9 27

9_26

9_28

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 27's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 27 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X11,1,12,18 X5,13,6,12 X13,17,14,16 X7,14,8,15 X15,6,16,7 X17,9,18,8 X9,2,10,3
Gauss code	-1, 9, -2, 1, -4, 7, -6, 8, -9, 2, -3, 4, -5, 6, -7, 5, -8, 3
Dowker-Thistlethwaite code	4 10 12 14 2 18 16 6 8
Conway Notation	[212112]

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

[edit Notes on presentations of 9 27]

Computer Talk

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 27"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3,10,4,11 X11,1,12,18 X5,13,6,12 X13,17,14,16 X7,14,8,15 X15,6,16,7 X17,9,18,8 X9,2,10,3

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 12 14 2 18 16 6 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[212112]

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]=

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

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 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 [-6][-5] Hyperbolic Volume 11. A-Polynomial See Data:9 27/A-polynomial

[edit Notes for 9 27'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 0} 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 0} 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 0} Rasmussen s-Invariant 0

[edit Notes for 9 27's four dimensional invariants]

Polynomial invariants

Alexander 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 -t^3+5 t^2-11 t+15-11 t^{-1} +5 t^{-2} - t^{-3} }
Conway polynomial	$-z^{6}-z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 49, 0 }
Jones polynomial	$q^{4}-3q^{3}+5q^{2}-7q+9-8q^{-1}+7q^{-2}-5q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+2a^{2}z^{4}+z^{4}a^{-2}-4z^{4}-a^{4}z^{2}+5a^{2}z^{2}+2z^{2}a^{-2}-6z^{2}-a^{4}+3a^{2}+a^{-2}-2$
Kauffman polynomial (db, data sources)	$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+6az^{7}+3z^{7}a^{-1}+3a^{4}z^{6}+6a^{2}z^{6}+4z^{6}a^{-2}+7z^{6}+a^{5}z^{5}-4a^{3}z^{5}-8az^{5}+3z^{5}a^{-3}-7a^{4}z^{4}-17a^{2}z^{4}-5z^{4}a^{-2}+z^{4}a^{-4}-16z^{4}-2a^{5}z^{3}-2a^{3}z^{3}-4z^{3}a^{-1}-4z^{3}a^{-3}+4a^{4}z^{2}+12a^{2}z^{2}+3z^{2}a^{-2}-z^{2}a^{-4}+12z^{2}+a^{5}z+2a^{3}z+2az+2za^{-1}+za^{-3}-a^{4}-3a^{2}-a^{-2}-2$
The A2 invariant	$-q^{16}+q^{12}-q^{10}+2q^{8}+2q^{2}-1+2q^{-2}-2q^{-4}+q^{-8}-q^{-10}+q^{-12}$
The G2 invariant	$q^{80}-2q^{78}+5q^{76}-8q^{74}+7q^{72}-4q^{70}-6q^{68}+19q^{66}-29q^{64}+33q^{62}-29q^{60}+6q^{58}+22q^{56}-50q^{54}+65q^{52}-56q^{50}+32q^{48}+6q^{46}-42q^{44}+61q^{42}-58q^{40}+33q^{38}+3q^{36}-32q^{34}+41q^{32}-25q^{30}-q^{28}+36q^{26}-53q^{24}+50q^{22}-24q^{20}-20q^{18}+64q^{16}-89q^{14}+88q^{12}-53q^{10}+8q^{8}+45q^{6}-81q^{4}+87q^{2}-67+26q^{-2}+16q^{-4}-47q^{-6}+48q^{-8}-25q^{-10}-5q^{-12}+31q^{-14}-41q^{-16}+24q^{-18}+q^{-20}-35q^{-22}+58q^{-24}-59q^{-26}+43q^{-28}-9q^{-30}-22q^{-32}+45q^{-34}-51q^{-36}+45q^{-38}-28q^{-40}+7q^{-42}+11q^{-44}-23q^{-46}+24q^{-48}-18q^{-50}+13q^{-52}-4q^{-54}-2q^{-56}+4q^{-58}-6q^{-60}+4q^{-62}-2q^{-64}+q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 9_27.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-2q^{7}+2q^{5}-q^{3}+q+2q^{-1}-2q^{-3}+2q^{-5}-2q^{-7}+q^{-9}$
2	$q^{32}-2q^{30}-2q^{28}+7q^{26}-2q^{24}-9q^{22}+11q^{20}+2q^{18}-15q^{16}+9q^{14}+7q^{12}-13q^{10}+3q^{8}+8q^{6}-4q^{4}-4q^{2}+5+9q^{-2}-11q^{-4}-3q^{-6}+15q^{-8}-10q^{-10}-7q^{-12}+13q^{-14}-4q^{-16}-5q^{-18}+6q^{-20}-q^{-22}-2q^{-24}+q^{-26}$
3	$-q^{63}+2q^{61}+2q^{59}-3q^{57}-7q^{55}+2q^{53}+16q^{51}+q^{49}-23q^{47}-11q^{45}+29q^{43}+26q^{41}-31q^{39}-42q^{37}+26q^{35}+55q^{33}-13q^{31}-65q^{29}-q^{27}+67q^{25}+14q^{23}-63q^{21}-25q^{19}+52q^{17}+34q^{15}-39q^{13}-36q^{11}+22q^{9}+39q^{7}-4q^{5}-36q^{3}-18q+34q^{-1}+42q^{-3}-27q^{-5}-55q^{-7}+13q^{-9}+68q^{-11}-2q^{-13}-69q^{-15}-12q^{-17}+62q^{-19}+20q^{-21}-48q^{-23}-23q^{-25}+34q^{-27}+21q^{-29}-21q^{-31}-16q^{-33}+11q^{-35}+11q^{-37}-7q^{-39}-5q^{-41}+3q^{-43}+3q^{-45}-q^{-47}-2q^{-49}+q^{-51}$
4	$q^{104}-2q^{102}-2q^{100}+3q^{98}+3q^{96}+7q^{94}-9q^{92}-16q^{90}-q^{88}+11q^{86}+41q^{84}-2q^{82}-47q^{80}-41q^{78}-7q^{76}+100q^{74}+62q^{72}-42q^{70}-117q^{68}-108q^{66}+117q^{64}+173q^{62}+62q^{60}-141q^{58}-259q^{56}+16q^{54}+223q^{52}+230q^{50}-46q^{48}-346q^{46}-148q^{44}+156q^{42}+333q^{40}+100q^{38}-305q^{36}-255q^{34}+30q^{32}+318q^{30}+195q^{28}-190q^{26}-261q^{24}-69q^{22}+225q^{20}+214q^{18}-58q^{16}-208q^{14}-135q^{12}+100q^{10}+195q^{8}+88q^{6}-124q^{4}-197q^{2}-61+154q^{-2}+254q^{-4}-2q^{-6}-232q^{-8}-236q^{-10}+52q^{-12}+355q^{-14}+151q^{-16}-171q^{-18}-343q^{-20}-94q^{-22}+321q^{-24}+243q^{-26}-34q^{-28}-299q^{-30}-185q^{-32}+176q^{-34}+204q^{-36}+72q^{-38}-160q^{-40}-163q^{-42}+51q^{-44}+96q^{-46}+80q^{-48}-49q^{-50}-85q^{-52}+7q^{-54}+23q^{-56}+41q^{-58}-9q^{-60}-29q^{-62}+4q^{-64}+12q^{-68}-2q^{-70}-8q^{-72}+3q^{-74}+3q^{-78}-q^{-80}-2q^{-82}+q^{-84}$
5	$-q^{155}+2q^{153}+2q^{151}-3q^{149}-3q^{147}-3q^{145}+9q^{141}+16q^{139}+q^{137}-20q^{135}-29q^{133}-19q^{131}+20q^{129}+61q^{127}+62q^{125}-11q^{123}-97q^{121}-121q^{119}-47q^{117}+108q^{115}+220q^{113}+161q^{111}-78q^{109}-309q^{107}-326q^{105}-59q^{103}+348q^{101}+541q^{99}+291q^{97}-284q^{95}-722q^{93}-609q^{91}+62q^{89}+811q^{87}+963q^{85}+295q^{83}-745q^{81}-1264q^{79}-732q^{77}+497q^{75}+1431q^{73}+1192q^{71}-123q^{69}-1433q^{67}-1558q^{65}-315q^{63}+1258q^{61}+1785q^{59}+742q^{57}-978q^{55}-1843q^{53}-1070q^{51}+644q^{49}+1750q^{47}+1273q^{45}-324q^{43}-1556q^{41}-1345q^{39}+50q^{37}+1315q^{35}+1314q^{33}+149q^{31}-1043q^{29}-1228q^{27}-316q^{25}+799q^{23}+1121q^{21}+450q^{19}-541q^{17}-1013q^{15}-623q^{13}+280q^{11}+929q^{9}+805q^{7}+27q^{5}-818q^{3}-1032q-389q^{-1}+672q^{-3}+1265q^{-5}+782q^{-7}-438q^{-9}-1410q^{-11}-1218q^{-13}+110q^{-15}+1477q^{-17}+1587q^{-19}+274q^{-21}-1350q^{-23}-1852q^{-25}-701q^{-27}+1098q^{-29}+1925q^{-31}+1056q^{-33}-717q^{-35}-1801q^{-37}-1283q^{-39}+301q^{-41}+1502q^{-43}+1336q^{-45}+71q^{-47}-1114q^{-49}-1217q^{-51}-315q^{-53}+702q^{-55}+984q^{-57}+430q^{-59}-366q^{-61}-710q^{-63}-417q^{-65}+137q^{-67}+449q^{-69}+335q^{-71}-3q^{-73}-256q^{-75}-239q^{-77}-36q^{-79}+127q^{-81}+138q^{-83}+47q^{-85}-51q^{-87}-81q^{-89}-32q^{-91}+23q^{-93}+34q^{-95}+17q^{-97}-5q^{-99}-13q^{-101}-10q^{-103}+3q^{-105}+8q^{-107}-q^{-109}-3q^{-111}+3q^{-119}-q^{-121}-2q^{-123}+q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{12}-q^{10}+2q^{8}+2q^{2}-1+2q^{-2}-2q^{-4}+q^{-8}-q^{-10}+q^{-12}$
1,1	$q^{44}-4q^{42}+12q^{40}-28q^{38}+52q^{36}-86q^{34}+130q^{32}-180q^{30}+222q^{28}-248q^{26}+258q^{24}-236q^{22}+176q^{20}-88q^{18}-24q^{16}+144q^{14}-267q^{12}+376q^{10}-450q^{8}+492q^{6}-483q^{4}+440q^{2}-352+244q^{-2}-120q^{-4}-4q^{-6}+102q^{-8}-176q^{-10}+222q^{-12}-238q^{-14}+228q^{-16}-196q^{-18}+162q^{-20}-126q^{-22}+88q^{-24}-58q^{-26}+36q^{-28}-20q^{-30}+10q^{-32}-4q^{-34}+q^{-36}$
2,0	$q^{42}-2q^{38}-2q^{36}+2q^{34}+4q^{32}-3q^{30}-3q^{28}+4q^{26}+4q^{24}-5q^{22}-5q^{20}+5q^{18}+2q^{16}-6q^{14}+6q^{10}-2q^{8}-q^{6}+6q^{4}+q^{2}-2+3q^{-2}+5q^{-4}-7q^{-6}-5q^{-8}+7q^{-10}+2q^{-12}-7q^{-14}+6q^{-18}-3q^{-22}+2q^{-26}-q^{-28}-q^{-30}+q^{-32}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-2q^{32}+q^{30}+3q^{28}-7q^{26}+3q^{24}+4q^{22}-12q^{20}+5q^{18}+8q^{16}-10q^{14}+5q^{12}+10q^{10}-5q^{8}-q^{6}+4q^{4}+q^{2}-4-4q^{-2}+9q^{-4}-5q^{-6}-8q^{-8}+13q^{-10}-2q^{-12}-8q^{-14}+10q^{-16}-6q^{-20}+4q^{-22}-2q^{-26}+q^{-28}$
1,0,0	$-q^{21}-q^{17}+q^{15}-q^{13}+3q^{11}+2q^{7}+q^{3}-q^{-1}+q^{-3}-2q^{-5}+q^{-7}-q^{-9}+2q^{-11}-q^{-13}+q^{-15}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{44}-q^{40}+q^{38}+q^{36}-3q^{34}-3q^{32}+2q^{30}-8q^{26}-2q^{24}+9q^{22}+q^{20}-5q^{18}+9q^{16}+11q^{14}-3q^{12}-2q^{10}+7q^{8}-10q^{4}+2q^{2}+4-9q^{-2}-4q^{-4}+10q^{-6}-2q^{-8}-7q^{-10}+7q^{-12}+8q^{-14}-4q^{-16}-4q^{-18}+6q^{-20}+2q^{-22}-5q^{-24}-q^{-26}+3q^{-28}-q^{-30}-q^{-32}+q^{-34}$
1,0,0,0	$-q^{26}-q^{22}-q^{20}+q^{18}-q^{16}+3q^{14}+q^{12}+2q^{10}+2q^{8}+q^{4}-q^{2}-2q^{-2}+q^{-4}-2q^{-6}+q^{-8}+2q^{-14}-q^{-16}+q^{-18}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+2q^{32}-5q^{30}+7q^{28}-9q^{26}+11q^{24}-12q^{22}+12q^{20}-9q^{18}+6q^{16}-5q^{12}+12q^{10}-17q^{8}+21q^{6}-22q^{4}+23q^{2}-20+16q^{-2}-9q^{-4}+3q^{-6}+2q^{-8}-7q^{-10}+10q^{-12}-12q^{-14}+12q^{-16}-10q^{-18}+8q^{-20}-6q^{-22}+4q^{-24}-2q^{-26}+q^{-28}$
1,0	$q^{56}-2q^{52}-2q^{50}+3q^{48}+5q^{46}-2q^{44}-8q^{42}-3q^{40}+9q^{38}+8q^{36}-7q^{34}-13q^{32}+13q^{28}+7q^{26}-9q^{24}-9q^{22}+5q^{20}+11q^{18}-8q^{14}-q^{12}+8q^{10}+3q^{8}-7q^{6}-5q^{4}+7q^{2}+7-5q^{-2}-9q^{-4}+4q^{-6}+10q^{-8}-q^{-10}-12q^{-12}-4q^{-14}+11q^{-16}+9q^{-18}-6q^{-20}-12q^{-22}+11q^{-26}+6q^{-28}-5q^{-30}-7q^{-32}+5q^{-36}+2q^{-38}-2q^{-40}-2q^{-42}+q^{-46}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{46}-2q^{44}+3q^{42}-4q^{40}+6q^{38}-8q^{36}+7q^{34}-10q^{32}+9q^{30}-11q^{28}+6q^{26}-6q^{24}+6q^{22}-q^{18}+9q^{16}-4q^{14}+15q^{12}-14q^{10}+16q^{8}-16q^{6}+18q^{4}-19q^{2}+12-15q^{-2}+10q^{-4}-6q^{-6}+q^{-8}-q^{-10}-2q^{-12}+9q^{-14}-6q^{-16}+8q^{-18}-9q^{-20}+11q^{-22}-7q^{-24}+6q^{-26}-7q^{-28}+5q^{-30}-3q^{-32}+2q^{-34}-2q^{-36}+q^{-38}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-2q^{78}+5q^{76}-8q^{74}+7q^{72}-4q^{70}-6q^{68}+19q^{66}-29q^{64}+33q^{62}-29q^{60}+6q^{58}+22q^{56}-50q^{54}+65q^{52}-56q^{50}+32q^{48}+6q^{46}-42q^{44}+61q^{42}-58q^{40}+33q^{38}+3q^{36}-32q^{34}+41q^{32}-25q^{30}-q^{28}+36q^{26}-53q^{24}+50q^{22}-24q^{20}-20q^{18}+64q^{16}-89q^{14}+88q^{12}-53q^{10}+8q^{8}+45q^{6}-81q^{4}+87q^{2}-67+26q^{-2}+16q^{-4}-47q^{-6}+48q^{-8}-25q^{-10}-5q^{-12}+31q^{-14}-41q^{-16}+24q^{-18}+q^{-20}-35q^{-22}+58q^{-24}-59q^{-26}+43q^{-28}-9q^{-30}-22q^{-32}+45q^{-34}-51q^{-36}+45q^{-38}-28q^{-40}+7q^{-42}+11q^{-44}-23q^{-46}+24q^{-48}-18q^{-50}+13q^{-52}-4q^{-54}-2q^{-56}+4q^{-58}-6q^{-60}+4q^{-62}-2q^{-64}+q^{-66}$

.

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 27"];

In[4]:=

Alexander[K][t]

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

Out[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 -t^3+5 t^2-11 t+15-11 t^{-1} +5 t^{-2} - t^{-3} }

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 49, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+6az^{7}+3z^{7}a^{-1}+3a^{4}z^{6}+6a^{2}z^{6}+4z^{6}a^{-2}+7z^{6}+a^{5}z^{5}-4a^{3}z^{5}-8az^{5}+3z^{5}a^{-3}-7a^{4}z^{4}-17a^{2}z^{4}-5z^{4}a^{-2}+z^{4}a^{-4}-16z^{4}-2a^{5}z^{3}-2a^{3}z^{3}-4z^{3}a^{-1}-4z^{3}a^{-3}+4a^{4}z^{2}+12a^{2}z^{2}+3z^{2}a^{-2}-z^{2}a^{-4}+12z^{2}+a^{5}z+2a^{3}z+2az+2za^{-1}+za^{-3}-a^{4}-3a^{2}-a^{-2}-2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n4, K11n21, K11n172,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n83,}

Computer Talk

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 27"];

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]=

{ $-t^{3}+5t^{2}-11t+15-11t^{-1}+5t^{-2}-t^{-3}$, $q^{4}-3q^{3}+5q^{2}-7q+9-8q^{-1}+7q^{-2}-5q^{-3}+3q^{-4}-q^{-5}$ }

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]=

{K11n4, K11n21, K11n172,}

In[6]:=

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

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

Out[6]=

{K11n83,}

Vassiliev invariants

V2 and V3:

(0, -1)

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 $0$ $-8$ $0$ 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 16} 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 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 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 \frac{16}{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 \frac{64}{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 -8} $0$ 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 32} 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 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 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 56} 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 56} 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 -\frac{152}{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 \frac{40}{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 -24}

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 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 t^rq^j} are shown, along with their alternating sums 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 \chi} (fixed 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} , 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=} 0 is the signature of 9 27. 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 9                   1 1 7                 2   -2 5               3 1   2 3             4 2     -2 1           5 3       2 -1         4 5         1 -3       3 4           -1 -5     2 4             2 -7   1 3               -2 -9   2                 2 -11 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}} $i=-1$ $i=1$ 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=-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}} 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=-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}^{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}} 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^{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=-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}^{4}\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}^{4}\oplus{\mathbb Z}_2^{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}^{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 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}^{5}\oplus{\mathbb Z}_2^{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}^{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 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}^{3}\oplus{\mathbb Z}_2^{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}^{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 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}\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=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}\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=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}_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^{12}-3 q^{11}+q^{10}+8 q^9-14 q^8+2 q^7+25 q^6-34 q^5-q^4+50 q^3-52 q^2-9 q+70-56 q^{-1} -18 q^{-2} +70 q^{-3} -44 q^{-4} -23 q^{-5} +54 q^{-6} -24 q^{-7} -21 q^{-8} +30 q^{-9} -7 q^{-10} -12 q^{-11} +10 q^{-12} -3 q^{-14} + q^{-15} }
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^{24}-3 q^{23}+q^{22}+4 q^{21}+q^{20}-11 q^{19}-q^{18}+22 q^{17}+q^{16}-38 q^{15}-6 q^{14}+64 q^{13}+14 q^{12}-95 q^{11}-31 q^{10}+132 q^9+56 q^8-169 q^7-88 q^6+199 q^5+126 q^4-224 q^3-156 q^2+227 q+195-232 q^{-1} -208 q^{-2} +209 q^{-3} +227 q^{-4} -189 q^{-5} -225 q^{-6} +151 q^{-7} +224 q^{-8} -116 q^{-9} -207 q^{-10} +74 q^{-11} +186 q^{-12} -39 q^{-13} -154 q^{-14} +6 q^{-15} +122 q^{-16} +13 q^{-17} -86 q^{-18} -23 q^{-19} +54 q^{-20} +24 q^{-21} -29 q^{-22} -20 q^{-23} +14 q^{-24} +12 q^{-25} -5 q^{-26} -5 q^{-27} +3 q^{-29} - q^{-30} }
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^{40}-3 q^{39}+q^{38}+4 q^{37}-3 q^{36}+4 q^{35}-14 q^{34}+7 q^{33}+18 q^{32}-15 q^{31}+8 q^{30}-47 q^{29}+27 q^{28}+68 q^{27}-33 q^{26}-8 q^{25}-139 q^{24}+63 q^{23}+197 q^{22}-17 q^{21}-53 q^{20}-353 q^{19}+66 q^{18}+429 q^{17}+115 q^{16}-81 q^{15}-714 q^{14}-48 q^{13}+694 q^{12}+392 q^{11}-3 q^{10}-1129 q^9-297 q^8+866 q^7+714 q^6+201 q^5-1432 q^4-585 q^3+870 q^2+944 q+457-1532 q^{-1} -800 q^{-2} +734 q^{-3} +1017 q^{-4} +669 q^{-5} -1425 q^{-6} -895 q^{-7} +499 q^{-8} +944 q^{-9} +819 q^{-10} -1153 q^{-11} -884 q^{-12} +205 q^{-13} +752 q^{-14} +890 q^{-15} -768 q^{-16} -761 q^{-17} -83 q^{-18} +467 q^{-19} +840 q^{-20} -363 q^{-21} -528 q^{-22} -260 q^{-23} +163 q^{-24} +642 q^{-25} -63 q^{-26} -252 q^{-27} -267 q^{-28} -44 q^{-29} +367 q^{-30} +55 q^{-31} -49 q^{-32} -156 q^{-33} -100 q^{-34} +142 q^{-35} +46 q^{-36} +26 q^{-37} -52 q^{-38} -62 q^{-39} +35 q^{-40} +12 q^{-41} +20 q^{-42} -7 q^{-43} -19 q^{-44} +5 q^{-45} +5 q^{-47} -3 q^{-49} + q^{-50} }
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^{60}-3 q^{59}+q^{58}+4 q^{57}-3 q^{56}+q^{54}-6 q^{53}+3 q^{52}+13 q^{51}-8 q^{50}-13 q^{49}-2 q^{48}+2 q^{47}+25 q^{46}+30 q^{45}-19 q^{44}-68 q^{43}-51 q^{42}+32 q^{41}+123 q^{40}+121 q^{39}-30 q^{38}-231 q^{37}-254 q^{36}+15 q^{35}+376 q^{34}+459 q^{33}+84 q^{32}-543 q^{31}-808 q^{30}-278 q^{29}+720 q^{28}+1255 q^{27}+638 q^{26}-825 q^{25}-1825 q^{24}-1180 q^{23}+823 q^{22}+2440 q^{21}+1903 q^{20}-659 q^{19}-3026 q^{18}-2764 q^{17}+305 q^{16}+3524 q^{15}+3676 q^{14}+210 q^{13}-3853 q^{12}-4563 q^{11}-846 q^{10}+4026 q^9+5300 q^8+1523 q^7-3963 q^6-5930 q^5-2174 q^4+3834 q^3+6272 q^2+2743 q-3480-6523 q^{-1} -3235 q^{-2} +3191 q^{-3} +6486 q^{-4} +3588 q^{-5} -2702 q^{-6} -6399 q^{-7} -3884 q^{-8} +2288 q^{-9} +6096 q^{-10} +4060 q^{-11} -1711 q^{-12} -5728 q^{-13} -4206 q^{-14} +1173 q^{-15} +5184 q^{-16} +4245 q^{-17} -519 q^{-18} -4563 q^{-19} -4205 q^{-20} -92 q^{-21} +3789 q^{-22} +4034 q^{-23} +713 q^{-24} -2966 q^{-25} -3728 q^{-26} -1198 q^{-27} +2075 q^{-28} +3261 q^{-29} +1578 q^{-30} -1246 q^{-31} -2685 q^{-32} -1725 q^{-33} +502 q^{-34} +2018 q^{-35} +1703 q^{-36} +64 q^{-37} -1370 q^{-38} -1486 q^{-39} -432 q^{-40} +789 q^{-41} +1171 q^{-42} +583 q^{-43} -330 q^{-44} -818 q^{-45} -584 q^{-46} +40 q^{-47} +500 q^{-48} +470 q^{-49} +108 q^{-50} -243 q^{-51} -334 q^{-52} -153 q^{-53} +93 q^{-54} +203 q^{-55} +125 q^{-56} -12 q^{-57} -95 q^{-58} -94 q^{-59} -19 q^{-60} +48 q^{-61} +51 q^{-62} +12 q^{-63} -9 q^{-64} -21 q^{-65} -20 q^{-66} +7 q^{-67} +12 q^{-68} +2 q^{-69} -5 q^{-72} +3 q^{-74} - q^{-75} }
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^{84}-3 q^{83}+q^{82}+4 q^{81}-3 q^{80}-3 q^{78}+9 q^{77}-10 q^{76}-2 q^{75}+20 q^{74}-18 q^{73}-7 q^{72}-7 q^{71}+40 q^{70}-12 q^{69}-5 q^{68}+48 q^{67}-73 q^{66}-57 q^{65}-29 q^{64}+139 q^{63}+38 q^{62}+43 q^{61}+114 q^{60}-242 q^{59}-269 q^{58}-176 q^{57}+330 q^{56}+284 q^{55}+350 q^{54}+381 q^{53}-586 q^{52}-932 q^{51}-826 q^{50}+414 q^{49}+873 q^{48}+1420 q^{47}+1430 q^{46}-839 q^{45}-2346 q^{44}-2782 q^{43}-428 q^{42}+1474 q^{41}+3757 q^{40}+4381 q^{39}+103 q^{38}-4115 q^{37}-6737 q^{36}-3643 q^{35}+708 q^{34}+6888 q^{33}+10003 q^{32}+3935 q^{31}-4578 q^{30}-11996 q^{29}-9990 q^{28}-3275 q^{27}+8867 q^{26}+17250 q^{25}+11236 q^{24}-1796 q^{23}-16228 q^{22}-18033 q^{21}-10774 q^{20}+7676 q^{19}+23398 q^{18}+20147 q^{17}+4318 q^{16}-17280 q^{15}-24815 q^{14}-19617 q^{13}+3325 q^{12}+26221 q^{11}+27600 q^{10}+11576 q^9-15114 q^8-28245 q^7-26869 q^6-2200 q^5+25655 q^4+31734 q^3+17494 q^2-11382 q-28381-31040 q^{-1} -6955 q^{-2} +23051 q^{-3} +32709 q^{-4} +21116 q^{-5} -7491 q^{-6} -26411 q^{-7} -32451 q^{-8} -10430 q^{-9} +19418 q^{-10} +31516 q^{-11} +22950 q^{-12} -3607 q^{-13} -23043 q^{-14} -31967 q^{-15} -13222 q^{-16} +14740 q^{-17} +28621 q^{-18} +23696 q^{-19} +801 q^{-20} -18128 q^{-21} -29822 q^{-22} -15749 q^{-23} +8657 q^{-24} +23742 q^{-25} +23215 q^{-26} +5727 q^{-27} -11405 q^{-28} -25504 q^{-29} -17374 q^{-30} +1624 q^{-31} +16648 q^{-32} +20570 q^{-33} +9923 q^{-34} -3615 q^{-35} -18663 q^{-36} -16662 q^{-37} -4583 q^{-38} +8272 q^{-39} +15172 q^{-40} +11451 q^{-41} +3148 q^{-42} -10332 q^{-43} -12819 q^{-44} -7710 q^{-45} +946 q^{-46} +8115 q^{-47} +9410 q^{-48} +6546 q^{-49} -3018 q^{-50} -7051 q^{-51} -6939 q^{-52} -2991 q^{-53} +1982 q^{-54} +5182 q^{-55} +6032 q^{-56} +1013 q^{-57} -2007 q^{-58} -3888 q^{-59} -3236 q^{-60} -1163 q^{-61} +1418 q^{-62} +3455 q^{-63} +1696 q^{-64} +487 q^{-65} -1130 q^{-66} -1693 q^{-67} -1520 q^{-68} -326 q^{-69} +1202 q^{-70} +870 q^{-71} +799 q^{-72} +80 q^{-73} -392 q^{-74} -797 q^{-75} -496 q^{-76} +203 q^{-77} +167 q^{-78} +380 q^{-79} +208 q^{-80} +65 q^{-81} -238 q^{-82} -227 q^{-83} -37 q^{-85} +89 q^{-86} +80 q^{-87} +79 q^{-88} -44 q^{-89} -58 q^{-90} - q^{-91} -29 q^{-92} +9 q^{-93} +12 q^{-94} +29 q^{-95} -7 q^{-96} -12 q^{-97} +5 q^{-98} -7 q^{-99} +5 q^{-102} -3 q^{-104} + q^{-105} }
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^{112}-3 q^{111}+q^{110}+4 q^{109}-3 q^{108}-3 q^{106}+5 q^{105}+5 q^{104}-15 q^{103}+5 q^{102}+10 q^{101}-12 q^{100}-q^{99}-8 q^{98}+23 q^{97}+34 q^{96}-39 q^{95}-2 q^{94}-2 q^{93}-52 q^{92}-4 q^{91}-17 q^{90}+92 q^{89}+152 q^{88}-35 q^{87}-30 q^{86}-116 q^{85}-239 q^{84}-75 q^{83}-27 q^{82}+301 q^{81}+560 q^{80}+197 q^{79}+q^{78}-497 q^{77}-943 q^{76}-585 q^{75}-243 q^{74}+792 q^{73}+1766 q^{72}+1377 q^{71}+671 q^{70}-1120 q^{69}-2892 q^{68}-2758 q^{67}-1804 q^{66}+1140 q^{65}+4477 q^{64}+5209 q^{63}+4045 q^{62}-629 q^{61}-6374 q^{60}-8732 q^{59}-7897 q^{58}-1353 q^{57}+7965 q^{56}+13612 q^{55}+14172 q^{54}+5467 q^{53}-8715 q^{52}-19328 q^{51}-22895 q^{50}-12804 q^{49}+7179 q^{48}+25168 q^{47}+34266 q^{46}+24009 q^{45}-2397 q^{44}-29861 q^{43}-47360 q^{42}-39195 q^{41}-6809 q^{40}+31862 q^{39}+60924 q^{38}+57869 q^{37}+20902 q^{36}-29940 q^{35}-73286 q^{34}-78632 q^{33}-39407 q^{32}+23183 q^{31}+82637 q^{30}+99660 q^{29}+61251 q^{28}-11558 q^{27}-87846 q^{26}-118947 q^{25}-84539 q^{24}-4031 q^{23}+88237 q^{22}+134783 q^{21}+107278 q^{20}+22113 q^{19}-84114 q^{18}-146119 q^{17}-127548 q^{16}-41030 q^{15}+76399 q^{14}+152853 q^{13}+144158 q^{12}+58779 q^{11}-66536 q^{10}-154976 q^9-156376 q^8-74618 q^7+55602 q^6+154039 q^5+164612 q^4+87139 q^3-45095 q^2-150246 q-168875-97203 q^{-1} +35059 q^{-2} +145379 q^{-3} +170719 q^{-4} +104201 q^{-5} -26401 q^{-6} -139098 q^{-7} -170054 q^{-8} -109677 q^{-9} +17979 q^{-10} +132477 q^{-11} +168325 q^{-12} +113539 q^{-13} -10181 q^{-14} -124683 q^{-15} -164926 q^{-16} -116950 q^{-17} +1602 q^{-18} +115820 q^{-19} +160641 q^{-20} +119763 q^{-21} +7502 q^{-22} -105075 q^{-23} -154443 q^{-24} -122222 q^{-25} -18015 q^{-26} +92245 q^{-27} +146427 q^{-28} +123742 q^{-29} +29301 q^{-30} -76916 q^{-31} -135600 q^{-32} -123865 q^{-33} -41201 q^{-34} +59434 q^{-35} +121883 q^{-36} +121506 q^{-37} +52501 q^{-38} -40112 q^{-39} -104949 q^{-40} -116130 q^{-41} -62182 q^{-42} +20326 q^{-43} +85363 q^{-44} +106842 q^{-45} +68749 q^{-46} -1236 q^{-47} -63810 q^{-48} -93870 q^{-49} -71343 q^{-50} -15247 q^{-51} +42006 q^{-52} +77551 q^{-53} +69026 q^{-54} +27826 q^{-55} -21387 q^{-56} -59219 q^{-57} -62302 q^{-58} -35356 q^{-59} +4051 q^{-60} +40538 q^{-61} +51752 q^{-62} +37531 q^{-63} +8960 q^{-64} -23359 q^{-65} -39145 q^{-66} -34988 q^{-67} -16789 q^{-68} +9320 q^{-69} +26200 q^{-70} +29011 q^{-71} +19680 q^{-72} +628 q^{-73} -14663 q^{-74} -21232 q^{-75} -18685 q^{-76} -6407 q^{-77} +5811 q^{-78} +13504 q^{-79} +15164 q^{-80} +8382 q^{-81} +7 q^{-82} -6901 q^{-83} -10683 q^{-84} -7935 q^{-85} -2965 q^{-86} +2343 q^{-87} +6480 q^{-88} +6056 q^{-89} +3688 q^{-90} +353 q^{-91} -3173 q^{-92} -3898 q^{-93} -3265 q^{-94} -1477 q^{-95} +1172 q^{-96} +2100 q^{-97} +2246 q^{-98} +1532 q^{-99} -61 q^{-100} -810 q^{-101} -1330 q^{-102} -1265 q^{-103} -306 q^{-104} +231 q^{-105} +640 q^{-106} +754 q^{-107} +306 q^{-108} +120 q^{-109} -213 q^{-110} -468 q^{-111} -246 q^{-112} -120 q^{-113} +78 q^{-114} +196 q^{-115} +93 q^{-116} +117 q^{-117} +43 q^{-118} -97 q^{-119} -75 q^{-120} -65 q^{-121} -5 q^{-122} +43 q^{-123} -3 q^{-124} +29 q^{-125} +29 q^{-126} -9 q^{-127} -12 q^{-128} -20 q^{-129} -2 q^{-130} +12 q^{-131} -5 q^{-132} +7 q^{-134} -5 q^{-137} +3 q^{-139} - q^{-140} }

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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