9 27

9_26

9_28

Visit 9 27's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 27's page at Knotilus! Visit 9 27's page at the original Knot Atlas! 9 27 Quick Notes

9 27 Further Notes and Views

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

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 2 Super bridge index $\{4,6\}$ 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 $0$ Topological 4 genus $0$ Concordance genus $0$ Rasmussen s-Invariant 0

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

Polynomial invariants

Alexander polynomial	$-t^{3}+5t^{2}-11t+15-11t^{-1}+5t^{-2}-t^{-3}$
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]=

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

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, ...}

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 $0$ $-8$ $0$ $16$ $8$ $0$ ${\frac {16}{3}}$ ${\frac {64}{3}}$ $-8$ $0$ $32$ $0$ $0$ $56$ $56$ $-{\frac {152}{3}}$ ${\frac {40}{3}}$ $-24$

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 $t^{r}q^{j}$ 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=$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)

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

The Coloured Jones Polynomials

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

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

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, 27]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 1, 12, 18], X[5, 13, 6, 12], X[13, 17, 14, 16], X[7, 14, 8, 15], X[15, 6, 16, 7], X[17, 9, 18, 8], X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[9, 27]]
Out[3]=	GaussCode[-1, 9, -2, 1, -4, 7, -6, 8, -9, 2, -3, 4, -5, 6, -7, 5, -8, 3]
In[4]:=	DTCode[Knot[9, 27]]
Out[4]=	DTCode[4, 10, 12, 14, 2, 18, 16, 6, 8]
In[5]:=	br = BR[Knot[9, 27]]
Out[5]=	BR[4, {-1, -1, 2, -1, 2, 2, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 27]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 27]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 27]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 2, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 27]][t]
Out[10]=	-3 5 11 2 3 15 - t + -- - -- - 11 t + 5 t - t 2 t t
In[11]:=	Conway[Knot[9, 27]][z]
Out[11]=	4 6 1 - z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 27], Knot[11, NonAlternating, 4], Knot[11, NonAlternating, 21], Knot[11, NonAlternating, 172]}
In[13]:=	{KnotDet[Knot[9, 27]], KnotSignature[Knot[9, 27]]}
Out[13]=	{49, 0}
In[14]:=	Jones[Knot[9, 27]][q]
Out[14]=	-5 3 5 7 8 2 3 4 9 - q + -- - -- + -- - - - 7 q + 5 q - 3 q + q 4 3 2 q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 27], Knot[11, NonAlternating, 83]}
In[16]:=	A2Invariant[Knot[9, 27]][q]
Out[16]=	-16 -12 -10 2 2 2 4 8 10 12 -1 - q + q - q + -- + -- + 2 q - 2 q + q - q + q 8 2 q q
In[17]:=	HOMFLYPT[Knot[9, 27]][a, z]
Out[17]=	2 4 -2 2 4 2 2 z 2 2 4 2 4 z -2 + a + 3 a - a - 6 z + ---- + 5 a z - a z - 4 z + -- + 2 2 a a 2 4 6 2 a z - z
In[18]:=	Kauffman[Knot[9, 27]][a, z]
Out[18]=	2 -2 2 4 z 2 z 3 5 2 z -2 - a - 3 a - a + -- + --- + 2 a z + 2 a z + a z + 12 z - -- + 3 a 4 a a 2 3 3 3 z 2 2 4 2 4 z 4 z 3 3 5 3 4 ---- + 12 a z + 4 a z - ---- - ---- - 2 a z - 2 a z - 16 z + 2 3 a a a 4 4 5 z 5 z 2 4 4 4 3 z 5 3 5 5 5 -- - ---- - 17 a z - 7 a z + ---- - 8 a z - 4 a z + a z + 4 2 3 a a a 6 7 6 4 z 2 6 4 6 3 z 7 3 7 8 2 8 7 z + ---- + 6 a z + 3 a z + ---- + 6 a z + 3 a z + z + a z 2 a a
In[19]:=	{Vassiliev[2][Knot[9, 27]], Vassiliev[3][Knot[9, 27]]}
Out[19]=	{0, -1}
In[20]:=	Kh[Knot[9, 27]][q, t]
Out[20]=	5 1 2 1 3 2 4 3 - + 5 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 4 4 3 3 2 5 2 5 3 7 3 ---- + --- + 3 q t + 4 q t + 2 q t + 3 q t + q t + 2 q t + 3 q t q t 9 4 q t
In[21]:=	ColouredJones[Knot[9, 27], 2][q]
Out[21]=	-15 3 10 12 7 30 21 24 54 23 44 70 70 + q - --- + --- - --- - --- + -- - -- - -- + -- - -- - -- + -- - 14 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q 18 56 2 3 4 5 6 7 8 -- - -- - 9 q - 52 q + 50 q - q - 34 q + 25 q + 2 q - 14 q + 2 q q 9 10 11 12 8 q + q - 3 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

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