10 78

10_77

10_79

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10 78 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3849 X5,14,6,15 X11,17,12,16 X15,13,16,12 X17,20,18,1 X9,18,10,19 X19,10,20,11 X13,6,14,7 X7283
Gauss code	-1, 10, -2, 1, -3, 9, -10, 2, -7, 8, -4, 5, -9, 3, -5, 4, -6, 7, -8, 6
Dowker-Thistlethwaite code	4 8 14 2 18 16 6 12 20 10
Conway Notation	[21,21,2++]

Minimum Braid Representative: Length is 12, width is 5. Braid index is 5.

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][1] Hyperbolic Volume 12.5021 A-Polynomial See Data:10 78/A-polynomial

[edit Notes for 10 78'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 78's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+7t^{2}-16t+21-16t^{-1}+7t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 69, -4 }
Jones polynomial	$1-3q^{-1}+6q^{-2}-8q^{-3}+11q^{-4}-11q^{-5}+11q^{-6}-9q^{-7}+5q^{-8}-3q^{-9}+q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$a^{10}-3z^{2}a^{8}-4a^{8}+3z^{4}a^{6}+7z^{2}a^{6}+4a^{6}-z^{6}a^{4}-3z^{4}a^{4}-3z^{2}a^{4}-a^{4}+z^{4}a^{2}+2z^{2}a^{2}+a^{2}$
Kauffman polynomial (db, data sources)	$z^{4}a^{12}-z^{2}a^{12}+3z^{5}a^{11}-4z^{3}a^{11}+2za^{11}+4z^{6}a^{10}-3z^{4}a^{10}+z^{2}a^{10}-a^{10}+4z^{7}a^{9}-7z^{3}a^{9}+6za^{9}+3z^{8}a^{8}+2z^{6}a^{8}-10z^{4}a^{8}+10z^{2}a^{8}-4a^{8}+z^{9}a^{7}+7z^{7}a^{7}-15z^{5}a^{7}+5z^{3}a^{7}+2za^{7}+6z^{8}a^{6}-8z^{6}a^{6}-7z^{4}a^{6}+11z^{2}a^{6}-4a^{6}+z^{9}a^{5}+6z^{7}a^{5}-21z^{5}a^{5}+15z^{3}a^{5}-3za^{5}+3z^{8}a^{4}-5z^{6}a^{4}-4z^{4}a^{4}+6z^{2}a^{4}-a^{4}+3z^{7}a^{3}-9z^{5}a^{3}+7z^{3}a^{3}-za^{3}+z^{6}a^{2}-3z^{4}a^{2}+3z^{2}a^{2}-a^{2}$
The A2 invariant	$q^{32}+q^{30}-2q^{28}-q^{26}-q^{24}-3q^{22}+2q^{20}+2q^{16}+2q^{14}-q^{12}+3q^{10}-2q^{8}+q^{6}+q^{4}-q^{2}+1$
The G2 invariant	$q^{162}-2q^{160}+4q^{158}-6q^{156}+4q^{154}-3q^{152}-4q^{150}+12q^{148}-18q^{146}+25q^{144}-24q^{142}+17q^{140}-19q^{136}+43q^{134}-58q^{132}+62q^{130}-53q^{128}+24q^{126}+19q^{124}-62q^{122}+98q^{120}-102q^{118}+79q^{116}-33q^{114}-31q^{112}+75q^{110}-98q^{108}+78q^{106}-31q^{104}-31q^{102}+66q^{100}-68q^{98}+24q^{96}+39q^{94}-100q^{92}+120q^{90}-93q^{88}+18q^{86}+76q^{84}-150q^{82}+184q^{80}-149q^{78}+68q^{76}+34q^{74}-116q^{72}+160q^{70}-143q^{68}+84q^{66}-3q^{64}-64q^{62}+99q^{60}-81q^{58}+29q^{56}+38q^{54}-84q^{52}+89q^{50}-52q^{48}-18q^{46}+89q^{44}-129q^{42}+125q^{40}-73q^{38}-4q^{36}+76q^{34}-115q^{32}+116q^{30}-79q^{28}+25q^{26}+22q^{24}-54q^{22}+58q^{20}-43q^{18}+24q^{16}-2q^{14}-9q^{12}+12q^{10}-10q^{8}+6q^{6}-2q^{4}+q^{2}$

Further Quantum Invariants

Further quantum knot invariants for 10_78.

A1 Invariants.

Weight	Invariant
1	$q^{21}-2q^{19}+2q^{17}-4q^{15}+2q^{13}+3q^{7}-2q^{5}+3q^{3}-2q+q^{-1}$
2	$q^{58}-2q^{56}-q^{54}+5q^{52}-6q^{50}+13q^{46}-14q^{44}-3q^{42}+22q^{40}-16q^{38}-10q^{36}+20q^{34}-5q^{32}-13q^{30}+5q^{28}+9q^{26}-8q^{24}-10q^{22}+17q^{20}+2q^{18}-20q^{16}+16q^{14}+11q^{12}-21q^{10}+6q^{8}+14q^{6}-12q^{4}-2q^{2}+8-2q^{-2}-2q^{-4}+q^{-6}$
3	$q^{111}-2q^{109}-q^{107}+2q^{105}+3q^{103}-3q^{101}-3q^{99}+8q^{97}-q^{95}-13q^{93}+q^{91}+27q^{89}-6q^{87}-45q^{85}+4q^{83}+69q^{81}+2q^{79}-88q^{77}-21q^{75}+106q^{73}+43q^{71}-106q^{69}-63q^{67}+83q^{65}+87q^{63}-55q^{61}-88q^{59}+12q^{57}+86q^{55}+23q^{53}-68q^{51}-60q^{49}+51q^{47}+81q^{45}-31q^{43}-101q^{41}+6q^{39}+109q^{37}+18q^{35}-110q^{33}-45q^{31}+105q^{29}+68q^{27}-81q^{25}-90q^{23}+56q^{21}+97q^{19}-21q^{17}-88q^{15}-7q^{13}+71q^{11}+29q^{9}-47q^{7}-33q^{5}+21q^{3}+29q-4q^{-1}-19q^{-3}-2q^{-5}+8q^{-7}+3q^{-9}-2q^{-11}-2q^{-13}+q^{-15}$
4	$q^{180}-2q^{178}-q^{176}+2q^{174}+6q^{170}-6q^{168}-2q^{166}+3q^{164}-10q^{162}+12q^{160}-6q^{158}+12q^{156}+13q^{154}-43q^{152}-7q^{150}-4q^{148}+71q^{146}+65q^{144}-107q^{142}-104q^{140}-41q^{138}+196q^{136}+227q^{134}-152q^{132}-308q^{130}-204q^{128}+316q^{126}+519q^{124}-45q^{122}-500q^{120}-523q^{118}+241q^{116}+772q^{114}+266q^{112}-440q^{110}-779q^{108}-87q^{106}+687q^{104}+551q^{102}-71q^{100}-692q^{98}-424q^{96}+268q^{94}+557q^{92}+319q^{90}-316q^{88}-527q^{86}-187q^{84}+349q^{82}+523q^{80}+69q^{78}-467q^{76}-485q^{74}+134q^{72}+582q^{70}+352q^{68}-366q^{66}-672q^{64}-61q^{62}+564q^{60}+582q^{58}-195q^{56}-760q^{54}-308q^{52}+397q^{50}+748q^{48}+115q^{46}-645q^{44}-538q^{42}+44q^{40}+690q^{38}+427q^{36}-283q^{34}-542q^{32}-325q^{30}+358q^{28}+491q^{26}+109q^{24}-266q^{22}-424q^{20}-13q^{18}+262q^{16}+243q^{14}+36q^{12}-241q^{10}-143q^{8}+13q^{6}+126q^{4}+118q^{2}-41-68q^{-2}-53q^{-4}+10q^{-6}+53q^{-8}+11q^{-10}-4q^{-12}-19q^{-14}-9q^{-16}+8q^{-18}+3q^{-20}+3q^{-22}-2q^{-24}-2q^{-26}+q^{-28}$
5	$q^{265}-2q^{263}-q^{261}+2q^{259}+3q^{255}+3q^{253}-5q^{251}-7q^{249}-3q^{245}+6q^{243}+16q^{241}+8q^{239}-8q^{237}-25q^{235}-28q^{233}-6q^{231}+47q^{229}+81q^{227}+28q^{225}-87q^{223}-151q^{221}-88q^{219}+108q^{217}+303q^{215}+223q^{213}-168q^{211}-509q^{209}-440q^{207}+148q^{205}+817q^{203}+838q^{201}-65q^{199}-1195q^{197}-1407q^{195}-211q^{193}+1572q^{191}+2190q^{189}+730q^{187}-1806q^{185}-3106q^{183}-1576q^{181}+1789q^{179}+3974q^{177}+2663q^{175}-1325q^{173}-4556q^{171}-3897q^{169}+436q^{167}+4664q^{165}+4936q^{163}+794q^{161}-4138q^{159}-5525q^{157}-2164q^{155}+3048q^{153}+5537q^{151}+3261q^{149}-1587q^{147}-4837q^{145}-3983q^{143}+26q^{141}+3737q^{139}+4139q^{137}+1286q^{135}-2316q^{133}-3875q^{131}-2296q^{129}+1003q^{127}+3322q^{125}+2898q^{123}+149q^{121}-2715q^{119}-3268q^{117}-994q^{115}+2180q^{113}+3488q^{111}+1657q^{109}-1799q^{107}-3696q^{105}-2184q^{103}+1487q^{101}+3935q^{99}+2738q^{97}-1166q^{95}-4170q^{93}-3343q^{91}+707q^{89}+4279q^{87}+4013q^{85}-11q^{83}-4165q^{81}-4642q^{79}-907q^{77}+3685q^{75}+5042q^{73}+2003q^{71}-2793q^{69}-5134q^{67}-3037q^{65}+1563q^{63}+4686q^{61}+3831q^{59}-121q^{57}-3787q^{55}-4162q^{53}-1207q^{51}+2473q^{49}+3923q^{47}+2205q^{45}-1027q^{43}-3155q^{41}-2666q^{39}-251q^{37}+2062q^{35}+2544q^{33}+1116q^{31}-891q^{29}-1975q^{27}-1480q^{25}-21q^{23}+1199q^{21}+1354q^{19}+562q^{17}-450q^{15}-973q^{13}-715q^{11}-35q^{9}+510q^{7}+583q^{5}+265q^{3}-145q-363q^{-1}-278q^{-3}-37q^{-5}+156q^{-7}+184q^{-9}+93q^{-11}-28q^{-13}-99q^{-15}-73q^{-17}-10q^{-19}+33q^{-21}+35q^{-23}+20q^{-25}-4q^{-27}-19q^{-29}-9q^{-31}+q^{-33}+3q^{-35}+3q^{-37}+3q^{-39}-2q^{-41}-2q^{-43}+q^{-45}$

A2 Invariants.

Weight	Invariant
1,0	$q^{32}+q^{30}-2q^{28}-q^{26}-q^{24}-3q^{22}+2q^{20}+2q^{16}+2q^{14}-q^{12}+3q^{10}-2q^{8}+q^{6}+q^{4}-q^{2}+1$
2,0	$q^{80}+q^{78}-q^{76}-4q^{74}-2q^{72}+2q^{70}-2q^{66}+5q^{64}+10q^{62}-7q^{58}+3q^{56}+8q^{54}-10q^{52}-10q^{50}+4q^{48}+3q^{46}-9q^{44}-4q^{42}+8q^{40}-3q^{38}-4q^{36}+7q^{34}+2q^{32}-8q^{30}+4q^{28}+9q^{26}-5q^{24}-6q^{22}+9q^{20}+8q^{18}-7q^{16}-4q^{14}+9q^{12}+2q^{10}-5q^{8}-q^{6}+3q^{4}+2q^{2}-2-q^{-2}+q^{-4}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{68}-2q^{66}+4q^{62}-7q^{60}-2q^{58}+12q^{56}-7q^{54}-5q^{52}+22q^{50}-6q^{48}-12q^{46}+14q^{44}-9q^{42}-16q^{40}+3q^{38}+q^{36}-4q^{34}-3q^{32}+11q^{30}+8q^{28}-13q^{26}+9q^{24}+12q^{22}-16q^{20}+4q^{18}+12q^{16}-13q^{14}+4q^{12}+7q^{10}-7q^{8}+3q^{6}+2q^{4}-2q^{2}+1$
1,0,0	$q^{43}+q^{41}+q^{39}-2q^{37}-q^{35}-4q^{33}-q^{31}-3q^{29}+3q^{27}+3q^{23}+2q^{21}+q^{19}+q^{17}-q^{15}+2q^{13}-2q^{11}+2q^{9}-q^{7}+2q^{5}-q^{3}+q$

B2 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

$q^{162}-2q^{160}+4q^{158}-6q^{156}+4q^{154}-3q^{152}-4q^{150}+12q^{148}-18q^{146}+25q^{144}-24q^{142}+17q^{140}-19q^{136}+43q^{134}-58q^{132}+62q^{130}-53q^{128}+24q^{126}+19q^{124}-62q^{122}+98q^{120}-102q^{118}+79q^{116}-33q^{114}-31q^{112}+75q^{110}-98q^{108}+78q^{106}-31q^{104}-31q^{102}+66q^{100}-68q^{98}+24q^{96}+39q^{94}-100q^{92}+120q^{90}-93q^{88}+18q^{86}+76q^{84}-150q^{82}+184q^{80}-149q^{78}+68q^{76}+34q^{74}-116q^{72}+160q^{70}-143q^{68}+84q^{66}-3q^{64}-64q^{62}+99q^{60}-81q^{58}+29q^{56}+38q^{54}-84q^{52}+89q^{50}-52q^{48}-18q^{46}+89q^{44}-129q^{42}+125q^{40}-73q^{38}-4q^{36}+76q^{34}-115q^{32}+116q^{30}-79q^{28}+25q^{26}+22q^{24}-54q^{22}+58q^{20}-43q^{18}+24q^{16}-2q^{14}-9q^{12}+12q^{10}-10q^{8}+6q^{6}-2q^{4}+q^{2}$

.

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 78"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+7t^{2}-16t+21-16t^{-1}+7t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{6}+z^{4}+3z^{2}+1$

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]=

{ 69, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$1-3q^{-1}+6q^{-2}-8q^{-3}+11q^{-4}-11q^{-5}+11q^{-6}-9q^{-7}+5q^{-8}-3q^{-9}+q^{-10}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{10}-3z^{2}a^{8}-4a^{8}+3z^{4}a^{6}+7z^{2}a^{6}+4a^{6}-z^{6}a^{4}-3z^{4}a^{4}-3z^{2}a^{4}-a^{4}+z^{4}a^{2}+2z^{2}a^{2}+a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{4}a^{12}-z^{2}a^{12}+3z^{5}a^{11}-4z^{3}a^{11}+2za^{11}+4z^{6}a^{10}-3z^{4}a^{10}+z^{2}a^{10}-a^{10}+4z^{7}a^{9}-7z^{3}a^{9}+6za^{9}+3z^{8}a^{8}+2z^{6}a^{8}-10z^{4}a^{8}+10z^{2}a^{8}-4a^{8}+z^{9}a^{7}+7z^{7}a^{7}-15z^{5}a^{7}+5z^{3}a^{7}+2za^{7}+6z^{8}a^{6}-8z^{6}a^{6}-7z^{4}a^{6}+11z^{2}a^{6}-4a^{6}+z^{9}a^{5}+6z^{7}a^{5}-21z^{5}a^{5}+15z^{3}a^{5}-3za^{5}+3z^{8}a^{4}-5z^{6}a^{4}-4z^{4}a^{4}+6z^{2}a^{4}-a^{4}+3z^{7}a^{3}-9z^{5}a^{3}+7z^{3}a^{3}-za^{3}+z^{6}a^{2}-3z^{4}a^{2}+3z^{2}a^{2}-a^{2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n98, K11n105, ...}

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

Vassiliev invariants

V2 and V3:

(3, -5)

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 $12$ $-40$ $72$ $158$ $26$ $-480$ $-{\frac {2128}{3}}$ $-{\frac {352}{3}}$ $-104$ $288$ $800$ $1896$ $312$ ${\frac {32751}{10}}$ ${\frac {1214}{15}}$ ${\frac {6274}{5}}$ ${\frac {59}{2}}$ ${\frac {1711}{10}}$

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 78. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{2}-3q+11q^{-1}-13q^{-2}-10q^{-3}+37q^{-4}-21q^{-5}-37q^{-6}+69q^{-7}-16q^{-8}-73q^{-9}+91q^{-10}-q^{-11}-100q^{-12}+93q^{-13}+16q^{-14}-104q^{-15}+75q^{-16}+24q^{-17}-79q^{-18}+45q^{-19}+18q^{-20}-41q^{-21}+20q^{-22}+7q^{-23}-14q^{-24}+7q^{-25}+q^{-26}-3q^{-27}+q^{-28}$
3	$q^{6}-3q^{5}+5q^{3}+6q^{2}-13q-17+20q^{-1}+39q^{-2}-21q^{-3}-71q^{-4}+6q^{-5}+115q^{-6}+21q^{-7}-149q^{-8}-75q^{-9}+182q^{-10}+139q^{-11}-190q^{-12}-221q^{-13}+191q^{-14}+288q^{-15}-153q^{-16}-371q^{-17}+126q^{-18}+416q^{-19}-62q^{-20}-474q^{-21}+19q^{-22}+486q^{-23}+50q^{-24}-504q^{-25}-92q^{-26}+478q^{-27}+141q^{-28}-441q^{-29}-166q^{-30}+378q^{-31}+174q^{-32}-299q^{-33}-170q^{-34}+232q^{-35}+131q^{-36}-150q^{-37}-107q^{-38}+105q^{-39}+64q^{-40}-60q^{-41}-40q^{-42}+40q^{-43}+15q^{-44}-21q^{-45}-7q^{-46}+14q^{-47}+q^{-48}-9q^{-49}+2q^{-50}+3q^{-51}+q^{-52}-3q^{-53}+q^{-54}$
4	$q^{12}-3q^{11}+5q^{9}+6q^{7}-20q^{6}-10q^{5}+20q^{4}+15q^{3}+48q^{2}-63q-73+5q^{-1}+42q^{-2}+207q^{-3}-55q^{-4}-186q^{-5}-151q^{-6}-56q^{-7}+484q^{-8}+152q^{-9}-167q^{-10}-426q^{-11}-467q^{-12}+642q^{-13}+527q^{-14}+215q^{-15}-559q^{-16}-1150q^{-17}+425q^{-18}+786q^{-19}+925q^{-20}-296q^{-21}-1796q^{-22}-157q^{-23}+679q^{-24}+1685q^{-25}+337q^{-26}-2147q^{-27}-862q^{-28}+227q^{-29}+2250q^{-30}+1114q^{-31}-2165q^{-32}-1487q^{-33}-384q^{-34}+2556q^{-35}+1832q^{-36}-1935q^{-37}-1935q^{-38}-1003q^{-39}+2574q^{-40}+2368q^{-41}-1481q^{-42}-2109q^{-43}-1539q^{-44}+2234q^{-45}+2579q^{-46}-846q^{-47}-1871q^{-48}-1828q^{-49}+1542q^{-50}+2311q^{-51}-225q^{-52}-1249q^{-53}-1692q^{-54}+768q^{-55}+1619q^{-56}+114q^{-57}-543q^{-58}-1186q^{-59}+237q^{-60}+855q^{-61}+137q^{-62}-88q^{-63}-622q^{-64}+34q^{-65}+335q^{-66}+33q^{-67}+68q^{-68}-243q^{-69}+3q^{-70}+98q^{-71}-30q^{-72}+65q^{-73}-71q^{-74}+9q^{-75}+23q^{-76}-33q^{-77}+29q^{-78}-15q^{-79}+8q^{-80}+5q^{-81}-15q^{-82}+7q^{-83}-2q^{-84}+3q^{-85}+q^{-86}-3q^{-87}+q^{-88}$
5	$q^{20}-3q^{19}+5q^{17}-q^{14}-13q^{13}-10q^{12}+20q^{11}+24q^{10}+15q^{9}-3q^{8}-56q^{7}-73q^{6}-6q^{5}+95q^{4}+136q^{3}+88q^{2}-84q-266-247q^{-1}+10q^{-2}+354q^{-3}+498q^{-4}+234q^{-5}-339q^{-6}-792q^{-7}-670q^{-8}+96q^{-9}+1021q^{-10}+1246q^{-11}+453q^{-12}-947q^{-13}-1890q^{-14}-1363q^{-15}+526q^{-16}+2330q^{-17}+2460q^{-18}+481q^{-19}-2372q^{-20}-3676q^{-21}-1889q^{-22}+1841q^{-23}+4588q^{-24}+3713q^{-25}-654q^{-26}-5126q^{-27}-5569q^{-28}-1114q^{-29}+4963q^{-30}+7379q^{-31}+3298q^{-32}-4271q^{-33}-8692q^{-34}-5714q^{-35}+2866q^{-36}+9720q^{-37}+8094q^{-38}-1232q^{-39}-10049q^{-40}-10306q^{-41}-869q^{-42}+10197q^{-43}+12248q^{-44}+2792q^{-45}-9783q^{-46}-13878q^{-47}-4919q^{-48}+9370q^{-49}+15252q^{-50}+6696q^{-51}-8586q^{-52}-16326q^{-53}-8590q^{-54}+7858q^{-55}+17149q^{-56}+10152q^{-57}-6755q^{-58}-17634q^{-59}-11764q^{-60}+5584q^{-61}+17702q^{-62}+13016q^{-63}-4006q^{-64}-17210q^{-65}-14083q^{-66}+2285q^{-67}+16123q^{-68}+14575q^{-69}-404q^{-70}-14357q^{-71}-14485q^{-72}-1426q^{-73}+12114q^{-74}+13721q^{-75}+2846q^{-76}-9509q^{-77}-12209q^{-78}-3915q^{-79}+6902q^{-80}+10360q^{-81}+4233q^{-82}-4577q^{-83}-8067q^{-84}-4187q^{-85}+2674q^{-86}+6027q^{-87}+3574q^{-88}-1346q^{-89}-4079q^{-90}-2876q^{-91}+489q^{-92}+2662q^{-93}+2044q^{-94}-46q^{-95}-1543q^{-96}-1416q^{-97}-129q^{-98}+879q^{-99}+848q^{-100}+166q^{-101}-413q^{-102}-513q^{-103}-150q^{-104}+210q^{-105}+260q^{-106}+97q^{-107}-72q^{-108}-122q^{-109}-70q^{-110}+15q^{-111}+64q^{-112}+34q^{-113}-8q^{-114}-7q^{-115}-17q^{-116}-19q^{-117}+11q^{-118}+12q^{-119}-5q^{-120}+10q^{-121}-q^{-122}-11q^{-123}+q^{-124}+3q^{-125}-2q^{-126}+3q^{-127}+q^{-128}-3q^{-129}+q^{-130}$
6	$q^{30}-3q^{29}+5q^{27}-7q^{24}+6q^{23}-13q^{22}-10q^{21}+29q^{20}+15q^{19}+15q^{18}-27q^{17}+4q^{16}-67q^{15}-73q^{14}+59q^{13}+86q^{12}+135q^{11}+15q^{10}+72q^{9}-230q^{8}-365q^{7}-121q^{6}+67q^{5}+422q^{4}+376q^{3}+666q^{2}-130q-844-955q^{-1}-799q^{-2}+100q^{-3}+744q^{-4}+2323q^{-5}+1444q^{-6}-63q^{-7}-1674q^{-8}-2939q^{-9}-2571q^{-10}-1234q^{-11}+3406q^{-12}+4632q^{-13}+4209q^{-14}+1127q^{-15}-3451q^{-16}-7082q^{-17}-8074q^{-18}-771q^{-19}+5024q^{-20}+10428q^{-21}+9987q^{-22}+3545q^{-23}-7307q^{-24}-16786q^{-25}-12513q^{-26}-4203q^{-27}+10689q^{-28}+20320q^{-29}+19700q^{-30}+4209q^{-31}-17836q^{-32}-25635q^{-33}-23755q^{-34}-2739q^{-35}+21464q^{-36}+37278q^{-37}+27093q^{-38}-3643q^{-39}-28813q^{-40}-44708q^{-41}-28428q^{-42}+6501q^{-43}+44747q^{-44}+51694q^{-45}+23351q^{-46}-16022q^{-47}-55896q^{-48}-56423q^{-49}-21155q^{-50}+37030q^{-51}+67591q^{-52}+53235q^{-53}+8682q^{-54}-53313q^{-55}-77196q^{-56}-51845q^{-57}+18441q^{-58}+71658q^{-59}+77422q^{-60}+36285q^{-61}-41293q^{-62}-88186q^{-63}-77893q^{-64}-3050q^{-65}+67801q^{-66}+93735q^{-67}+60290q^{-68}-26534q^{-69}-92520q^{-70}-97465q^{-71}-22458q^{-72}+61055q^{-73}+104478q^{-74}+79605q^{-75}-12244q^{-76}-93390q^{-77}-112292q^{-78}-39902q^{-79}+52519q^{-80}+111176q^{-81}+95965q^{-82}+3219q^{-83}-89748q^{-84}-122656q^{-85}-57530q^{-86}+38902q^{-87}+110931q^{-88}+108825q^{-89}+22424q^{-90}-76852q^{-91}-124197q^{-92}-73977q^{-93}+17489q^{-94}+98016q^{-95}+112533q^{-96}+42840q^{-97}-52366q^{-98}-110636q^{-99}-82112q^{-100}-7603q^{-101}+71011q^{-102}+100433q^{-103}+55677q^{-104}-22378q^{-105}-81793q^{-106}-74838q^{-107}-25953q^{-108}+38032q^{-109}+73539q^{-110}+53566q^{-111}+1206q^{-112}-47522q^{-113}-53940q^{-114}-30082q^{-115}+11806q^{-116}+42539q^{-117}+38927q^{-118}+11098q^{-119}-20559q^{-120}-29958q^{-121}-22676q^{-122}-1132q^{-123}+19006q^{-124}+21589q^{-125}+10109q^{-126}-6202q^{-127}-12561q^{-128}-12368q^{-129}-3839q^{-130}+6575q^{-131}+9300q^{-132}+5608q^{-133}-1184q^{-134}-3864q^{-135}-5131q^{-136}-2622q^{-137}+1863q^{-138}+3229q^{-139}+2249q^{-140}-160q^{-141}-773q^{-142}-1700q^{-143}-1236q^{-144}+513q^{-145}+947q^{-146}+724q^{-147}-75q^{-148}-10q^{-149}-466q^{-150}-507q^{-151}+164q^{-152}+240q^{-153}+206q^{-154}-56q^{-155}+86q^{-156}-105q^{-157}-192q^{-158}+53q^{-159}+45q^{-160}+57q^{-161}-31q^{-162}+56q^{-163}-16q^{-164}-64q^{-165}+16q^{-166}+16q^{-168}-12q^{-169}+20q^{-170}+q^{-171}-17q^{-172}+5q^{-173}-3q^{-174}+3q^{-175}-2q^{-176}+3q^{-177}+q^{-178}-3q^{-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, 78]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 14, 6, 15], X[11, 17, 12, 16], X[15, 13, 16, 12], X[17, 20, 18, 1], X[9, 18, 10, 19], X[19, 10, 20, 11], X[13, 6, 14, 7], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 78]]
Out[3]=	GaussCode[-1, 10, -2, 1, -3, 9, -10, 2, -7, 8, -4, 5, -9, 3, -5, 4, -6, 7, -8, 6]
In[4]:=	DTCode[Knot[10, 78]]
Out[4]=	DTCode[4, 8, 14, 2, 18, 16, 6, 12, 20, 10]
In[5]:=	br = BR[Knot[10, 78]]
Out[5]=	BR[5, {-1, -1, -2, 1, -2, -1, 3, -2, -4, 3, -4, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 78]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 78]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 78]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 78]][t]
Out[10]=	-3 7 16 2 3 21 - t + -- - -- - 16 t + 7 t - t 2 t t
In[11]:=	Conway[Knot[10, 78]][z]
Out[11]=	2 4 6 1 + 3 z + z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 78], Knot[11, NonAlternating, 98], Knot[11, NonAlternating, 105]}
In[13]:=	{KnotDet[Knot[10, 78]], KnotSignature[Knot[10, 78]]}
Out[13]=	{69, -4}
In[14]:=	Jones[Knot[10, 78]][q]
Out[14]=	-10 3 5 9 11 11 11 8 6 3 1 + q - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q 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, 78]}
In[16]:=	A2Invariant[Knot[10, 78]][q]
Out[16]=	-32 -30 2 -26 -24 3 2 2 2 -12 1 + q + q - --- - q - q - --- + --- + --- + --- - q + 28 22 20 16 14 q q q q q 3 2 -6 -4 -2 --- - -- + q + q - q 10 8 q q
In[17]:=	HOMFLYPT[Knot[10, 78]][a, z]
Out[17]=	2 4 6 8 10 2 2 4 2 6 2 8 2 a - a + 4 a - 4 a + a + 2 a z - 3 a z + 7 a z - 3 a z + 2 4 4 4 6 4 4 6 a z - 3 a z + 3 a z - a z
In[18]:=	Kauffman[Knot[10, 78]][a, z]
Out[18]=	2 4 6 8 10 3 5 7 9 -a - a - 4 a - 4 a - a - a z - 3 a z + 2 a z + 6 a z + 11 2 2 4 2 6 2 8 2 10 2 12 2 2 a z + 3 a z + 6 a z + 11 a z + 10 a z + a z - a z + 3 3 5 3 7 3 9 3 11 3 2 4 7 a z + 15 a z + 5 a z - 7 a z - 4 a z - 3 a z - 4 4 6 4 8 4 10 4 12 4 3 5 4 a z - 7 a z - 10 a z - 3 a z + a z - 9 a z - 5 5 7 5 11 5 2 6 4 6 6 6 21 a z - 15 a z + 3 a z + a z - 5 a z - 8 a z + 8 6 10 6 3 7 5 7 7 7 9 7 2 a z + 4 a z + 3 a z + 6 a z + 7 a z + 4 a z + 4 8 6 8 8 8 5 9 7 9 3 a z + 6 a z + 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 78]], Vassiliev[3][Knot[10, 78]]}
Out[19]=	{3, -5}
In[20]:=	Kh[Knot[10, 78]][q, t]
Out[20]=	3 4 1 2 1 3 2 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 q q q t q t q t q t q t q t 3 5 6 6 5 5 6 3 ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 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 5 t 2 t 2 ---- + -- + --- + q t 5 3 q q t q
In[21]:=	ColouredJones[Knot[10, 78], 2][q]
Out[21]=	-28 3 -26 7 14 7 20 41 18 45 79 q - --- + q + --- - --- + --- + --- - --- + --- + --- - --- + 27 25 24 23 22 21 20 19 18 q q q q q q q q q 24 75 104 16 93 100 -11 91 73 16 69 37 --- + --- - --- + --- + --- - --- - q + --- - -- - -- + -- - -- - 17 16 15 14 13 12 10 9 8 7 6 q q q q q q q q q q q 21 37 10 13 11 2 -- + -- - -- - -- + -- - 3 q + q 5 4 3 2 q q q q q

See/edit the Rolfsen_Splice_Template. Back to the top.

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