10 127

10_126

10_128

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10 127 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3849 X14,6,15,5 X15,20,16,1 X9,16,10,17 X11,18,12,19 X17,10,18,11 X19,12,20,13 X6,14,7,13 X7283
Gauss code	-1, 10, -2, 1, 3, -9, -10, 2, -5, 7, -6, 8, 9, -3, -4, 5, -7, 6, -8, 4
Dowker-Thistlethwaite code	4 8 -14 2 16 18 -6 20 10 12
Conway Notation	[41,21,2-]

Minimum Braid Representative: Length is 10, width is 3. Braid index is 3.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-13][3] Hyperbolic Volume 8.89682 A-Polynomial See Data:10 127/A-polynomial

[edit Notes for 10 127's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 127's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+4t^{2}-6t+7-6t^{-1}+4t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 29, -4 }
Jones polynomial	$2q^{-2}-2q^{-3}+4q^{-4}-5q^{-5}+5q^{-6}-5q^{-7}+3q^{-8}-2q^{-9}+q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{8}+3z^{2}a^{8}+2a^{8}-z^{6}a^{6}-5z^{4}a^{6}-9z^{2}a^{6}-6a^{6}+2z^{4}a^{4}+7z^{2}a^{4}+5a^{4}$
Kauffman polynomial (db, data sources)	$z^{4}a^{12}-2z^{2}a^{12}+2z^{5}a^{11}-4z^{3}a^{11}+za^{11}+2z^{6}a^{10}-3z^{4}a^{10}+z^{2}a^{10}+2z^{7}a^{9}-5z^{5}a^{9}+7z^{3}a^{9}-2za^{9}+z^{8}a^{8}-2z^{6}a^{8}+4z^{4}a^{8}-2z^{2}a^{8}+2a^{8}+3z^{7}a^{7}-10z^{5}a^{7}+16z^{3}a^{7}-8za^{7}+z^{8}a^{6}-4z^{6}a^{6}+11z^{4}a^{6}-14z^{2}a^{6}+6a^{6}+z^{7}a^{5}-3z^{5}a^{5}+5z^{3}a^{5}-5za^{5}+3z^{4}a^{4}-9z^{2}a^{4}+5a^{4}$
The A2 invariant	$q^{30}+q^{26}-2q^{22}-q^{20}-3q^{18}+q^{12}+3q^{10}+q^{8}+2q^{6}$
The G2 invariant	$q^{162}-q^{160}+2q^{158}-3q^{156}+q^{154}-3q^{150}+5q^{148}-6q^{146}+6q^{144}-5q^{142}+q^{140}+3q^{138}-8q^{136}+12q^{134}-11q^{132}+9q^{130}-3q^{128}-5q^{126}+13q^{124}-13q^{122}+12q^{120}-3q^{118}-5q^{116}+11q^{114}-8q^{112}+2q^{110}+9q^{108}-14q^{106}+16q^{104}-8q^{102}-5q^{100}+15q^{98}-22q^{96}+21q^{94}-16q^{92}+2q^{90}+7q^{88}-17q^{86}+17q^{84}-17q^{82}+5q^{80}-11q^{76}+9q^{74}-9q^{72}+7q^{68}-13q^{66}+10q^{64}-4q^{62}-7q^{60}+17q^{58}-17q^{56}+14q^{54}-3q^{52}-4q^{50}+14q^{48}-12q^{46}+13q^{44}-4q^{42}+q^{40}+6q^{38}-5q^{36}+5q^{34}+2q^{30}+q^{28}$

Further Quantum Invariants

Further quantum knot invariants for 10_127.

A1 Invariants.

Weight	Invariant
1	$q^{21}-q^{19}+q^{17}-2q^{15}-q^{9}+2q^{7}+2q^{3}$
2	$q^{58}-q^{56}-q^{54}+2q^{52}-2q^{50}-2q^{48}+5q^{46}-q^{44}-5q^{42}+6q^{40}+q^{38}-4q^{36}+2q^{34}+2q^{32}-q^{30}-4q^{28}+2q^{26}+2q^{24}-6q^{22}+q^{20}+4q^{18}-5q^{16}+5q^{12}-q^{10}-q^{8}+3q^{6}+q^{4}$
3	$q^{111}-q^{109}-q^{107}+q^{103}-q^{99}+2q^{97}+2q^{95}-3q^{93}-5q^{91}+4q^{89}+9q^{87}-2q^{85}-15q^{83}-q^{81}+17q^{79}+7q^{77}-19q^{75}-10q^{73}+14q^{71}+12q^{69}-8q^{67}-12q^{65}+4q^{63}+9q^{61}+3q^{59}-7q^{57}-7q^{55}+5q^{53}+12q^{51}-4q^{49}-13q^{47}+q^{45}+17q^{43}-q^{41}-17q^{39}-5q^{37}+17q^{35}+7q^{33}-14q^{31}-12q^{29}+7q^{27}+13q^{25}-3q^{23}-10q^{21}-2q^{19}+8q^{17}+5q^{15}-2q^{13}-4q^{11}+2q^{9}+2q^{7}+2q^{5}$
5	$q^{265}-q^{263}-q^{261}-q^{257}+q^{255}+3q^{253}+2q^{251}-q^{249}-q^{247}-4q^{245}-5q^{243}+q^{241}+6q^{239}+6q^{237}+4q^{235}-2q^{233}-12q^{231}-14q^{229}-2q^{227}+14q^{225}+24q^{223}+18q^{221}-6q^{219}-38q^{217}-44q^{215}-10q^{213}+40q^{211}+75q^{209}+54q^{207}-26q^{205}-107q^{203}-112q^{201}-19q^{199}+119q^{197}+179q^{195}+89q^{193}-96q^{191}-231q^{189}-182q^{187}+38q^{185}+250q^{183}+257q^{181}+46q^{179}-214q^{177}-308q^{175}-128q^{173}+156q^{171}+297q^{169}+187q^{167}-71q^{165}-248q^{163}-207q^{161}+178q^{157}+187q^{155}+48q^{153}-99q^{151}-148q^{149}-78q^{147}+42q^{145}+101q^{143}+76q^{141}+4q^{139}-67q^{137}-79q^{135}-29q^{133}+49q^{131}+83q^{129}+46q^{127}-44q^{125}-104q^{123}-61q^{121}+51q^{119}+132q^{117}+93q^{115}-58q^{113}-171q^{111}-122q^{109}+50q^{107}+201q^{105}+177q^{103}-29q^{101}-224q^{99}-217q^{97}-17q^{95}+208q^{93}+262q^{91}+79q^{89}-174q^{87}-270q^{85}-143q^{83}+98q^{81}+250q^{79}+196q^{77}-14q^{75}-194q^{73}-214q^{71}-69q^{69}+109q^{67}+190q^{65}+129q^{63}-21q^{61}-134q^{59}-139q^{57}-55q^{55}+59q^{53}+114q^{51}+88q^{49}+9q^{47}-65q^{45}-88q^{43}-47q^{41}+15q^{39}+52q^{37}+58q^{35}+21q^{33}-23q^{31}-38q^{29}-26q^{27}-5q^{25}+17q^{23}+25q^{21}+11q^{19}-2q^{17}-6q^{15}-10q^{13}-4q^{11}+2q^{9}+4q^{7}+2q^{5}+2q^{3}$
6	$q^{366}-q^{364}-q^{362}-q^{358}+q^{356}+q^{354}+5q^{352}-3q^{348}-q^{346}-5q^{344}-4q^{342}-2q^{340}+10q^{338}+6q^{336}+2q^{334}+4q^{332}-6q^{330}-13q^{328}-15q^{326}+7q^{324}+11q^{322}+13q^{320}+22q^{318}+4q^{316}-24q^{314}-45q^{312}-17q^{310}+2q^{308}+30q^{306}+71q^{304}+56q^{302}-7q^{300}-90q^{298}-104q^{296}-90q^{294}-3q^{292}+146q^{290}+216q^{288}+154q^{286}-40q^{284}-218q^{282}-351q^{280}-265q^{278}+60q^{276}+396q^{274}+542q^{272}+336q^{270}-83q^{268}-596q^{266}-783q^{264}-445q^{262}+219q^{260}+834q^{258}+957q^{256}+530q^{254}-374q^{252}-1085q^{250}-1118q^{248}-441q^{246}+556q^{244}+1206q^{242}+1176q^{240}+301q^{238}-737q^{236}-1274q^{234}-1003q^{232}-114q^{230}+778q^{228}+1199q^{226}+772q^{224}-70q^{222}-778q^{220}-934q^{218}-517q^{216}+144q^{214}+678q^{212}+677q^{210}+301q^{208}-193q^{206}-479q^{204}-449q^{202}-176q^{200}+178q^{198}+338q^{196}+295q^{194}+79q^{192}-135q^{190}-255q^{188}-205q^{186}-29q^{184}+159q^{182}+245q^{180}+135q^{178}-60q^{176}-248q^{174}-246q^{172}-71q^{170}+219q^{168}+383q^{166}+233q^{164}-103q^{162}-440q^{160}-466q^{158}-173q^{156}+355q^{154}+683q^{152}+500q^{150}-49q^{148}-645q^{146}-812q^{144}-461q^{142}+322q^{140}+926q^{138}+888q^{136}+259q^{134}-593q^{132}-1055q^{130}-888q^{128}-48q^{126}+821q^{124}+1135q^{122}+742q^{120}-130q^{118}-884q^{116}-1141q^{114}-614q^{112}+250q^{110}+909q^{108}+1006q^{106}+507q^{104}-240q^{102}-863q^{100}-877q^{98}-423q^{96}+222q^{94}+688q^{92}+745q^{90}+406q^{88}-170q^{86}-528q^{84}-598q^{82}-356q^{80}+34q^{78}+376q^{76}+487q^{74}+305q^{72}+48q^{70}-220q^{68}-345q^{66}-303q^{64}-96q^{62}+121q^{60}+223q^{58}+241q^{56}+128q^{54}-24q^{52}-158q^{50}-175q^{48}-111q^{46}-24q^{44}+76q^{42}+118q^{40}+100q^{38}+26q^{36}-28q^{34}-59q^{32}-64q^{30}-33q^{28}+5q^{26}+34q^{24}+31q^{22}+21q^{20}+8q^{18}-8q^{16}-15q^{14}-11q^{12}-3q^{10}+q^{8}+3q^{6}+3q^{4}+3q^{2}+1$

A2 Invariants.

Weight	Invariant
1,0	$q^{30}+q^{26}-2q^{22}-q^{20}-3q^{18}+q^{12}+3q^{10}+q^{8}+2q^{6}$
1,1	$q^{84}-2q^{82}+4q^{80}-8q^{78}+11q^{76}-14q^{74}+18q^{72}-24q^{70}+29q^{68}-30q^{66}+34q^{64}-36q^{62}+28q^{60}-16q^{58}-2q^{56}+20q^{54}-44q^{52}+62q^{50}-72q^{48}+82q^{46}-74q^{44}+72q^{42}-50q^{40}+38q^{38}-17q^{36}-10q^{34}+18q^{32}-44q^{30}+38q^{28}-52q^{26}+40q^{24}-30q^{22}+26q^{20}-10q^{18}+12q^{16}+4q^{14}+4q^{12}+2q^{10}$
2,0	$q^{76}-q^{68}-q^{66}-q^{64}+q^{62}-q^{60}-3q^{58}+2q^{54}+2q^{50}+6q^{48}+5q^{46}+q^{44}-4q^{38}-6q^{36}-2q^{34}-4q^{32}-5q^{30}-q^{28}-q^{24}+5q^{20}+4q^{18}+2q^{16}+2q^{14}+4q^{12}+q^{10}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{68}-q^{66}+q^{62}-3q^{60}+3q^{56}-4q^{54}-q^{52}+5q^{50}-3q^{48}+q^{46}+5q^{44}+3q^{42}+3q^{40}+2q^{38}+q^{36}-5q^{34}-10q^{32}-5q^{30}-6q^{28}-8q^{26}+4q^{24}+5q^{22}+2q^{20}+7q^{18}+5q^{16}+q^{14}+3q^{12}$
1,0,0	$q^{39}+2q^{35}+q^{31}-2q^{29}-2q^{27}-4q^{25}-3q^{23}-q^{21}+3q^{17}+2q^{15}+4q^{13}+q^{11}+2q^{9}$
1,0,1	$q^{110}-2q^{108}+3q^{106}-3q^{104}-q^{102}+6q^{100}-9q^{98}+9q^{96}-4q^{94}-6q^{92}+16q^{90}-19q^{88}+13q^{86}+6q^{84}-28q^{82}+38q^{80}-30q^{78}+5q^{76}+24q^{74}-41q^{72}+39q^{70}-30q^{68}+q^{66}+3q^{64}-16q^{62}+6q^{60}+7q^{58}+11q^{56}+4q^{54}+50q^{52}-29q^{50}+48q^{48}-15q^{46}-26q^{44}+18q^{42}-70q^{40}+8q^{38}-37q^{36}-15q^{34}+11q^{32}-11q^{30}+25q^{28}+11q^{26}+14q^{24}+16q^{22}+7q^{20}+5q^{18}+2q^{16}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{86}+2q^{80}-4q^{76}-q^{74}-5q^{70}-5q^{68}+q^{64}-q^{62}+5q^{60}+12q^{58}+9q^{56}+9q^{54}+14q^{52}+5q^{50}-5q^{48}-7q^{46}-12q^{44}-22q^{42}-19q^{40}-12q^{38}-7q^{36}-4q^{34}+5q^{32}+12q^{30}+9q^{28}+10q^{26}+9q^{24}+6q^{22}+2q^{20}+3q^{18}$
1,0,0,0	$q^{48}+2q^{44}+q^{42}+q^{40}+q^{38}-2q^{36}-2q^{34}-5q^{32}-4q^{30}-4q^{28}-q^{26}+3q^{22}+4q^{20}+3q^{18}+4q^{16}+q^{14}+2q^{12}$

B2 Invariants.

Weight	Invariant
0,1	$q^{68}-q^{66}+2q^{64}-3q^{62}+3q^{60}-4q^{58}+5q^{56}-4q^{54}+3q^{52}-q^{50}-q^{48}+3q^{46}-5q^{44}+7q^{42}-9q^{40}+8q^{38}-9q^{36}+5q^{34}-6q^{32}+q^{30}-2q^{26}+4q^{24}-3q^{22}+6q^{20}-3q^{18}+5q^{16}-q^{14}+3q^{12}$
1,0	$q^{110}-q^{106}-q^{104}+q^{102}+2q^{100}-q^{98}-3q^{96}-2q^{94}+2q^{92}+4q^{90}-q^{88}-5q^{86}-2q^{84}+4q^{82}+4q^{80}-2q^{78}-3q^{76}+2q^{74}+5q^{72}+q^{70}-2q^{68}+q^{66}+5q^{64}+2q^{62}-2q^{60}-3q^{58}+q^{56}-4q^{52}-7q^{50}-2q^{48}+q^{46}-2q^{44}-6q^{42}-4q^{40}+4q^{38}+5q^{36}+q^{34}-3q^{32}+q^{30}+5q^{28}+5q^{26}+q^{20}+3q^{18}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{94}-q^{92}+q^{90}-2q^{88}+2q^{86}-3q^{84}+2q^{82}-3q^{80}+4q^{78}-4q^{76}+2q^{74}-2q^{72}+2q^{70}-2q^{66}+3q^{64}-q^{62}+9q^{60}-2q^{58}+10q^{56}-3q^{54}+9q^{52}-6q^{50}+q^{48}-12q^{46}-6q^{44}-10q^{42}-8q^{40}-6q^{38}-5q^{36}+4q^{34}+q^{32}+9q^{30}+3q^{28}+10q^{26}+2q^{24}+6q^{22}+3q^{18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{162}-q^{160}+2q^{158}-3q^{156}+q^{154}-3q^{150}+5q^{148}-6q^{146}+6q^{144}-5q^{142}+q^{140}+3q^{138}-8q^{136}+12q^{134}-11q^{132}+9q^{130}-3q^{128}-5q^{126}+13q^{124}-13q^{122}+12q^{120}-3q^{118}-5q^{116}+11q^{114}-8q^{112}+2q^{110}+9q^{108}-14q^{106}+16q^{104}-8q^{102}-5q^{100}+15q^{98}-22q^{96}+21q^{94}-16q^{92}+2q^{90}+7q^{88}-17q^{86}+17q^{84}-17q^{82}+5q^{80}-11q^{76}+9q^{74}-9q^{72}+7q^{68}-13q^{66}+10q^{64}-4q^{62}-7q^{60}+17q^{58}-17q^{56}+14q^{54}-3q^{52}-4q^{50}+14q^{48}-12q^{46}+13q^{44}-4q^{42}+q^{40}+6q^{38}-5q^{36}+5q^{34}+2q^{30}+q^{28}$

.

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["10 127"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+4t^{2}-6t+7-6t^{-1}+4t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 29, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$2q^{-2}-2q^{-3}+4q^{-4}-5q^{-5}+5q^{-6}-5q^{-7}+3q^{-8}-2q^{-9}+q^{-10}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{4}a^{8}+3z^{2}a^{8}+2a^{8}-z^{6}a^{6}-5z^{4}a^{6}-9z^{2}a^{6}-6a^{6}+2z^{4}a^{4}+7z^{2}a^{4}+5a^{4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{4}a^{12}-2z^{2}a^{12}+2z^{5}a^{11}-4z^{3}a^{11}+za^{11}+2z^{6}a^{10}-3z^{4}a^{10}+z^{2}a^{10}+2z^{7}a^{9}-5z^{5}a^{9}+7z^{3}a^{9}-2za^{9}+z^{8}a^{8}-2z^{6}a^{8}+4z^{4}a^{8}-2z^{2}a^{8}+2a^{8}+3z^{7}a^{7}-10z^{5}a^{7}+16z^{3}a^{7}-8za^{7}+z^{8}a^{6}-4z^{6}a^{6}+11z^{4}a^{6}-14z^{2}a^{6}+6a^{6}+z^{7}a^{5}-3z^{5}a^{5}+5z^{3}a^{5}-5za^{5}+3z^{4}a^{4}-9z^{2}a^{4}+5a^{4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_150, K11n51, ...}

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

Vassiliev invariants

V2 and V3:

(1, 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 $4$ $8$ $8$ $-{\frac {130}{3}}$ ${\frac {58}{3}}$ $32$ ${\frac {272}{3}}$ $-{\frac {160}{3}}$ $-88$ ${\frac {32}{3}}$ $32$ $-{\frac {520}{3}}$ ${\frac {232}{3}}$ ${\frac {2191}{30}}$ $-{\frac {3782}{15}}$ ${\frac {38822}{45}}$ ${\frac {1073}{18}}$ ${\frac {2671}{30}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$-4 is the signature of 10 127. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                 2 2 -5               1 1 0 -7             3 1   2 -9           2 1     -1 -11         3 3       0 -13       2 2         0 -15     1 3           -2 -17   1 2             1 -19   1               -1 -21 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{-3}+2q^{-4}-4q^{-5}+q^{-6}+8q^{-7}-9q^{-8}-4q^{-9}+17q^{-10}-12q^{-11}-11q^{-12}+25q^{-13}-12q^{-14}-17q^{-15}+28q^{-16}-9q^{-17}-17q^{-18}+22q^{-19}-4q^{-20}-12q^{-21}+11q^{-22}-6q^{-24}+4q^{-25}-2q^{-27}+q^{-28}$
3	$2q^{-4}-6q^{-7}+4q^{-8}+7q^{-9}+3q^{-10}-16q^{-11}-4q^{-12}+14q^{-13}+19q^{-14}-22q^{-15}-23q^{-16}+12q^{-17}+40q^{-18}-12q^{-19}-45q^{-20}+56q^{-22}+6q^{-23}-61q^{-24}-14q^{-25}+65q^{-26}+22q^{-27}-68q^{-28}-26q^{-29}+65q^{-30}+32q^{-31}-62q^{-32}-31q^{-33}+49q^{-34}+36q^{-35}-42q^{-36}-29q^{-37}+25q^{-38}+27q^{-39}-16q^{-40}-19q^{-41}+7q^{-42}+13q^{-43}-3q^{-44}-8q^{-45}+2q^{-46}+4q^{-47}-q^{-48}-3q^{-49}+2q^{-50}+q^{-51}-2q^{-53}+q^{-54}$
4	$q^{-4}+2q^{-5}-4q^{-7}-2q^{-8}-3q^{-9}+9q^{-10}+12q^{-11}-5q^{-12}-8q^{-13}-25q^{-14}+6q^{-15}+33q^{-16}+11q^{-17}+8q^{-18}-57q^{-19}-27q^{-20}+32q^{-21}+34q^{-22}+63q^{-23}-62q^{-24}-69q^{-25}-13q^{-26}+26q^{-27}+138q^{-28}-24q^{-29}-85q^{-30}-80q^{-31}-24q^{-32}+196q^{-33}+35q^{-34}-68q^{-35}-136q^{-36}-89q^{-37}+229q^{-38}+84q^{-39}-41q^{-40}-170q^{-41}-140q^{-42}+242q^{-43}+115q^{-44}-15q^{-45}-187q^{-46}-172q^{-47}+234q^{-48}+131q^{-49}+13q^{-50}-176q^{-51}-189q^{-52}+187q^{-53}+125q^{-54}+51q^{-55}-128q^{-56}-181q^{-57}+109q^{-58}+85q^{-59}+75q^{-60}-54q^{-61}-133q^{-62}+38q^{-63}+25q^{-64}+64q^{-65}+q^{-66}-68q^{-67}+11q^{-68}-14q^{-69}+31q^{-70}+14q^{-71}-24q^{-72}+11q^{-73}-17q^{-74}+8q^{-75}+6q^{-76}-9q^{-77}+11q^{-78}-7q^{-79}+q^{-80}+q^{-81}-5q^{-82}+5q^{-83}-q^{-84}+q^{-85}-2q^{-87}+q^{-88}$
5	$2q^{-4}+2q^{-6}-2q^{-7}-6q^{-8}-6q^{-9}+6q^{-10}+4q^{-11}+15q^{-12}+12q^{-13}-14q^{-14}-28q^{-15}-15q^{-16}-8q^{-17}+30q^{-18}+56q^{-19}+23q^{-20}-34q^{-21}-52q^{-22}-70q^{-23}-11q^{-24}+79q^{-25}+97q^{-26}+45q^{-27}-26q^{-28}-125q^{-29}-125q^{-30}-5q^{-31}+102q^{-32}+158q^{-33}+124q^{-34}-64q^{-35}-206q^{-36}-183q^{-37}-43q^{-38}+178q^{-39}+304q^{-40}+146q^{-41}-152q^{-42}-335q^{-43}-284q^{-44}+51q^{-45}+400q^{-46}+399q^{-47}+31q^{-48}-389q^{-49}-509q^{-50}-149q^{-51}+393q^{-52}+594q^{-53}+237q^{-54}-365q^{-55}-660q^{-56}-321q^{-57}+344q^{-58}+707q^{-59}+388q^{-60}-326q^{-61}-741q^{-62}-433q^{-63}+301q^{-64}+767q^{-65}+478q^{-66}-289q^{-67}-775q^{-68}-511q^{-69}+251q^{-70}+779q^{-71}+549q^{-72}-217q^{-73}-750q^{-74}-570q^{-75}+131q^{-76}+709q^{-77}+598q^{-78}-70q^{-79}-611q^{-80}-579q^{-81}-47q^{-82}+502q^{-83}+557q^{-84}+107q^{-85}-353q^{-86}-469q^{-87}-188q^{-88}+218q^{-89}+377q^{-90}+201q^{-91}-93q^{-92}-258q^{-93}-195q^{-94}+6q^{-95}+157q^{-96}+152q^{-97}+42q^{-98}-73q^{-99}-105q^{-100}-54q^{-101}+19q^{-102}+59q^{-103}+47q^{-104}+8q^{-105}-25q^{-106}-33q^{-107}-16q^{-108}+9q^{-109}+13q^{-110}+14q^{-111}+7q^{-112}-9q^{-113}-10q^{-114}-q^{-115}-3q^{-116}+2q^{-117}+9q^{-118}+q^{-119}-4q^{-120}+q^{-121}-3q^{-122}-3q^{-123}+3q^{-124}+2q^{-125}-q^{-126}+q^{-127}-2q^{-129}+q^{-130}$
6	$q^{-3}+2q^{-4}-2q^{-7}-4q^{-8}-8q^{-9}-3q^{-10}+9q^{-11}+16q^{-12}+13q^{-13}+8q^{-14}-q^{-15}-37q^{-16}-41q^{-17}-22q^{-18}+21q^{-19}+44q^{-20}+62q^{-21}+73q^{-22}-19q^{-23}-83q^{-24}-122q^{-25}-66q^{-26}-20q^{-27}+79q^{-28}+207q^{-29}+133q^{-30}+30q^{-31}-140q^{-32}-168q^{-33}-237q^{-34}-128q^{-35}+165q^{-36}+258q^{-37}+298q^{-38}+117q^{-39}+14q^{-40}-348q^{-41}-470q^{-42}-225q^{-43}+16q^{-44}+368q^{-45}+475q^{-46}+590q^{-47}-9q^{-48}-527q^{-49}-691q^{-50}-629q^{-51}-86q^{-52}+489q^{-53}+1213q^{-54}+738q^{-55}-28q^{-56}-788q^{-57}-1288q^{-58}-950q^{-59}-38q^{-60}+1470q^{-61}+1492q^{-62}+844q^{-63}-395q^{-64}-1602q^{-65}-1819q^{-66}-878q^{-67}+1303q^{-68}+1954q^{-69}+1696q^{-70}+234q^{-71}-1564q^{-72}-2423q^{-73}-1661q^{-74}+952q^{-75}+2121q^{-76}+2292q^{-77}+783q^{-78}-1381q^{-79}-2751q^{-80}-2189q^{-81}+659q^{-82}+2147q^{-83}+2629q^{-84}+1119q^{-85}-1231q^{-86}-2915q^{-87}-2479q^{-88}+484q^{-89}+2145q^{-90}+2817q^{-91}+1314q^{-92}-1121q^{-93}-3001q^{-94}-2667q^{-95}+308q^{-96}+2095q^{-97}+2937q^{-98}+1528q^{-99}-905q^{-100}-2958q^{-101}-2827q^{-102}-46q^{-103}+1822q^{-104}+2907q^{-105}+1814q^{-106}-411q^{-107}-2582q^{-108}-2826q^{-109}-580q^{-110}+1161q^{-111}+2490q^{-112}+1970q^{-113}+297q^{-114}-1740q^{-115}-2399q^{-116}-1001q^{-117}+269q^{-118}+1601q^{-119}+1699q^{-120}+834q^{-121}-702q^{-122}-1524q^{-123}-971q^{-124}-381q^{-125}+604q^{-126}+1022q^{-127}+867q^{-128}+9q^{-129}-620q^{-130}-544q^{-131}-504q^{-132}-11q^{-133}+358q^{-134}+529q^{-135}+196q^{-136}-107q^{-137}-125q^{-138}-298q^{-139}-157q^{-140}+22q^{-141}+204q^{-142}+110q^{-143}+26q^{-144}+53q^{-145}-104q^{-146}-95q^{-147}-48q^{-148}+55q^{-149}+23q^{-150}+12q^{-151}+67q^{-152}-21q^{-153}-32q^{-154}-33q^{-155}+14q^{-156}-5q^{-157}-7q^{-158}+39q^{-159}-4q^{-161}-15q^{-162}+5q^{-163}-7q^{-164}-11q^{-165}+17q^{-166}+2q^{-167}+3q^{-168}-5q^{-169}+3q^{-170}-3q^{-171}-7q^{-172}+5q^{-173}+2q^{-175}-q^{-176}+q^{-177}-2q^{-179}+q^{-180}$
7	Not Available

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[10, 127]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[14, 6, 15, 5], X[15, 20, 16, 1], X[9, 16, 10, 17], X[11, 18, 12, 19], X[17, 10, 18, 11], X[19, 12, 20, 13], X[6, 14, 7, 13], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 127]]
Out[3]=	GaussCode[-1, 10, -2, 1, 3, -9, -10, 2, -5, 7, -6, 8, 9, -3, -4, 5, -7, 6, -8, 4]
In[4]:=	DTCode[Knot[10, 127]]
Out[4]=	DTCode[4, 8, -14, 2, 16, 18, -6, 20, 10, 12]
In[5]:=	br = BR[Knot[10, 127]]
Out[5]=	BR[3, {-1, -1, -1, -1, -1, -2, 1, 1, -2, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 127]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 127]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 127]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 127]][t]
Out[10]=	-3 4 6 2 3 7 - t + -- - - - 6 t + 4 t - t 2 t t
In[11]:=	Conway[Knot[10, 127]][z]
Out[11]=	2 4 6 1 + z - 2 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 127], Knot[10, 150], Knot[11, NonAlternating, 51]}
In[13]:=	{KnotDet[Knot[10, 127]], KnotSignature[Knot[10, 127]]}
Out[13]=	{29, -4}
In[14]:=	Jones[Knot[10, 127]][q]
Out[14]=	-10 2 3 5 5 5 4 2 2 q - -- + -- - -- + -- - -- + -- - -- + -- 9 8 7 6 5 4 3 2 q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 127]}
In[16]:=	A2Invariant[Knot[10, 127]][q]
Out[16]=	-30 -26 2 -20 3 -12 3 -8 2 q + q - --- - q - --- + q + --- + q + -- 22 18 10 6 q q q q
In[17]:=	HOMFLYPT[Knot[10, 127]][a, z]
Out[17]=	4 6 8 4 2 6 2 8 2 4 4 6 4 5 a - 6 a + 2 a + 7 a z - 9 a z + 3 a z + 2 a z - 5 a z + 8 4 6 6 a z - a z
In[18]:=	Kauffman[Knot[10, 127]][a, z]
Out[18]=	4 6 8 5 7 9 11 4 2 5 a + 6 a + 2 a - 5 a z - 8 a z - 2 a z + a z - 9 a z - 6 2 8 2 10 2 12 2 5 3 7 3 14 a z - 2 a z + a z - 2 a z + 5 a z + 16 a z + 9 3 11 3 4 4 6 4 8 4 10 4 7 a z - 4 a z + 3 a z + 11 a z + 4 a z - 3 a z + 12 4 5 5 7 5 9 5 11 5 6 6 a z - 3 a z - 10 a z - 5 a z + 2 a z - 4 a z - 8 6 10 6 5 7 7 7 9 7 6 8 8 8 2 a z + 2 a z + a z + 3 a z + 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 127]], Vassiliev[3][Knot[10, 127]]}
Out[19]=	{1, 1}
In[20]:=	Kh[Knot[10, 127]][q, t]
Out[20]=	-5 2 1 1 1 2 1 3 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + 3 21 8 19 7 17 7 17 6 15 6 15 5 q q t q t q t q t q t q t 2 2 3 3 2 1 3 1 ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 q t q t q t q t q t q t q t q t 1 ---- 5 q t
In[21]:=	ColouredJones[Knot[10, 127], 2][q]
Out[21]=	-28 2 4 6 11 12 4 22 17 9 28 q - --- + --- - --- + --- - --- - --- + --- - --- - --- + --- - 27 25 24 22 21 20 19 18 17 16 q q q q q q q q q q 17 12 25 11 12 17 4 9 8 -6 4 2 -3 --- - --- + --- - --- - --- + --- - -- - -- + -- + q - -- + -- + q 15 14 13 12 11 10 9 8 7 5 4 q q q q q q q q q q q

See/edit the Rolfsen_Splice_Template. Back to the top.

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