9 29

9_28

9_30

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 29's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 29 at Knotilus!

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	A Morse Link Presentation	An Arc Presentation
Length is 9, width is 4, Braid index is 4		[{2, 4}, {1, 3}, {12, 5}, {4, 9}, {10, 6}, {5, 7}, {9, 11}, {6, 8}, {7, 2}, {3, 10}, {8, 12}, {11, 1}]

[edit Notes on presentations of 9 29]

Computer Talk[show]

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["9 29"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

1, -4, 3, -7, 5, -1, 6, -9, 7, -3, 2, -8, 9, -5, 4, -2, 8, -6

In[6]:=

DTCode[K]

Out[6]=

6 10 14 18 4 16 8 2 12

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[.2.20.2]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

${\textrm {BR}}(4,\{1,-2,-2,3,-2,1,-2,3,-2\})$

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

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

Out[10]=

{ 4, 9, 4 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{2, 4}, {1, 3}, {12, 5}, {4, 9}, {10, 6}, {5, 7}, {9, 11}, {6, 8}, {7, 2}, {3, 10}, {8, 12}, {11, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

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[show]

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[show]

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, $q\leftrightarrow q^{-1}$): {}

Computer Talk[show]

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 29"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

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

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

Out[4]=

{ $t^{3}-5t^{2}+12t-15+12t^{-1}-5t^{-2}+t^{-3}$, $q^{3}-3q^{2}+5q-7+9q^{-1}-8q^{-2}+8q^{-3}-6q^{-4}+3q^{-5}-q^{-6}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{9_28, 10_163, K11n87,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

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 $sl(2)$)[show]

$n$	$J_{n}$
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	$q^{55}-3q^{54}-q^{53}+5q^{52}+3q^{51}+3q^{50}-q^{49}-17q^{48}-24q^{47}+6q^{46}+31q^{45}+49q^{44}+40q^{43}-30q^{42}-116q^{41}-121q^{40}-14q^{39}+141q^{38}+254q^{37}+183q^{36}-97q^{35}-393q^{34}-436q^{33}-119q^{32}+397q^{31}+745q^{30}+547q^{29}-187q^{28}-946q^{27}-1087q^{26}-342q^{25}+854q^{24}+1603q^{23}+1140q^{22}-372q^{21}-1852q^{20}-2026q^{19}-532q^{18}+1651q^{17}+2778q^{16}+1718q^{15}-939q^{14}-3154q^{13}-2961q^{12}-221q^{11}+3013q^{10}+4016q^{9}+1672q^{8}-2353q^{7}-4724q^{6}-3172q^{5}+1288q^{4}+4966q^{3}+4547q^{2}+62q-4825-5707q^{-1}-1432q^{-2}+4371q^{-3}+6521q^{-4}+2836q^{-5}-3748q^{-6}-7193q^{-7}-4013q^{-8}+3081q^{-9}+7581q^{-10}+5147q^{-11}-2392q^{-12}-8012q^{-13}-6080q^{-14}+1787q^{-15}+8239q^{-16}+7044q^{-17}-1117q^{-18}-8550q^{-19}-7887q^{-20}+434q^{-21}+8574q^{-22}+8763q^{-23}+451q^{-24}-8492q^{-25}-9411q^{-26}-1445q^{-27}+7895q^{-28}+9875q^{-29}+2584q^{-30}-6985q^{-31}-9832q^{-32}-3624q^{-33}+5569q^{-34}+9284q^{-35}+4462q^{-36}-3979q^{-37}-8171q^{-38}-4816q^{-39}+2348q^{-40}+6628q^{-41}+4672q^{-42}-964q^{-43}-4922q^{-44}-4076q^{-45}+42q^{-46}+3291q^{-47}+3159q^{-48}+473q^{-49}-1978q^{-50}-2219q^{-51}-579q^{-52}+1066q^{-53}+1371q^{-54}+490q^{-55}-511q^{-56}-764q^{-57}-323q^{-58}+220q^{-59}+392q^{-60}+170q^{-61}-98q^{-62}-172q^{-63}-71q^{-64}+33q^{-65}+71q^{-66}+37q^{-67}-28q^{-68}-28q^{-69}+4q^{-70}+3q^{-71}+2q^{-72}+9q^{-73}-6q^{-74}-6q^{-75}+7q^{-76}-q^{-77}-3q^{-78}+3q^{-79}-q^{-80}$
6	$q^{78}-3q^{77}-q^{76}+5q^{75}+3q^{74}+3q^{73}-7q^{72}+q^{71}-20q^{70}-22q^{69}+18q^{68}+32q^{67}+54q^{66}+18q^{65}+24q^{64}-90q^{63}-164q^{62}-101q^{61}-6q^{60}+180q^{59}+241q^{58}+393q^{57}+74q^{56}-324q^{55}-581q^{54}-647q^{53}-271q^{52}+207q^{51}+1217q^{50}+1242q^{49}+702q^{48}-376q^{47}-1558q^{46}-2121q^{45}-1870q^{44}+492q^{43}+2230q^{42}+3431q^{41}+2741q^{40}+412q^{39}-2883q^{38}-5620q^{37}-4131q^{36}-1195q^{35}+3704q^{34}+6942q^{33}+7131q^{32}+2474q^{31}-5071q^{30}-8939q^{29}-9751q^{28}-3787q^{27}+4702q^{26}+12713q^{25}+12976q^{24}+4715q^{23}-5335q^{22}-15491q^{21}-16099q^{20}-7864q^{19}+8110q^{18}+19072q^{17}+18793q^{16}+9107q^{15}-9630q^{14}-22746q^{13}-24071q^{12}-7497q^{11}+12872q^{10}+26434q^{9}+26530q^{8}+7049q^{7}-17140q^{6}-33580q^{5}-25683q^{4}-3472q^{3}+22509q^{2}+37520q+26018-2422q^{-1}-32653q^{-2}-38225q^{-3}-21762q^{-4}+10767q^{-5}+39573q^{-6}+40200q^{-7}+13842q^{-8}-25216q^{-9}-43540q^{-10}-36056q^{-11}-2035q^{-12}+36528q^{-13}+48571q^{-14}+26735q^{-15}-17057q^{-16}-45092q^{-17}-45769q^{-18}-12014q^{-19}+33152q^{-20}+54119q^{-21}+36146q^{-22}-10914q^{-23}-46555q^{-24}-53611q^{-25}-19825q^{-26}+30914q^{-27}+59547q^{-28}+44895q^{-29}-5020q^{-30}-48015q^{-31}-61616q^{-32}-28849q^{-33}+26676q^{-34}+63623q^{-35}+54687q^{-36}+4746q^{-37}-44936q^{-38}-67352q^{-39}-40453q^{-40}+15693q^{-41}+60548q^{-42}+61882q^{-43}+19047q^{-44}-32354q^{-45}-63985q^{-46}-49510q^{-47}-1443q^{-48}+45690q^{-49}+58777q^{-50}+30939q^{-51}-12630q^{-52}-47747q^{-53}-47621q^{-54}-15894q^{-55}+23543q^{-56}+42812q^{-57}+31684q^{-58}+3796q^{-59}-25265q^{-60}-33676q^{-61}-19131q^{-62}+5635q^{-63}+22233q^{-64}+21648q^{-65}+9222q^{-66}-8177q^{-67}-16791q^{-68}-12912q^{-69}-1642q^{-70}+7705q^{-71}+9869q^{-72}+6473q^{-73}-987q^{-74}-5801q^{-75}-5525q^{-76}-1818q^{-77}+1708q^{-78}+2954q^{-79}+2617q^{-80}+272q^{-81}-1472q^{-82}-1551q^{-83}-611q^{-84}+282q^{-85}+558q^{-86}+695q^{-87}+117q^{-88}-337q^{-89}-294q^{-90}-77q^{-91}+68q^{-92}+43q^{-93}+137q^{-94}+12q^{-95}-81q^{-96}-32q^{-97}+8q^{-98}+24q^{-99}-18q^{-100}+23q^{-101}+3q^{-102}-20q^{-103}+3q^{-104}+3q^{-105}+6q^{-106}-7q^{-107}+q^{-108}+3q^{-109}-3q^{-110}+q^{-111}$
7	$q^{105}-3q^{104}-q^{103}+5q^{102}+3q^{101}+3q^{100}-7q^{99}-5q^{98}-2q^{97}-18q^{96}-10q^{95}+19q^{94}+37q^{93}+57q^{92}+19q^{91}-20q^{90}-35q^{89}-133q^{88}-152q^{87}-89q^{86}+42q^{85}+268q^{84}+342q^{83}+314q^{82}+226q^{81}-187q^{80}-602q^{79}-875q^{78}-884q^{77}-221q^{76}+540q^{75}+1308q^{74}+1944q^{73}+1583q^{72}+493q^{71}-1225q^{70}-3091q^{69}-3567q^{68}-2798q^{67}-617q^{66}+2878q^{65}+5492q^{64}+6556q^{63}+4805q^{62}-155q^{61}-5479q^{60}-9858q^{59}-10883q^{58}-6485q^{57}+1127q^{56}+10330q^{55}+16875q^{54}+15988q^{53}+8580q^{52}-4633q^{51}-18514q^{50}-25401q^{49}-22994q^{48}-8982q^{47}+12056q^{46}+29687q^{45}+37723q^{44}+29101q^{43}+5120q^{42}-23262q^{41}-46602q^{40}-51133q^{39}-31706q^{38}+3323q^{37}+43079q^{36}+67283q^{35}+62149q^{34}+29443q^{33}-22796q^{32}-69988q^{31}-88369q^{30}-69424q^{29}-13652q^{28}+53920q^{27}+101325q^{26}+107869q^{25}+61475q^{24}-18257q^{23}-94844q^{22}-135477q^{21}-112112q^{20}-32813q^{19}+66891q^{18}+144984q^{17}+155949q^{16}+91657q^{15}-20006q^{14}-132996q^{13}-185483q^{12}-149479q^{11}-39161q^{10}+100893q^{9}+196107q^{8}+198060q^{7}+102902q^{6}-53039q^{5}-187532q^{4}-232727q^{3}-163734q^{2}-3501q+162847+251327q^{-1}+215796q^{-2}+62457q^{-3}-126646q^{-4}-255411q^{-5}-256846q^{-6}-118052q^{-7}+85277q^{-8}+248069q^{-9}+285755q^{-10}+166696q^{-11}-42882q^{-12}-233495q^{-13}-304972q^{-14}-206976q^{-15}+4237q^{-16}+216215q^{-17}+316237q^{-18}+238621q^{-19}+29174q^{-20}-199228q^{-21}-323576q^{-22}-263598q^{-23}-55747q^{-24}+185744q^{-25}+329272q^{-26}+283380q^{-27}+76834q^{-28}-176163q^{-29}-336443q^{-30}-301601q^{-31}-93702q^{-32}+171077q^{-33}+345916q^{-34}+320128q^{-35}+109928q^{-36}-167583q^{-37}-358376q^{-38}-341928q^{-39}-128398q^{-40}+163029q^{-41}+371166q^{-42}+366896q^{-43}+153100q^{-44}-152125q^{-45}-381130q^{-46}-394208q^{-47}-184955q^{-48}+131056q^{-49}+381977q^{-50}+419089q^{-51}+223868q^{-52}-96262q^{-53}-368829q^{-54}-435963q^{-55}-264871q^{-56}+48726q^{-57}+336921q^{-58}+437136q^{-59}+301531q^{-60}+8157q^{-61}-286250q^{-62}-418086q^{-63}-325264q^{-64}-65768q^{-65}+220557q^{-66}+376410q^{-67}+329393q^{-68}+115480q^{-69}-147897q^{-70}-316152q^{-71}-311118q^{-72}-148664q^{-73}+78719q^{-74}+244798q^{-75}+272303q^{-76}+161037q^{-77}-21770q^{-78}-172351q^{-79}-220234q^{-80}-153536q^{-81}-16964q^{-82}+108723q^{-83}+163644q^{-84}+130712q^{-85}+36969q^{-86}-59298q^{-87}-111331q^{-88}-100865q^{-89}-41494q^{-90}+26537q^{-91}+69152q^{-92}+70437q^{-93}+36131q^{-94}-7869q^{-95}-39001q^{-96}-44848q^{-97}-26780q^{-98}-477q^{-99}+20036q^{-100}+26152q^{-101}+17297q^{-102}+2775q^{-103}-9348q^{-104}-13916q^{-105}-9910q^{-106}-2628q^{-107}+4013q^{-108}+6942q^{-109}+5121q^{-110}+1609q^{-111}-1701q^{-112}-3180q^{-113}-2289q^{-114}-809q^{-115}+627q^{-116}+1386q^{-117}+983q^{-118}+359q^{-119}-325q^{-120}-623q^{-121}-283q^{-122}-74q^{-123}+97q^{-124}+201q^{-125}+102q^{-126}+64q^{-127}-71q^{-128}-128q^{-129}+9q^{-130}+27q^{-131}+12q^{-132}+11q^{-133}-10q^{-134}+22q^{-135}-9q^{-136}-29q^{-137}+15q^{-138}+8q^{-139}-3q^{-141}-6q^{-142}+7q^{-143}-q^{-144}-3q^{-145}+3q^{-146}-q^{-147}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

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