4q^{30}-9q^{28}+3q^{26}+11q^{24}-15q^{22}-q^{20}+14q^{18}-9q^{16}-6q^{14}+10q^{12}+4q^{10}-8q^{8}+11q^{4}-5q^{2}-10+13q^{-2}+q^{-4}-15q^{-6}+9q^{-8}+8q^{-10}-11q^{-12}+q^{-14}+7q^{-16}-3q^{-18}-2q^{-20}+q^{-22}$
3	$-q^{69}+2q^{67}-q^{65}-2q^{63}+3q^{61}+3q^{59}-6q^{57}-8q^{55}+17q^{53}+16q^{51}-30q^{49}-29q^{47}+36q^{45}+50q^{43}-36q^{41}-71q^{39}+25q^{37}+78q^{35}-76q^{31}-23q^{29}+55q^{27}+45q^{25}-32q^{23}-53q^{21}+6q^{19}+58q^{17}+17q^{15}-56q^{13}-32q^{11}+52q^{9}+46q^{7}-46q^{5}-56q^{3}+32q+70q^{-1}-19q^{-3}-74q^{-5}-4q^{-7}+74q^{-9}+28q^{-11}-61q^{-13}-45q^{-15}+40q^{-17}+54q^{-19}-14q^{-21}-50q^{-23}-7q^{-25}+34q^{-27}+17q^{-29}-16q^{-31}-18q^{-33}+4q^{-35}+10q^{-37}+2q^{-39}-3q^{-41}-2q^{-43}+q^{-45}$
4	$q^{112}-2q^{110}+q^{108}+2q^{106}-5q^{104}+3q^{102}+6q^{98}-3q^{96}-26q^{94}+13q^{92}+27q^{90}+31q^{88}-28q^{86}-108q^{84}+116q^{80}+152q^{78}-38q^{76}-292q^{74}-141q^{72}+179q^{70}+390q^{68}+125q^{66}-400q^{64}-409q^{62}+9q^{60}+492q^{58}+406q^{56}-200q^{54}-488q^{52}-290q^{50}+251q^{48}+476q^{46}+141q^{44}-244q^{42}-395q^{40}-96q^{38}+269q^{36}+317q^{34}+61q^{32}-296q^{30}-286q^{28}+46q^{26}+330q^{24}+214q^{22}-185q^{20}-349q^{18}-75q^{16}+332q^{14}+294q^{12}-117q^{10}-407q^{8}-186q^{6}+320q^{4}+392q^{2}+25-417q^{-2}-365q^{-4}+167q^{-6}+440q^{-8}+273q^{-10}-247q^{-12}-476q^{-14}-129q^{-16}+271q^{-18}+433q^{-20}+77q^{-22}-324q^{-24}-318q^{-26}-57q^{-28}+298q^{-30}+267q^{-32}-10q^{-34}-206q^{-36}-224q^{-38}+24q^{-40}+162q^{-42}+130q^{-44}+8q^{-46}-122q^{-48}-76q^{-50}+57q^{-54}+56q^{-56}-8q^{-58}-24q^{-60}-23q^{-62}-3q^{-64}+13q^{-66}+5q^{-68}+2q^{-70}-3q^{-72}-2q^{-74}+q^{-76}$
5	$-q^{165}+2q^{163}-q^{161}-2q^{159}+5q^{157}-q^{155}-6q^{153}+5q^{149}+9q^{147}+6q^{145}-16q^{143}-38q^{141}-10q^{139}+59q^{137}+89q^{135}+14q^{133}-130q^{131}-200q^{129}-67q^{127}+254q^{125}+441q^{123}+189q^{121}-403q^{119}-816q^{117}-496q^{115}+483q^{113}+1329q^{111}+1073q^{109}-382q^{107}-1849q^{105}-1866q^{103}-78q^{101}+2147q^{99}+2768q^{97}+911q^{95}-2033q^{93}-3470q^{91}-1957q^{89}+1380q^{87}+3686q^{85}+2946q^{83}-303q^{81}-3318q^{79}-3528q^{77}-872q^{75}+2349q^{73}+3541q^{71}+1880q^{69}-1126q^{67}-3004q^{65}-2413q^{63}-87q^{61}+2092q^{59}+2513q^{57}+1006q^{55}-1127q^{53}-2239q^{51}-1560q^{49}+319q^{47}+1843q^{45}+1785q^{43}+217q^{41}-1502q^{39}-1837q^{37}-484q^{35}+1323q^{33}+1861q^{31}+586q^{29}-1311q^{27}-1969q^{25}-675q^{23}+1392q^{21}+2211q^{19}+855q^{17}-1456q^{15}-2516q^{13}-1226q^{11}+1355q^{9}+2841q^{7}+1764q^{5}-1010q^{3}-2984q-2389q^{-1}+331q^{-3}+2866q^{-5}+2967q^{-7}+552q^{-9}-2323q^{-11}-3273q^{-13}-1548q^{-15}+1403q^{-17}+3166q^{-19}+2365q^{-21}-246q^{-23}-2544q^{-25}-2779q^{-27}-907q^{-29}+1522q^{-31}+2650q^{-33}+1737q^{-35}-353q^{-37}-1991q^{-39}-2039q^{-41}-653q^{-43}+1031q^{-45}+1796q^{-47}+1222q^{-49}-105q^{-51}-1161q^{-53}-1270q^{-55}-531q^{-57}+437q^{-59}+947q^{-61}+741q^{-63}+97q^{-65}-465q^{-67}-608q^{-69}-348q^{-71}+74q^{-73}+346q^{-75}+327q^{-77}+114q^{-79}-100q^{-81}-192q^{-83}-147q^{-85}-20q^{-87}+72q^{-89}+85q^{-91}+44q^{-93}-2q^{-95}-30q^{-97}-31q^{-99}-8q^{-101}+6q^{-103}+8q^{-105}+5q^{-107}+2q^{-109}-3q^{-111}-2q^{-113}+q^{-115}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{18}+q^{16}-2q^{14}-q^{12}+2q^{10}+4q^{6}+q^{2}-2q^{-2}+q^{-4}-q^{-6}+q^{-10}$
1,1	$q^{52}-4q^{50}+8q^{48}-12q^{46}+22q^{44}-36q^{42}+48q^{40}-64q^{38}+93q^{36}-118q^{34}+142q^{32}-176q^{30}+194q^{28}-204q^{26}+174q^{24}-128q^{22}+52q^{20}+62q^{18}-170q^{16}+292q^{14}-375q^{12}+438q^{10}-454q^{8}+428q^{6}-372q^{4}+276q^{2}-162+38q^{-2}+70q^{-4}-164q^{-6}+234q^{-8}-258q^{-10}+250q^{-12}-212q^{-14}+168q^{-16}-114q^{-18}+66q^{-20}-36q^{-22}+14q^{-24}-4q^{-26}+q^{-28}$
2,0	$q^{46}-q^{44}+q^{42}+2q^{40}-2q^{38}-2q^{36}+3q^{34}+3q^{32}-8q^{30}-8q^{28}+3q^{26}+3q^{24}-7q^{22}+2q^{20}+11q^{18}+4q^{16}+q^{14}+3q^{12}+2q^{10}-4q^{8}-q^{6}+2q^{4}-6q^{2}-4+5q^{-2}+2q^{-4}-6q^{-6}+q^{-8}+7q^{-10}+2q^{-12}-4q^{-14}-q^{-16}+5q^{-18}+q^{-20}-3q^{-22}-2q^{-24}+q^{-28}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{42}-2q^{40}-q^{38}+6q^{36}-3q^{34}-6q^{32}+10q^{30}-3q^{28}-10q^{26}+10q^{24}-3q^{22}-10q^{20}+6q^{18}+q^{16}-2q^{14}+3q^{12}+7q^{10}+7q^{8}-6q^{6}+3q^{4}+7q^{2}-11-2q^{-2}+8q^{-4}-9q^{-6}+q^{-8}+7q^{-10}-6q^{-12}+2q^{-14}+2q^{-16}-2q^{-18}+q^{-20}$
1,0,0	$-q^{23}+q^{21}-3q^{19}-2q^{15}+2q^{13}+q^{11}+4q^{9}+3q^{7}+q^{5}+q^{3}-2q-3q^{-3}+q^{-5}-q^{-7}+q^{-9}+q^{-13}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{52}-q^{50}-2q^{48}+4q^{46}+3q^{44}-6q^{42}-q^{40}+7q^{38}-10q^{34}+5q^{30}-10q^{28}-11q^{26}+6q^{24}+3q^{22}-7q^{20}+11q^{18}+15q^{16}+2q^{14}+2q^{12}+12q^{10}-13q^{6}-q^{4}+6q^{2}-8-7q^{-2}+8q^{-4}+4q^{-6}-5q^{-8}+4q^{-12}-q^{-14}-3q^{-16}+q^{-18}+2q^{-20}-q^{-22}+q^{-26}$
1,0,0,0	$-q^{28}+q^{26}-3q^{24}-q^{22}-q^{20}-2q^{18}+2q^{16}+q^{14}+5q^{12}+3q^{10}+4q^{8}+q^{6}+q^{4}-2q^{2}-2-q^{-2}-3q^{-4}+q^{-6}-q^{-8}+q^{-10}+q^{-12}+q^{-16}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{42}+2q^{40}-3q^{38}+6q^{36}-9q^{34}+10q^{32}-14q^{30}+13q^{28}-12q^{26}+8q^{24}-3q^{22}-2q^{20}+10q^{18}-15q^{16}+22q^{14}-23q^{12}+27q^{10}-23q^{8}+20q^{6}-13q^{4}+7q^{2}-1-6q^{-2}+10q^{-4}-13q^{-6}+13q^{-8}-13q^{-10}+10q^{-12}-8q^{-14}+6q^{-16}-2q^{-18}+q^{-20}$
1,0	$q^{68}-2q^{64}-2q^{62}+q^{60}+6q^{58}+3q^{56}-6q^{54}-8q^{52}+12q^{48}+5q^{46}-10q^{44}-11q^{42}+4q^{40}+13q^{38}-13q^{34}-6q^{32}+10q^{30}+7q^{28}-7q^{26}-9q^{24}+5q^{22}+11q^{20}+q^{18}-8q^{16}+2q^{14}+10q^{12}+3q^{10}-9q^{8}-4q^{6}+9q^{4}+9q^{2}-7-15q^{-2}+14q^{-6}+6q^{-8}-11q^{-10}-11q^{-12}+6q^{-14}+12q^{-16}-8q^{-20}-4q^{-22}+5q^{-24}+4q^{-26}-2q^{-28}-2q^{-30}+q^{-34}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{58}-2q^{56}+q^{54}-2q^{52}+6q^{50}-6q^{48}+5q^{46}-8q^{44}+12q^{42}-10q^{40}+8q^{38}-11q^{36}+7q^{34}-6q^{32}-2q^{28}-8q^{26}+9q^{24}-11q^{22}+17q^{20}-15q^{18}+24q^{16}-14q^{14}+24q^{12}-14q^{10}+16q^{8}-10q^{6}+7q^{4}-7q^{2}-2+2q^{-2}-8q^{-4}+6q^{-6}-11q^{-8}+12q^{-10}-10q^{-12}+10q^{-14}-8q^{-16}+8q^{-18}-5q^{-20}+4q^{-22}-2q^{-24}+q^{-26}$

G2 Invariants.

Weight

Invariant

1,0

$q^{100}-2q^{98}+3q^{96}-4q^{94}+3q^{92}-2q^{90}-q^{88}+8q^{86}-12q^{84}+16q^{82}-19q^{80}+13q^{78}-6q^{76}-9q^{74}+28q^{72}-39q^{70}+44q^{68}-37q^{66}+16q^{64}+13q^{62}-46q^{60}+63q^{58}-61q^{56}+31q^{54}+5q^{52}-41q^{50}+57q^{48}-42q^{46}+9q^{44}+30q^{42}-59q^{40}+56q^{38}-21q^{36}-30q^{34}+78q^{32}-91q^{30}+77q^{28}-24q^{26}-29q^{24}+77q^{22}-96q^{20}+88q^{18}-50q^{16}+47q^{12}-69q^{10}+69q^{8}-36q^{6}-5q^{4}+36q^{2}-55+41q^{-2}-7q^{-4}-39q^{-6}+68q^{-8}-70q^{-10}+39q^{-12}+10q^{-14}-56q^{-16}+77q^{-18}-70q^{-20}+40q^{-22}-3q^{-24}-32q^{-26}+47q^{-28}-40q^{-30}+27q^{-32}-6q^{-34}-7q^{-36}+11q^{-38}-10q^{-40}+6q^{-42}-2q^{-44}+q^{-46}$

.

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 29"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-5t^{2}+12t-15+12t^{-1}-5t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}+z^{4}+z^{2}+1$

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]=

{ 51, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{2}z^{6}-a^{4}z^{4}+4a^{2}z^{4}-2z^{4}-2a^{4}z^{2}+7a^{2}z^{2}+z^{2}a^{-2}-5z^{2}-2a^{4}+5a^{2}+a^{-2}-3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{2}z^{8}+2z^{8}+6a^{3}z^{7}+9az^{7}+3z^{7}a^{-1}+8a^{4}z^{6}+6a^{2}z^{6}+z^{6}a^{-2}-z^{6}+6a^{5}z^{5}-8a^{3}z^{5}-24az^{5}-10z^{5}a^{-1}+3a^{6}z^{4}-13a^{4}z^{4}-24a^{2}z^{4}-3z^{4}a^{-2}-11z^{4}+a^{7}z^{3}-5a^{5}z^{3}-a^{3}z^{3}+14az^{3}+9z^{3}a^{-1}+8a^{4}z^{2}+17a^{2}z^{2}+3z^{2}a^{-2}+12z^{2}+2a^{5}z+2a^{3}z-az-za^{-1}-2a^{4}-5a^{2}-a^{-2}-3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_28, 10_163, K11n87, ...}

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

Vassiliev invariants

V2 and V3:

(1, -2)

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 $4$ $-16$ $8$ ${\frac {62}{3}}$ $-{\frac {14}{3}}$ $-64$ $-{\frac {352}{3}}$ ${\frac {128}{3}}$ $-80$ ${\frac {32}{3}}$ $128$ ${\frac {248}{3}}$ $-{\frac {56}{3}}$ ${\frac {11311}{30}}$ $-{\frac {634}{5}}$ ${\frac {11822}{45}}$ ${\frac {209}{18}}$ ${\frac {751}{30}}$

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

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{10}-3q^{9}-q^{8}+11q^{7}-9q^{6}-13q^{5}+30q^{4}-8q^{3}-37q^{2}+46q+4-60q^{-1}+51q^{-2}+20q^{-3}-71q^{-4}+43q^{-5}+32q^{-6}-65q^{-7}+27q^{-8}+29q^{-9}-42q^{-10}+12q^{-11}+15q^{-12}-16q^{-13}+4q^{-14}+3q^{-15}-3q^{-16}+q^{-17}$
3	$q^{21}-3q^{20}-q^{19}+5q^{18}+9q^{17}-9q^{16}-23q^{15}+7q^{14}+42q^{13}+8q^{12}-64q^{11}-36q^{10}+78q^{9}+76q^{8}-78q^{7}-121q^{6}+62q^{5}+165q^{4}-32q^{3}-199q^{2}-8q+220+57q^{-1}-237q^{-2}-96q^{-3}+230q^{-4}+149q^{-5}-231q^{-6}-180q^{-7}+206q^{-8}+222q^{-9}-190q^{-10}-232q^{-11}+147q^{-12}+243q^{-13}-113q^{-14}-222q^{-15}+69q^{-16}+190q^{-17}-37q^{-18}-144q^{-19}+16q^{-20}+94q^{-21}-2q^{-22}-58q^{-23}+2q^{-24}+29q^{-25}-3q^{-26}-12q^{-27}+3q^{-28}+4q^{-29}-q^{-30}-3q^{-31}+3q^{-32}-q^{-33}$
4	$q^{36}-3q^{35}-q^{34}+5q^{33}+3q^{32}+9q^{31}-19q^{30}-21q^{29}+4q^{28}+19q^{27}+73q^{26}-18q^{25}-78q^{24}-72q^{23}-27q^{22}+203q^{21}+104q^{20}-46q^{19}-210q^{18}-275q^{17}+221q^{16}+300q^{15}+231q^{14}-179q^{13}-630q^{12}-40q^{11}+294q^{10}+632q^{9}+177q^{8}-792q^{7}-440q^{6}-53q^{5}+861q^{4}+697q^{3}-625q^{2}-713q-585+809q^{-1}+1139q^{-2}-258q^{-3}-785q^{-4}-1091q^{-5}+588q^{-6}+1429q^{-7}+153q^{-8}-747q^{-9}-1498q^{-10}+314q^{-11}+1593q^{-12}+552q^{-13}-631q^{-14}-1782q^{-15}-18q^{-16}+1583q^{-17}+909q^{-18}-375q^{-19}-1830q^{-20}-383q^{-21}+1284q^{-22}+1060q^{-23}+10q^{-24}-1495q^{-25}-608q^{-26}+743q^{-27}+862q^{-28}+298q^{-29}-889q^{-30}-522q^{-31}+260q^{-32}+444q^{-33}+307q^{-34}-364q^{-35}-257q^{-36}+49q^{-37}+124q^{-38}+156q^{-39}-110q^{-40}-67q^{-41}+13q^{-42}+8q^{-43}+48q^{-44}-30q^{-45}-8q^{-46}+9q^{-47}-6q^{-48}+9q^{-49}-7q^{-50}+q^{-51}+3q^{-52}-3q^{-53}+q^{-54}$
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{55}-3 q^{54}-q^{53}+5 q^{52}+3 q^{51}+3 q^{50}-q^{49}-17 q^{48}-24 q^{47}+6 q^{46}+31 q^{45}+49 q^{44}+40 q^{43}-30 q^{42}-116 q^{41}-121 q^{40}-14 q^{39}+141 q^{38}+254 q^{37}+183 q^{36}-97 q^{35}-393 q^{34}-436 q^{33}-119 q^{32}+397 q^{31}+745 q^{30}+547 q^{29}-187 q^{28}-946 q^{27}-1087 q^{26}-342 q^{25}+854 q^{24}+1603 q^{23}+1140 q^{22}-372 q^{21}-1852 q^{20}-2026 q^{19}-532 q^{18}+1651 q^{17}+2778 q^{16}+1718 q^{15}-939 q^{14}-3154 q^{13}-2961 q^{12}-221 q^{11}+3013 q^{10}+4016 q^9+1672 q^8-2353 q^7-4724 q^6-3172 q^5+1288 q^4+4966 q^3+4547 q^2+62 q-4825-5707 q^{-1} -1432 q^{-2} +4371 q^{-3} +6521 q^{-4} +2836 q^{-5} -3748 q^{-6} -7193 q^{-7} -4013 q^{-8} +3081 q^{-9} +7581 q^{-10} +5147 q^{-11} -2392 q^{-12} -8012 q^{-13} -6080 q^{-14} +1787 q^{-15} +8239 q^{-16} +7044 q^{-17} -1117 q^{-18} -8550 q^{-19} -7887 q^{-20} +434 q^{-21} +8574 q^{-22} +8763 q^{-23} +451 q^{-24} -8492 q^{-25} -9411 q^{-26} -1445 q^{-27} +7895 q^{-28} +9875 q^{-29} +2584 q^{-30} -6985 q^{-31} -9832 q^{-32} -3624 q^{-33} +5569 q^{-34} +9284 q^{-35} +4462 q^{-36} -3979 q^{-37} -8171 q^{-38} -4816 q^{-39} +2348 q^{-40} +6628 q^{-41} +4672 q^{-42} -964 q^{-43} -4922 q^{-44} -4076 q^{-45} +42 q^{-46} +3291 q^{-47} +3159 q^{-48} +473 q^{-49} -1978 q^{-50} -2219 q^{-51} -579 q^{-52} +1066 q^{-53} +1371 q^{-54} +490 q^{-55} -511 q^{-56} -764 q^{-57} -323 q^{-58} +220 q^{-59} +392 q^{-60} +170 q^{-61} -98 q^{-62} -172 q^{-63} -71 q^{-64} +33 q^{-65} +71 q^{-66} +37 q^{-67} -28 q^{-68} -28 q^{-69} +4 q^{-70} +3 q^{-71} +2 q^{-72} +9 q^{-73} -6 q^{-74} -6 q^{-75} +7 q^{-76} - q^{-77} -3 q^{-78} +3 q^{-79} - q^{-80} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{78}-3 q^{77}-q^{76}+5 q^{75}+3 q^{74}+3 q^{73}-7 q^{72}+q^{71}-20 q^{70}-22 q^{69}+18 q^{68}+32 q^{67}+54 q^{66}+18 q^{65}+24 q^{64}-90 q^{63}-164 q^{62}-101 q^{61}-6 q^{60}+180 q^{59}+241 q^{58}+393 q^{57}+74 q^{56}-324 q^{55}-581 q^{54}-647 q^{53}-271 q^{52}+207 q^{51}+1217 q^{50}+1242 q^{49}+702 q^{48}-376 q^{47}-1558 q^{46}-2121 q^{45}-1870 q^{44}+492 q^{43}+2230 q^{42}+3431 q^{41}+2741 q^{40}+412 q^{39}-2883 q^{38}-5620 q^{37}-4131 q^{36}-1195 q^{35}+3704 q^{34}+6942 q^{33}+7131 q^{32}+2474 q^{31}-5071 q^{30}-8939 q^{29}-9751 q^{28}-3787 q^{27}+4702 q^{26}+12713 q^{25}+12976 q^{24}+4715 q^{23}-5335 q^{22}-15491 q^{21}-16099 q^{20}-7864 q^{19}+8110 q^{18}+19072 q^{17}+18793 q^{16}+9107 q^{15}-9630 q^{14}-22746 q^{13}-24071 q^{12}-7497 q^{11}+12872 q^{10}+26434 q^9+26530 q^8+7049 q^7-17140 q^6-33580 q^5-25683 q^4-3472 q^3+22509 q^2+37520 q+26018-2422 q^{-1} -32653 q^{-2} -38225 q^{-3} -21762 q^{-4} +10767 q^{-5} +39573 q^{-6} +40200 q^{-7} +13842 q^{-8} -25216 q^{-9} -43540 q^{-10} -36056 q^{-11} -2035 q^{-12} +36528 q^{-13} +48571 q^{-14} +26735 q^{-15} -17057 q^{-16} -45092 q^{-17} -45769 q^{-18} -12014 q^{-19} +33152 q^{-20} +54119 q^{-21} +36146 q^{-22} -10914 q^{-23} -46555 q^{-24} -53611 q^{-25} -19825 q^{-26} +30914 q^{-27} +59547 q^{-28} +44895 q^{-29} -5020 q^{-30} -48015 q^{-31} -61616 q^{-32} -28849 q^{-33} +26676 q^{-34} +63623 q^{-35} +54687 q^{-36} +4746 q^{-37} -44936 q^{-38} -67352 q^{-39} -40453 q^{-40} +15693 q^{-41} +60548 q^{-42} +61882 q^{-43} +19047 q^{-44} -32354 q^{-45} -63985 q^{-46} -49510 q^{-47} -1443 q^{-48} +45690 q^{-49} +58777 q^{-50} +30939 q^{-51} -12630 q^{-52} -47747 q^{-53} -47621 q^{-54} -15894 q^{-55} +23543 q^{-56} +42812 q^{-57} +31684 q^{-58} +3796 q^{-59} -25265 q^{-60} -33676 q^{-61} -19131 q^{-62} +5635 q^{-63} +22233 q^{-64} +21648 q^{-65} +9222 q^{-66} -8177 q^{-67} -16791 q^{-68} -12912 q^{-69} -1642 q^{-70} +7705 q^{-71} +9869 q^{-72} +6473 q^{-73} -987 q^{-74} -5801 q^{-75} -5525 q^{-76} -1818 q^{-77} +1708 q^{-78} +2954 q^{-79} +2617 q^{-80} +272 q^{-81} -1472 q^{-82} -1551 q^{-83} -611 q^{-84} +282 q^{-85} +558 q^{-86} +695 q^{-87} +117 q^{-88} -337 q^{-89} -294 q^{-90} -77 q^{-91} +68 q^{-92} +43 q^{-93} +137 q^{-94} +12 q^{-95} -81 q^{-96} -32 q^{-97} +8 q^{-98} +24 q^{-99} -18 q^{-100} +23 q^{-101} +3 q^{-102} -20 q^{-103} +3 q^{-104} +3 q^{-105} +6 q^{-106} -7 q^{-107} + q^{-108} +3 q^{-109} -3 q^{-110} + q^{-111} }
7	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{105}-3 q^{104}-q^{103}+5 q^{102}+3 q^{101}+3 q^{100}-7 q^{99}-5 q^{98}-2 q^{97}-18 q^{96}-10 q^{95}+19 q^{94}+37 q^{93}+57 q^{92}+19 q^{91}-20 q^{90}-35 q^{89}-133 q^{88}-152 q^{87}-89 q^{86}+42 q^{85}+268 q^{84}+342 q^{83}+314 q^{82}+226 q^{81}-187 q^{80}-602 q^{79}-875 q^{78}-884 q^{77}-221 q^{76}+540 q^{75}+1308 q^{74}+1944 q^{73}+1583 q^{72}+493 q^{71}-1225 q^{70}-3091 q^{69}-3567 q^{68}-2798 q^{67}-617 q^{66}+2878 q^{65}+5492 q^{64}+6556 q^{63}+4805 q^{62}-155 q^{61}-5479 q^{60}-9858 q^{59}-10883 q^{58}-6485 q^{57}+1127 q^{56}+10330 q^{55}+16875 q^{54}+15988 q^{53}+8580 q^{52}-4633 q^{51}-18514 q^{50}-25401 q^{49}-22994 q^{48}-8982 q^{47}+12056 q^{46}+29687 q^{45}+37723 q^{44}+29101 q^{43}+5120 q^{42}-23262 q^{41}-46602 q^{40}-51133 q^{39}-31706 q^{38}+3323 q^{37}+43079 q^{36}+67283 q^{35}+62149 q^{34}+29443 q^{33}-22796 q^{32}-69988 q^{31}-88369 q^{30}-69424 q^{29}-13652 q^{28}+53920 q^{27}+101325 q^{26}+107869 q^{25}+61475 q^{24}-18257 q^{23}-94844 q^{22}-135477 q^{21}-112112 q^{20}-32813 q^{19}+66891 q^{18}+144984 q^{17}+155949 q^{16}+91657 q^{15}-20006 q^{14}-132996 q^{13}-185483 q^{12}-149479 q^{11}-39161 q^{10}+100893 q^9+196107 q^8+198060 q^7+102902 q^6-53039 q^5-187532 q^4-232727 q^3-163734 q^2-3501 q+162847+251327 q^{-1} +215796 q^{-2} +62457 q^{-3} -126646 q^{-4} -255411 q^{-5} -256846 q^{-6} -118052 q^{-7} +85277 q^{-8} +248069 q^{-9} +285755 q^{-10} +166696 q^{-11} -42882 q^{-12} -233495 q^{-13} -304972 q^{-14} -206976 q^{-15} +4237 q^{-16} +216215 q^{-17} +316237 q^{-18} +238621 q^{-19} +29174 q^{-20} -199228 q^{-21} -323576 q^{-22} -263598 q^{-23} -55747 q^{-24} +185744 q^{-25} +329272 q^{-26} +283380 q^{-27} +76834 q^{-28} -176163 q^{-29} -336443 q^{-30} -301601 q^{-31} -93702 q^{-32} +171077 q^{-33} +345916 q^{-34} +320128 q^{-35} +109928 q^{-36} -167583 q^{-37} -358376 q^{-38} -341928 q^{-39} -128398 q^{-40} +163029 q^{-41} +371166 q^{-42} +366896 q^{-43} +153100 q^{-44} -152125 q^{-45} -381130 q^{-46} -394208 q^{-47} -184955 q^{-48} +131056 q^{-49} +381977 q^{-50} +419089 q^{-51} +223868 q^{-52} -96262 q^{-53} -368829 q^{-54} -435963 q^{-55} -264871 q^{-56} +48726 q^{-57} +336921 q^{-58} +437136 q^{-59} +301531 q^{-60} +8157 q^{-61} -286250 q^{-62} -418086 q^{-63} -325264 q^{-64} -65768 q^{-65} +220557 q^{-66} +376410 q^{-67} +329393 q^{-68} +115480 q^{-69} -147897 q^{-70} -316152 q^{-71} -311118 q^{-72} -148664 q^{-73} +78719 q^{-74} +244798 q^{-75} +272303 q^{-76} +161037 q^{-77} -21770 q^{-78} -172351 q^{-79} -220234 q^{-80} -153536 q^{-81} -16964 q^{-82} +108723 q^{-83} +163644 q^{-84} +130712 q^{-85} +36969 q^{-86} -59298 q^{-87} -111331 q^{-88} -100865 q^{-89} -41494 q^{-90} +26537 q^{-91} +69152 q^{-92} +70437 q^{-93} +36131 q^{-94} -7869 q^{-95} -39001 q^{-96} -44848 q^{-97} -26780 q^{-98} -477 q^{-99} +20036 q^{-100} +26152 q^{-101} +17297 q^{-102} +2775 q^{-103} -9348 q^{-104} -13916 q^{-105} -9910 q^{-106} -2628 q^{-107} +4013 q^{-108} +6942 q^{-109} +5121 q^{-110} +1609 q^{-111} -1701 q^{-112} -3180 q^{-113} -2289 q^{-114} -809 q^{-115} +627 q^{-116} +1386 q^{-117} +983 q^{-118} +359 q^{-119} -325 q^{-120} -623 q^{-121} -283 q^{-122} -74 q^{-123} +97 q^{-124} +201 q^{-125} +102 q^{-126} +64 q^{-127} -71 q^{-128} -128 q^{-129} +9 q^{-130} +27 q^{-131} +12 q^{-132} +11 q^{-133} -10 q^{-134} +22 q^{-135} -9 q^{-136} -29 q^{-137} +15 q^{-138} +8 q^{-139} -3 q^{-141} -6 q^{-142} +7 q^{-143} - q^{-144} -3 q^{-145} +3 q^{-146} - q^{-147} }

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, 29]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 11, 17, 12], X[10, 4, 11, 3], X[2, 15, 3, 16], X[14, 5, 15, 6], X[18, 8, 1, 7], X[4, 10, 5, 9], X[12, 17, 13, 18], X[8, 13, 9, 14]]
In[3]:=	GaussCode[Knot[9, 29]]
Out[3]=	GaussCode[1, -4, 3, -7, 5, -1, 6, -9, 7, -3, 2, -8, 9, -5, 4, -2, 8, -6]
In[4]:=	DTCode[Knot[9, 29]]
Out[4]=	DTCode[6, 10, 14, 18, 4, 16, 8, 2, 12]
In[5]:=	br = BR[Knot[9, 29]]
Out[5]=	BR[4, {1, -2, -2, 3, -2, 1, -2, 3, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 29]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 29]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 29]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, {4, 7}, 1}
In[10]:=	alex = Alexander[Knot[9, 29]][t]
Out[10]=	-3 5 12 2 3 -15 + t - -- + -- + 12 t - 5 t + t 2 t t
In[11]:=	Conway[Knot[9, 29]][z]
Out[11]=	2 4 6 1 + z + z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 28], Knot[9, 29], Knot[10, 163], Knot[11, NonAlternating, 87]}
In[13]:=	{KnotDet[Knot[9, 29]], KnotSignature[Knot[9, 29]]}
Out[13]=	{51, -2}
In[14]:=	Jones[Knot[9, 29]][q]
Out[14]=	-6 3 6 8 8 9 2 3 -7 - q + -- - -- + -- - -- + - + 5 q - 3 q + q 5 4 3 2 q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 29]}
In[16]:=	A2Invariant[Knot[9, 29]][q]
Out[16]=	-18 -16 2 -12 2 4 -2 2 4 6 10 -q + q - --- - q + --- + -- + q - 2 q + q - q + q 14 10 6 q q q
In[17]:=	HOMFLYPT[Knot[9, 29]][a, z]
Out[17]=	2 -2 2 4 2 z 2 2 4 2 4 -3 + a + 5 a - 2 a - 5 z + -- + 7 a z - 2 a z - 2 z + 2 a 2 4 4 4 2 6 4 a z - a z + a z
In[18]:=	Kauffman[Knot[9, 29]][a, z]
Out[18]=	2 -2 2 4 z 3 5 2 3 z -3 - a - 5 a - 2 a - - - a z + 2 a z + 2 a z + 12 z + ---- + a 2 a 3 2 2 4 2 9 z 3 3 3 5 3 7 3 17 a z + 8 a z + ---- + 14 a z - a z - 5 a z + a z - a 4 5 4 3 z 2 4 4 4 6 4 10 z 5 11 z - ---- - 24 a z - 13 a z + 3 a z - ----- - 24 a z - 2 a a 6 7 3 5 5 5 6 z 2 6 4 6 3 z 7 8 a z + 6 a z - z + -- + 6 a z + 8 a z + ---- + 9 a z + 2 a a 3 7 8 2 8 6 a z + 2 z + 2 a z
In[19]:=	{Vassiliev[2][Knot[9, 29]], Vassiliev[3][Knot[9, 29]]}
Out[19]=	{1, -2}
In[20]:=	Kh[Knot[9, 29]][q, t]
Out[20]=	5 5 1 2 1 4 2 4 4 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 q q t q t q t q t q t q t q t 4 4 3 t 2 3 2 3 3 5 3 7 4 ---- + ---- + --- + 4 q t + 2 q t + 3 q t + q t + 2 q t + q t 5 3 q q t q t
In[21]:=	ColouredJones[Knot[9, 29], 2][q]
Out[21]=	-17 3 3 4 16 15 12 42 29 27 65 4 + q - --- + --- + --- - --- + --- + --- - --- + -- + -- - -- + 16 15 14 13 12 11 10 9 8 7 q q q q q q q q q q 32 43 71 20 51 60 2 3 4 5 -- + -- - -- + -- + -- - -- + 46 q - 37 q - 8 q + 30 q - 13 q - 6 5 4 3 2 q q q q q q 6 7 8 9 10 9 q + 11 q - q - 3 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

9_28 9_30