9 12

9_11

9_13

Visit 9 12's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 12's page at Knotilus! Visit 9 12's page at the original Knot Atlas! 9 12 Quick Notes

9 12 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X5,16,6,17 X11,1,12,18 X17,13,18,12 X7,14,8,15 X13,8,14,9 X15,6,16,7 X9,2,10,3
Gauss code	-1, 9, -2, 1, -3, 8, -6, 7, -9, 2, -4, 5, -7, 6, -8, 3, -5, 4
Dowker-Thistlethwaite code	4 10 16 14 2 18 8 6 12
Conway Notation	[4212]

Minimum Braid Representative: Length is 10, width is 5. Braid index is 5.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index 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 \{4,6\}} Nakanishi index 1 Maximal Thurston-Bennequin number [-10][-1] Hyperbolic Volume 8.83664 A-Polynomial See Data:9 12/A-polynomial

[edit Notes for 9 12's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 12's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{2}+9t-13+9t^{-1}-2t^{-2}$
Conway polynomial	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 -2 z^4+z^2+1}
2nd Alexander ideal (db, data sources)	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 \{1\}}
Determinant and Signature	{ 35, -2 }
Jones polynomial	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-2+4 q^{-1} -5 q^{-2} +6 q^{-3} -6 q^{-4} +5 q^{-5} -3 q^{-6} +2 q^{-7} - q^{-8} }
HOMFLY-PT polynomial (db, data sources)	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 -a^8+2 z^2 a^6+2 a^6-z^4 a^4-z^2 a^4-a^4-z^4 a^2-z^2 a^2+z^2+1}
Kauffman polynomial (db, data sources)	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 z^5 a^9-3 z^3 a^9+z a^9+2 z^6 a^8-6 z^4 a^8+4 z^2 a^8-a^8+2 z^7 a^7-5 z^5 a^7+3 z^3 a^7-z a^7+z^8 a^6-5 z^4 a^6+7 z^2 a^6-2 a^6+4 z^7 a^5-11 z^5 a^5+13 z^3 a^5-4 z a^5+z^8 a^4-z^4 a^4+3 z^2 a^4-a^4+2 z^7 a^3-3 z^5 a^3+4 z^3 a^3-2 z a^3+2 z^6 a^2-z^4 a^2-2 z^2 a^2+2 z^5 a-3 z^3 a+z^4-2 z^2+1}
The A2 invariant	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^{26}-q^{24}+q^{22}+q^{18}+2 q^{16}-q^{14}-q^{10}+q^6-q^4+2 q^2+ q^{-4} }
The G2 invariant	$q^{128}-q^{126}+2q^{124}-3q^{122}+2q^{120}-2q^{118}-2q^{116}+7q^{114}-10q^{112}+10q^{110}-10q^{108}+4q^{106}+4q^{104}-15q^{102}+20q^{100}-20q^{98}+14q^{96}-q^{94}-12q^{92}+19q^{90}-18q^{88}+15q^{86}-3q^{84}-10q^{82}+14q^{80}-11q^{78}+2q^{76}+10q^{74}-16q^{72}+18q^{70}-8q^{68}-4q^{66}+17q^{64}-26q^{62}+29q^{60}-21q^{58}+6q^{56}+11q^{54}-23q^{52}+30q^{50}-25q^{48}+13q^{46}-13q^{42}+16q^{40}-14q^{38}+3q^{36}+8q^{34}-13q^{32}+10q^{30}-2q^{28}-9q^{26}+17q^{24}-19q^{22}+15q^{20}-6q^{18}-5q^{16}+14q^{14}-16q^{12}+17q^{10}-10q^{8}+5q^{6}+q^{4}-7q^{2}+9-8q^{-2}+7q^{-4}-3q^{-6}+q^{-8}+2q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

Further Quantum Invariants

Further quantum knot invariants for 9_12.

A1 Invariants.

Weight	Invariant
1	$-q^{17}+q^{15}-q^{13}+2q^{11}-q^{9}+q^{5}-q^{3}+2q-q^{-1}+q^{-3}$
2	$q^{48}-q^{46}-q^{44}+3q^{42}-2q^{40}-4q^{38}+5q^{36}+q^{34}-6q^{32}+5q^{30}+3q^{28}-7q^{26}+2q^{24}+3q^{22}-3q^{20}-2q^{18}+2q^{16}+4q^{14}-5q^{12}+7q^{8}-5q^{6}-2q^{4}+7q^{2}-2-2q^{-2}+3q^{-4}-q^{-6}-q^{-8}+q^{-10}$
3	$-q^{93}+q^{91}+q^{89}-q^{87}-2q^{85}+2q^{83}+5q^{81}-2q^{79}-8q^{77}+10q^{73}+3q^{71}-12q^{69}-10q^{67}+12q^{65}+15q^{63}-6q^{61}-19q^{59}+3q^{57}+21q^{55}+2q^{53}-21q^{51}-7q^{49}+19q^{47}+7q^{45}-14q^{43}-9q^{41}+9q^{39}+10q^{37}-5q^{35}-10q^{33}-3q^{31}+11q^{29}+10q^{27}-10q^{25}-16q^{23}+9q^{21}+20q^{19}-5q^{17}-21q^{15}+q^{13}+20q^{11}+4q^{9}-15q^{7}-5q^{5}+11q^{3}+5q-5q^{-1}-4q^{-3}+3q^{-5}+2q^{-7}-2q^{-9}+2q^{-13}-q^{-17}-q^{-19}+q^{-21}$
4	$q^{152}-q^{150}-q^{148}+q^{146}+2q^{142}-4q^{140}-3q^{138}+4q^{136}+3q^{134}+8q^{132}-8q^{130}-13q^{128}+2q^{126}+8q^{124}+23q^{122}-4q^{120}-25q^{118}-17q^{116}-4q^{114}+40q^{112}+23q^{110}-12q^{108}-36q^{106}-41q^{104}+28q^{102}+50q^{100}+33q^{98}-24q^{96}-77q^{94}-14q^{92}+44q^{90}+69q^{88}+12q^{86}-80q^{84}-49q^{82}+16q^{80}+74q^{78}+38q^{76}-56q^{74}-51q^{72}-5q^{70}+52q^{68}+39q^{66}-25q^{64}-39q^{62}-15q^{60}+29q^{58}+32q^{56}+3q^{54}-30q^{52}-29q^{50}+31q^{46}+43q^{44}-17q^{42}-50q^{40}-36q^{38}+24q^{36}+79q^{34}+13q^{32}-50q^{30}-68q^{28}-5q^{26}+84q^{24}+42q^{22}-18q^{20}-67q^{18}-38q^{16}+48q^{14}+42q^{12}+18q^{10}-33q^{8}-40q^{6}+10q^{4}+14q^{2}+24-q^{-2}-18q^{-4}-2q^{-6}-5q^{-8}+10q^{-10}+4q^{-12}-3q^{-14}+3q^{-16}-6q^{-18}+4q^{-26}-q^{-28}-q^{-32}-q^{-34}+q^{-36}$
5	$-q^{225}+q^{223}+q^{221}-q^{219}+2q^{211}+2q^{209}-4q^{207}-6q^{205}-q^{203}+4q^{201}+9q^{199}+8q^{197}-4q^{195}-19q^{193}-17q^{191}+3q^{189}+23q^{187}+33q^{185}+13q^{183}-24q^{181}-50q^{179}-36q^{177}+12q^{175}+56q^{173}+66q^{171}+23q^{169}-47q^{167}-91q^{165}-71q^{163}+6q^{161}+91q^{159}+119q^{157}+58q^{155}-56q^{153}-149q^{151}-138q^{149}-4q^{147}+147q^{145}+202q^{143}+99q^{141}-106q^{139}-247q^{137}-186q^{135}+40q^{133}+245q^{131}+256q^{129}+44q^{127}-220q^{125}-299q^{123}-113q^{121}+170q^{119}+300q^{117}+164q^{115}-109q^{113}-278q^{111}-188q^{109}+65q^{107}+231q^{105}+181q^{103}-19q^{101}-182q^{99}-164q^{97}-3q^{95}+138q^{93}+134q^{91}+17q^{89}-96q^{87}-114q^{85}-35q^{83}+68q^{81}+106q^{79}+51q^{77}-34q^{75}-103q^{73}-93q^{71}+q^{69}+108q^{67}+140q^{65}+48q^{63}-107q^{61}-193q^{59}-114q^{57}+89q^{55}+238q^{53}+187q^{51}-47q^{49}-263q^{47}-256q^{45}-19q^{43}+249q^{41}+307q^{39}+93q^{37}-201q^{35}-316q^{33}-160q^{31}+122q^{29}+287q^{27}+207q^{25}-35q^{23}-224q^{21}-210q^{19}-36q^{17}+139q^{15}+184q^{13}+84q^{11}-65q^{9}-135q^{7}-90q^{5}+7q^{3}+80q+81q^{-1}+26q^{-3}-39q^{-5}-57q^{-7}-32q^{-9}+8q^{-11}+33q^{-13}+28q^{-15}+7q^{-17}-15q^{-19}-20q^{-21}-8q^{-23}+3q^{-25}+9q^{-27}+9q^{-29}+4q^{-31}-5q^{-33}-6q^{-35}-2q^{-37}-2q^{-39}+2q^{-41}+4q^{-43}+q^{-45}-q^{-47}-q^{-51}-q^{-53}+q^{-55}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{26}-q^{24}+q^{22}+q^{18}+2q^{16}-q^{14}-q^{10}+q^{6}-q^{4}+2q^{2}+q^{-4}$
1,1	$q^{68}-2q^{66}+4q^{64}-8q^{62}+17q^{60}-24q^{58}+32q^{56}-48q^{54}+57q^{52}-64q^{50}+62q^{48}-58q^{46}+46q^{44}-18q^{42}-6q^{40}+38q^{38}-65q^{36}+90q^{34}-112q^{32}+116q^{30}-122q^{28}+110q^{26}-90q^{24}+68q^{22}-37q^{20}+10q^{18}+20q^{16}-36q^{14}+47q^{12}-54q^{10}+56q^{8}-48q^{6}+42q^{4}-36q^{2}+30-20q^{-2}+15q^{-4}-10q^{-6}+6q^{-8}-2q^{-10}+q^{-12}$
2,0	$q^{66}+q^{64}-2q^{60}-q^{58}+q^{56}-q^{54}-3q^{52}-q^{50}+4q^{48}+3q^{46}-2q^{44}+4q^{40}-5q^{36}-3q^{34}+2q^{32}-2q^{28}+2q^{26}+q^{24}-q^{22}+3q^{20}+2q^{18}-3q^{16}-q^{14}+4q^{12}+q^{10}-4q^{8}+6q^{4}-3+q^{-2}+2q^{-4}-q^{-8}+q^{-12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{54}-q^{52}+q^{48}-3q^{46}+q^{44}+2q^{42}-5q^{40}+2q^{38}+4q^{36}-6q^{34}+q^{32}+4q^{30}-3q^{28}+3q^{24}+q^{22}-q^{20}-q^{18}+4q^{16}-2q^{14}-4q^{12}+6q^{10}-q^{8}-5q^{6}+5q^{4}+q^{2}-2+3q^{-2}+q^{-4}-q^{-6}+q^{-8}$
1,0,0	$-q^{35}-q^{33}-q^{31}+q^{29}+2q^{25}+q^{23}+2q^{21}-q^{19}-q^{15}-q^{13}+q^{7}-q^{5}+2q^{3}+q^{-1}+q^{-5}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{54}+q^{52}-2q^{50}+3q^{48}-5q^{46}+5q^{44}-6q^{42}+5q^{40}-4q^{38}+4q^{36}-q^{32}+6q^{30}-7q^{28}+10q^{26}-11q^{24}+11q^{22}-11q^{20}+7q^{18}-6q^{16}+2q^{14}-2q^{10}+5q^{8}-5q^{6}+7q^{4}-5q^{2}+4-3q^{-2}+3q^{-4}-q^{-6}+q^{-8}$
1,0	$q^{88}-q^{84}-q^{82}+q^{80}+2q^{78}-q^{76}-4q^{74}-q^{72}+4q^{70}+4q^{68}-3q^{66}-6q^{64}+6q^{60}+4q^{58}-5q^{56}-5q^{54}+q^{52}+5q^{50}-4q^{46}-q^{44}+4q^{42}+2q^{40}-3q^{38}-2q^{36}+3q^{34}+4q^{32}-2q^{30}-4q^{28}+q^{26}+5q^{24}-5q^{20}-2q^{18}+5q^{16}+5q^{14}-3q^{12}-6q^{10}-q^{8}+6q^{6}+3q^{4}-2q^{2}-3+3q^{-4}+2q^{-6}-q^{-8}-q^{-10}+q^{-14}$

G2 Invariants.

Weight

Invariant

1,0

$q^{128}-q^{126}+2q^{124}-3q^{122}+2q^{120}-2q^{118}-2q^{116}+7q^{114}-10q^{112}+10q^{110}-10q^{108}+4q^{106}+4q^{104}-15q^{102}+20q^{100}-20q^{98}+14q^{96}-q^{94}-12q^{92}+19q^{90}-18q^{88}+15q^{86}-3q^{84}-10q^{82}+14q^{80}-11q^{78}+2q^{76}+10q^{74}-16q^{72}+18q^{70}-8q^{68}-4q^{66}+17q^{64}-26q^{62}+29q^{60}-21q^{58}+6q^{56}+11q^{54}-23q^{52}+30q^{50}-25q^{48}+13q^{46}-13q^{42}+16q^{40}-14q^{38}+3q^{36}+8q^{34}-13q^{32}+10q^{30}-2q^{28}-9q^{26}+17q^{24}-19q^{22}+15q^{20}-6q^{18}-5q^{16}+14q^{14}-16q^{12}+17q^{10}-10q^{8}+5q^{6}+q^{4}-7q^{2}+9-8q^{-2}+7q^{-4}-3q^{-6}+q^{-8}+2q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

.

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 12"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{2}+9t-13+9t^{-1}-2t^{-2}$

In[5]:=

Conway[K][z]

Out[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 -2 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]=

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 \{1\}}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 35, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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 -a^8+2 z^2 a^6+2 a^6-z^4 a^4-z^2 a^4-a^4-z^4 a^2-z^2 a^2+z^2+1}

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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 z^5 a^9-3 z^3 a^9+z a^9+2 z^6 a^8-6 z^4 a^8+4 z^2 a^8-a^8+2 z^7 a^7-5 z^5 a^7+3 z^3 a^7-z a^7+z^8 a^6-5 z^4 a^6+7 z^2 a^6-2 a^6+4 z^7 a^5-11 z^5 a^5+13 z^3 a^5-4 z a^5+z^8 a^4-z^4 a^4+3 z^2 a^4-a^4+2 z^7 a^3-3 z^5 a^3+4 z^3 a^3-2 z a^3+2 z^6 a^2-z^4 a^2-2 z^2 a^2+2 z^5 a-3 z^3 a+z^4-2 z^2+1}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n84, ...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n15, ...}

Vassiliev invariants

V2 and V3:

(1, -3)

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 $4$ $-24$ $8$ ${\frac {302}{3}}$ ${\frac {58}{3}}$ $-96$ $-400$ $-32$ $-120$ ${\frac {32}{3}}$ $288$ ${\frac {1208}{3}}$ ${\frac {232}{3}}$ ${\frac {49471}{30}}$ $-{\frac {5942}{15}}$ ${\frac {49622}{45}}$ ${\frac {1793}{18}}$ ${\frac {4831}{30}}$

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 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 r} ). The squares with yellow highlighting are those on the "critical diagonals", where 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 j-2r=s+1} or 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 j-2r=s-1} , where 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 s=} -2 is the signature of 9 12. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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 \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} 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 i=-3} 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 i=-1} 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 r=-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 {\mathbb Z}} 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 r=-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 {\mathbb Z}\oplus{\mathbb Z}_2} 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 {\mathbb Z}} 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 r=-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 {\mathbb Z}^{2}\oplus{\mathbb Z}_2} 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 {\mathbb Z}} $r=-4$ 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 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}} 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 {\mathbb Z}^{2}} 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 r=-3} 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 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}} 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 {\mathbb Z}^{3}} 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 r=-2} 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 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}} 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 {\mathbb Z}^{3}} 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 r=-1} 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 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}} 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 {\mathbb Z}^{3}} 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 r=0} 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 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}} 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 {\mathbb Z}^{3}} 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 r=1} 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 {\mathbb Z}\oplus{\mathbb Z}_2} 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 {\mathbb Z}} 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 r=2} 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 {\mathbb Z}_2} 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 {\mathbb Z}}

The Coloured Jones Polynomials

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

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 n}	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 J_n}
2	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^4-2 q^3+5 q-7+14 q^{-2} -16 q^{-3} -3 q^{-4} +26 q^{-5} -23 q^{-6} -8 q^{-7} +35 q^{-8} -25 q^{-9} -12 q^{-10} +34 q^{-11} -19 q^{-12} -13 q^{-13} +25 q^{-14} -9 q^{-15} -11 q^{-16} +14 q^{-17} -2 q^{-18} -7 q^{-19} +5 q^{-20} -2 q^{-22} + q^{-23} }
3	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^9-2 q^8+q^6+3 q^5-4 q^4-2 q^3+5 q^2+4 q-11-3 q^{-1} +15 q^{-2} +10 q^{-3} -27 q^{-4} -13 q^{-5} +34 q^{-6} +26 q^{-7} -46 q^{-8} -35 q^{-9} +50 q^{-10} +51 q^{-11} -57 q^{-12} -60 q^{-13} +56 q^{-14} +71 q^{-15} -56 q^{-16} -74 q^{-17} +49 q^{-18} +76 q^{-19} -41 q^{-20} -75 q^{-21} +31 q^{-22} +71 q^{-23} -20 q^{-24} -63 q^{-25} +5 q^{-26} +57 q^{-27} +3 q^{-28} -44 q^{-29} -13 q^{-30} +35 q^{-31} +16 q^{-32} -23 q^{-33} -16 q^{-34} +13 q^{-35} +14 q^{-36} -8 q^{-37} -9 q^{-38} +3 q^{-39} +6 q^{-40} -2 q^{-41} -2 q^{-42} +2 q^{-44} - q^{-45} }
4	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^{16}-2 q^{15}+q^{13}-q^{12}+6 q^{11}-6 q^{10}+q^8-7 q^7+15 q^6-12 q^5+7 q^4+7 q^3-22 q^2+18 q-28+24 q^{-1} +32 q^{-2} -32 q^{-3} +14 q^{-4} -78 q^{-5} +31 q^{-6} +83 q^{-7} -8 q^{-8} +20 q^{-9} -164 q^{-10} +2 q^{-11} +132 q^{-12} +52 q^{-13} +62 q^{-14} -253 q^{-15} -61 q^{-16} +150 q^{-17} +115 q^{-18} +128 q^{-19} -308 q^{-20} -121 q^{-21} +136 q^{-22} +148 q^{-23} +188 q^{-24} -320 q^{-25} -152 q^{-26} +107 q^{-27} +147 q^{-28} +221 q^{-29} -291 q^{-30} -155 q^{-31} +63 q^{-32} +123 q^{-33} +235 q^{-34} -227 q^{-35} -142 q^{-36} +6 q^{-37} +77 q^{-38} +230 q^{-39} -133 q^{-40} -106 q^{-41} -52 q^{-42} +12 q^{-43} +199 q^{-44} -41 q^{-45} -49 q^{-46} -77 q^{-47} -46 q^{-48} +136 q^{-49} +12 q^{-50} +8 q^{-51} -60 q^{-52} -68 q^{-53} +67 q^{-54} +17 q^{-55} +32 q^{-56} -25 q^{-57} -51 q^{-58} +23 q^{-59} +4 q^{-60} +24 q^{-61} -4 q^{-62} -24 q^{-63} +8 q^{-64} -2 q^{-65} +9 q^{-66} + q^{-67} -8 q^{-68} +3 q^{-69} - q^{-70} +2 q^{-71} -2 q^{-73} + q^{-74} }
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^{25}-2 q^{24}+q^{22}-q^{21}+2 q^{20}+4 q^{19}-4 q^{18}-4 q^{17}+q^{16}-5 q^{15}+3 q^{14}+13 q^{13}+q^{12}-4 q^{11}-5 q^{10}-16 q^9-9 q^8+18 q^7+23 q^6+17 q^5-41 q^3-49 q^2-7 q+41+82 q^{-1} +55 q^{-2} -42 q^{-3} -122 q^{-4} -104 q^{-5} -4 q^{-6} +152 q^{-7} +204 q^{-8} +58 q^{-9} -167 q^{-10} -279 q^{-11} -178 q^{-12} +138 q^{-13} +393 q^{-14} +300 q^{-15} -87 q^{-16} -444 q^{-17} -460 q^{-18} -18 q^{-19} +508 q^{-20} +594 q^{-21} +127 q^{-22} -502 q^{-23} -728 q^{-24} -255 q^{-25} +501 q^{-26} +810 q^{-27} +361 q^{-28} -451 q^{-29} -877 q^{-30} -458 q^{-31} +422 q^{-32} +896 q^{-33} +516 q^{-34} -359 q^{-35} -909 q^{-36} -565 q^{-37} +328 q^{-38} +886 q^{-39} +585 q^{-40} -274 q^{-41} -854 q^{-42} -603 q^{-43} +225 q^{-44} +807 q^{-45} +603 q^{-46} -161 q^{-47} -737 q^{-48} -599 q^{-49} +84 q^{-50} +646 q^{-51} +585 q^{-52} +2 q^{-53} -537 q^{-54} -549 q^{-55} -82 q^{-56} +393 q^{-57} +495 q^{-58} +171 q^{-59} -264 q^{-60} -413 q^{-61} -212 q^{-62} +110 q^{-63} +309 q^{-64} +250 q^{-65} -201 q^{-67} -223 q^{-68} -95 q^{-69} +83 q^{-70} +189 q^{-71} +141 q^{-72} +4 q^{-73} -120 q^{-74} -150 q^{-75} -68 q^{-76} +55 q^{-77} +130 q^{-78} +97 q^{-79} -6 q^{-80} -89 q^{-81} -96 q^{-82} -30 q^{-83} +53 q^{-84} +77 q^{-85} +38 q^{-86} -19 q^{-87} -53 q^{-88} -40 q^{-89} +9 q^{-90} +29 q^{-91} +24 q^{-92} +7 q^{-93} -16 q^{-94} -20 q^{-95} - q^{-96} +9 q^{-97} +4 q^{-98} +5 q^{-99} - q^{-100} -8 q^{-101} +4 q^{-103} - q^{-104} + q^{-106} -2 q^{-107} +2 q^{-109} - q^{-110} }
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^{36}-2 q^{35}+q^{33}-q^{32}+2 q^{31}+6 q^{29}-8 q^{28}-4 q^{27}+3 q^{26}-6 q^{25}+4 q^{24}+4 q^{23}+23 q^{22}-14 q^{21}-10 q^{20}+q^{19}-23 q^{18}-5 q^{17}+7 q^{16}+63 q^{15}-8 q^{14}-6 q^{13}+q^{12}-66 q^{11}-48 q^{10}-9 q^9+128 q^8+34 q^7+41 q^6+30 q^5-140 q^4-167 q^3-103 q^2+179 q+123+203 q^{-1} +184 q^{-2} -185 q^{-3} -384 q^{-4} -388 q^{-5} +74 q^{-6} +164 q^{-7} +514 q^{-8} +622 q^{-9} +3 q^{-10} -566 q^{-11} -904 q^{-12} -390 q^{-13} -102 q^{-14} +801 q^{-15} +1362 q^{-16} +649 q^{-17} -403 q^{-18} -1420 q^{-19} -1200 q^{-20} -893 q^{-21} +729 q^{-22} +2109 q^{-23} +1675 q^{-24} +284 q^{-25} -1581 q^{-26} -2006 q^{-27} -2038 q^{-28} +171 q^{-29} +2492 q^{-30} +2666 q^{-31} +1258 q^{-32} -1295 q^{-33} -2446 q^{-34} -3078 q^{-35} -598 q^{-36} +2439 q^{-37} +3266 q^{-38} +2081 q^{-39} -819 q^{-40} -2475 q^{-41} -3703 q^{-42} -1212 q^{-43} +2177 q^{-44} +3457 q^{-45} +2535 q^{-46} -430 q^{-47} -2300 q^{-48} -3934 q^{-49} -1547 q^{-50} +1910 q^{-51} +3419 q^{-52} +2693 q^{-53} -174 q^{-54} -2078 q^{-55} -3930 q^{-56} -1709 q^{-57} +1630 q^{-58} +3254 q^{-59} +2709 q^{-60} +79 q^{-61} -1770 q^{-62} -3764 q^{-63} -1836 q^{-64} +1206 q^{-65} +2904 q^{-66} +2635 q^{-67} +435 q^{-68} -1252 q^{-69} -3366 q^{-70} -1943 q^{-71} +568 q^{-72} +2263 q^{-73} +2379 q^{-74} +848 q^{-75} -500 q^{-76} -2642 q^{-77} -1884 q^{-78} -154 q^{-79} +1349 q^{-80} +1810 q^{-81} +1092 q^{-82} +303 q^{-83} -1635 q^{-84} -1485 q^{-85} -652 q^{-86} +403 q^{-87} +959 q^{-88} +938 q^{-89} +810 q^{-90} -626 q^{-91} -773 q^{-92} -670 q^{-93} -204 q^{-94} +121 q^{-95} +428 q^{-96} +794 q^{-97} +6 q^{-98} -73 q^{-99} -295 q^{-100} -283 q^{-101} -336 q^{-102} -82 q^{-103} +412 q^{-104} +124 q^{-105} +265 q^{-106} +83 q^{-107} -43 q^{-108} -333 q^{-109} -281 q^{-110} +63 q^{-111} -25 q^{-112} +223 q^{-113} +196 q^{-114} +145 q^{-115} -138 q^{-116} -201 q^{-117} -49 q^{-118} -120 q^{-119} +69 q^{-120} +115 q^{-121} +152 q^{-122} -17 q^{-123} -75 q^{-124} -19 q^{-125} -94 q^{-126} -6 q^{-127} +28 q^{-128} +82 q^{-129} +6 q^{-130} -20 q^{-131} +10 q^{-132} -40 q^{-133} -12 q^{-134} - q^{-135} +31 q^{-136} + q^{-137} -8 q^{-138} +11 q^{-139} -11 q^{-140} -4 q^{-141} -3 q^{-142} +10 q^{-143} - q^{-144} -5 q^{-145} +5 q^{-146} -2 q^{-147} - q^{-149} +2 q^{-150} -2 q^{-152} + q^{-153} }
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^{49}-2 q^{48}+q^{46}-q^{45}+2 q^{44}+2 q^{42}+2 q^{41}-8 q^{40}-2 q^{39}+2 q^{38}-5 q^{37}+6 q^{36}+2 q^{35}+10 q^{34}+14 q^{33}-20 q^{32}-8 q^{31}-3 q^{30}-18 q^{29}+5 q^{28}+q^{27}+27 q^{26}+47 q^{25}-22 q^{24}-15 q^{23}-17 q^{22}-52 q^{21}-3 q^{20}-13 q^{19}+47 q^{18}+115 q^{17}+2 q^{16}-10 q^{15}-54 q^{14}-129 q^{13}-46 q^{12}-45 q^{11}+87 q^{10}+245 q^9+105 q^8+56 q^7-107 q^6-328 q^5-232 q^4-190 q^3+107 q^2+497 q+437+394 q^{-1} -21 q^{-2} -633 q^{-3} -751 q^{-4} -785 q^{-5} -213 q^{-6} +719 q^{-7} +1108 q^{-8} +1367 q^{-9} +734 q^{-10} -604 q^{-11} -1499 q^{-12} -2163 q^{-13} -1532 q^{-14} +211 q^{-15} +1679 q^{-16} +3035 q^{-17} +2740 q^{-18} +666 q^{-19} -1634 q^{-20} -3948 q^{-21} -4153 q^{-22} -1911 q^{-23} +1075 q^{-24} +4556 q^{-25} +5758 q^{-26} +3676 q^{-27} -81 q^{-28} -4912 q^{-29} -7247 q^{-30} -5570 q^{-31} -1417 q^{-32} +4701 q^{-33} +8508 q^{-34} +7619 q^{-35} +3201 q^{-36} -4138 q^{-37} -9366 q^{-38} -9411 q^{-39} -5108 q^{-40} +3157 q^{-41} +9801 q^{-42} +10930 q^{-43} +6913 q^{-44} -2041 q^{-45} -9850 q^{-46} -11991 q^{-47} -8469 q^{-48} +875 q^{-49} +9628 q^{-50} +12691 q^{-51} +9659 q^{-52} +131 q^{-53} -9240 q^{-54} -13006 q^{-55} -10516 q^{-56} -995 q^{-57} +8840 q^{-58} +13141 q^{-59} +11041 q^{-60} +1579 q^{-61} -8434 q^{-62} -13063 q^{-63} -11373 q^{-64} -2057 q^{-65} +8099 q^{-66} +12985 q^{-67} +11532 q^{-68} +2363 q^{-69} -7775 q^{-70} -12798 q^{-71} -11617 q^{-72} -2689 q^{-73} +7427 q^{-74} +12593 q^{-75} +11682 q^{-76} +3005 q^{-77} -7008 q^{-78} -12276 q^{-79} -11681 q^{-80} -3415 q^{-81} +6397 q^{-82} +11827 q^{-83} +11667 q^{-84} +3922 q^{-85} -5618 q^{-86} -11181 q^{-87} -11530 q^{-88} -4511 q^{-89} +4590 q^{-90} +10267 q^{-91} +11255 q^{-92} +5150 q^{-93} -3337 q^{-94} -9083 q^{-95} -10777 q^{-96} -5738 q^{-97} +1963 q^{-98} +7609 q^{-99} +9957 q^{-100} +6175 q^{-101} -463 q^{-102} -5890 q^{-103} -8898 q^{-104} -6365 q^{-105} -848 q^{-106} +4064 q^{-107} +7429 q^{-108} +6152 q^{-109} +2011 q^{-110} -2213 q^{-111} -5819 q^{-112} -5594 q^{-113} -2696 q^{-114} +633 q^{-115} +4038 q^{-116} +4596 q^{-117} +2962 q^{-118} +657 q^{-119} -2385 q^{-120} -3432 q^{-121} -2731 q^{-122} -1386 q^{-123} +993 q^{-124} +2130 q^{-125} +2124 q^{-126} +1672 q^{-127} -4 q^{-128} -999 q^{-129} -1338 q^{-130} -1489 q^{-131} -512 q^{-132} +135 q^{-133} +529 q^{-134} +1043 q^{-135} +632 q^{-136} +370 q^{-137} +107 q^{-138} -522 q^{-139} -459 q^{-140} -520 q^{-141} -489 q^{-142} +56 q^{-143} +163 q^{-144} +444 q^{-145} +608 q^{-146} +232 q^{-147} +118 q^{-148} -241 q^{-149} -546 q^{-150} -343 q^{-151} -278 q^{-152} +30 q^{-153} +369 q^{-154} +315 q^{-155} +358 q^{-156} +105 q^{-157} -224 q^{-158} -217 q^{-159} -294 q^{-160} -160 q^{-161} +65 q^{-162} +109 q^{-163} +245 q^{-164} +166 q^{-165} -26 q^{-166} -45 q^{-167} -134 q^{-168} -113 q^{-169} -24 q^{-170} -19 q^{-171} +92 q^{-172} +93 q^{-173} +11 q^{-174} +10 q^{-175} -42 q^{-176} -35 q^{-177} -10 q^{-178} -32 q^{-179} +18 q^{-180} +36 q^{-181} +6 q^{-182} +8 q^{-183} -14 q^{-184} -6 q^{-185} +6 q^{-186} -16 q^{-187} +10 q^{-189} +2 q^{-190} +3 q^{-191} -6 q^{-192} - q^{-193} +6 q^{-194} -4 q^{-195} -2 q^{-196} +2 q^{-197} + q^{-199} -2 q^{-200} +2 q^{-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, 12]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[5, 16, 6, 17], X[11, 1, 12, 18], X[17, 13, 18, 12], X[7, 14, 8, 15], X[13, 8, 14, 9], X[15, 6, 16, 7], X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[9, 12]]
Out[3]=	GaussCode[-1, 9, -2, 1, -3, 8, -6, 7, -9, 2, -4, 5, -7, 6, -8, 3, -5, 4]
In[4]:=	DTCode[Knot[9, 12]]
Out[4]=	DTCode[4, 10, 16, 14, 2, 18, 8, 6, 12]
In[5]:=	br = BR[Knot[9, 12]]
Out[5]=	BR[5, {-1, -1, 2, -1, -3, 2, -3, -4, 3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 10}
In[7]:=	BraidIndex[Knot[9, 12]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[9, 12]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 12]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 2, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 12]][t]
Out[10]=	2 9 2 -13 - -- + - + 9 t - 2 t 2 t t
In[11]:=	Conway[Knot[9, 12]][z]
Out[11]=	2 4 1 + z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 12], Knot[11, NonAlternating, 84]}
In[13]:=	{KnotDet[Knot[9, 12]], KnotSignature[Knot[9, 12]]}
Out[13]=	{35, -2}
In[14]:=	Jones[Knot[9, 12]][q]
Out[14]=	-8 2 3 5 6 6 5 4 -2 - q + -- - -- + -- - -- + -- - -- + - + 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, 12], Knot[11, NonAlternating, 15]}
In[16]:=	A2Invariant[Knot[9, 12]][q]
Out[16]=	-26 -24 -22 -18 2 -14 -10 -6 -4 2 4 -q - q + q + q + --- - q - q + q - q + -- + q 16 2 q q
In[17]:=	HOMFLYPT[Knot[9, 12]][a, z]
Out[17]=	4 6 8 2 2 2 4 2 6 2 2 4 4 4 1 - a + 2 a - a + z - a z - a z + 2 a z - a z - a z
In[18]:=	Kauffman[Knot[9, 12]][a, z]
Out[18]=	4 6 8 3 5 7 9 2 2 2 1 - a - 2 a - a - 2 a z - 4 a z - a z + a z - 2 z - 2 a z + 4 2 6 2 8 2 3 3 3 5 3 7 3 3 a z + 7 a z + 4 a z - 3 a z + 4 a z + 13 a z + 3 a z - 9 3 4 2 4 4 4 6 4 8 4 5 3 5 3 a z + z - a z - a z - 5 a z - 6 a z + 2 a z - 3 a z - 5 5 7 5 9 5 2 6 8 6 3 7 5 7 11 a z - 5 a z + a z + 2 a z + 2 a z + 2 a z + 4 a z + 7 7 4 8 6 8 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[9, 12]], Vassiliev[3][Knot[9, 12]]}
Out[19]=	{1, -3}
In[20]:=	Kh[Knot[9, 12]][q, t]
Out[20]=	2 3 1 1 1 2 1 3 2 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 q q t q t q t q t q t q t q t 3 3 3 3 2 3 t 3 2 ----- + ----- + ----- + ----- + ---- + ---- + - + q t + q t 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t
In[21]:=	ColouredJones[Knot[9, 12], 2][q]
Out[21]=	-23 2 5 7 2 14 11 9 25 13 19 -7 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + 22 20 19 18 17 16 15 14 13 12 q q q q q q q q q q 34 12 25 35 8 23 26 3 16 14 3 4 --- - --- - -- + -- - -- - -- + -- - -- - -- + -- + 5 q - 2 q + q 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q

See/edit the Rolfsen_Splice_Template. Back to the top.

9_11 9_13