10 73

10_72

10_74

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10 73 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,6,11,5 X8394 X2,9,3,10 X16,14,17,13 X14,7,15,8 X6,15,7,16 X20,17,1,18 X18,11,19,12 X12,19,13,20
Gauss code	1, -4, 3, -1, 2, -7, 6, -3, 4, -2, 9, -10, 5, -6, 7, -5, 8, -9, 10, -8
Dowker-Thistlethwaite code	4 8 10 14 2 18 16 6 20 12
Conway Notation	[211,21,2+]

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 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-10][-2] Hyperbolic Volume 13.7069 A-Polynomial See Data:10 73/A-polynomial

[edit Notes for 10 73's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for 10 73's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-7t^{2}+20t-27+20t^{-1}-7t^{-2}+t^{-3}$
Conway polynomial	$z^{6}-z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 83, -2 }
Jones polynomial	$-q^{2}+4q-7+11q^{-1}-13q^{-2}+14q^{-3}-13q^{-4}+10q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{8}+3z^{2}a^{6}+3a^{6}-3z^{4}a^{4}-6z^{2}a^{4}-4a^{4}+z^{6}a^{2}+3z^{4}a^{2}+5z^{2}a^{2}+3a^{2}-z^{4}-z^{2}$
Kauffman polynomial (db, data sources)	$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-6z^{4}a^{8}+4z^{2}a^{8}-a^{8}+4z^{7}a^{7}-5z^{5}a^{7}+z^{3}a^{7}+3z^{8}a^{6}+3z^{6}a^{6}-14z^{4}a^{6}+12z^{2}a^{6}-3a^{6}+z^{9}a^{5}+10z^{7}a^{5}-21z^{5}a^{5}+14z^{3}a^{5}-3za^{5}+7z^{8}a^{4}-2z^{6}a^{4}-17z^{4}a^{4}+17z^{2}a^{4}-4a^{4}+z^{9}a^{3}+12z^{7}a^{3}-26z^{5}a^{3}+16z^{3}a^{3}-3za^{3}+4z^{8}a^{2}+2z^{6}a^{2}-16z^{4}a^{2}+12z^{2}a^{2}-3a^{2}+6z^{7}a-10z^{5}a+4z^{3}a-za+4z^{6}-7z^{4}+3z^{2}+z^{5}a^{-1}-z^{3}a^{-1}$
The A2 invariant	$-q^{26}-q^{24}+2q^{22}+3q^{16}-3q^{14}-q^{10}-q^{8}+3q^{6}-2q^{4}+4q^{2}-q^{-2}+2q^{-4}-q^{-6}$
The G2 invariant	$q^{128}-2q^{126}+5q^{124}-8q^{122}+8q^{120}-7q^{118}-2q^{116}+16q^{114}-31q^{112}+46q^{110}-52q^{108}+38q^{106}-8q^{104}-43q^{102}+98q^{100}-138q^{98}+143q^{96}-104q^{94}+19q^{92}+89q^{90}-183q^{88}+235q^{86}-206q^{84}+111q^{82}+22q^{80}-145q^{78}+206q^{76}-180q^{74}+88q^{72}+41q^{70}-135q^{68}+154q^{66}-84q^{64}-50q^{62}+184q^{60}-258q^{58}+227q^{56}-101q^{54}-87q^{52}+259q^{50}-356q^{48}+339q^{46}-215q^{44}+20q^{42}+165q^{40}-285q^{38}+298q^{36}-206q^{34}+58q^{32}+88q^{30}-170q^{28}+162q^{26}-69q^{24}-56q^{22}+161q^{20}-188q^{18}+126q^{16}+2q^{14}-141q^{12}+238q^{10}-247q^{8}+177q^{6}-51q^{4}-82q^{2}+173-199q^{-2}+165q^{-4}-88q^{-6}+7q^{-8}+53q^{-10}-83q^{-12}+78q^{-14}-52q^{-16}+26q^{-18}-q^{-20}-12q^{-22}+14q^{-24}-13q^{-26}+7q^{-28}-3q^{-30}+q^{-32}$

Further Quantum Invariants

Further quantum knot invariants for 10_73.

A1 Invariants.

Weight	Invariant
1	$-q^{17}+2q^{15}-3q^{13}+4q^{11}-3q^{9}+q^{7}+q^{5}-2q^{3}+4q-3q^{-1}+3q^{-3}-q^{-5}$
2	$q^{48}-2q^{46}-q^{44}+7q^{42}-7q^{40}-6q^{38}+21q^{36}-12q^{34}-20q^{32}+34q^{30}-6q^{28}-32q^{26}+29q^{24}+8q^{22}-26q^{20}+5q^{18}+16q^{16}-4q^{14}-20q^{12}+16q^{10}+19q^{8}-34q^{6}+6q^{4}+32q^{2}-29-6q^{-2}+27q^{-4}-12q^{-6}-9q^{-8}+11q^{-10}-q^{-12}-3q^{-14}+q^{-16}$
3	$-q^{93}+2q^{91}+q^{89}-3q^{87}-4q^{85}+7q^{83}+9q^{81}-15q^{79}-18q^{77}+24q^{75}+38q^{73}-34q^{71}-69q^{69}+39q^{67}+111q^{65}-31q^{63}-158q^{61}+2q^{59}+203q^{57}+40q^{55}-224q^{53}-94q^{51}+218q^{49}+147q^{47}-189q^{45}-179q^{43}+132q^{41}+190q^{39}-62q^{37}-182q^{35}-7q^{33}+154q^{31}+75q^{29}-117q^{27}-135q^{25}+70q^{23}+187q^{21}-22q^{19}-221q^{17}-33q^{15}+237q^{13}+88q^{11}-223q^{9}-136q^{7}+191q^{5}+168q^{3}-138q-173q^{-1}+75q^{-3}+160q^{-5}-26q^{-7}-124q^{-9}-8q^{-11}+82q^{-13}+23q^{-15}-44q^{-17}-24q^{-19}+22q^{-21}+13q^{-23}-6q^{-25}-7q^{-27}+q^{-29}+3q^{-31}-q^{-33}$
4	$q^{152}-2q^{150}-q^{148}+3q^{146}+4q^{142}-10q^{140}-4q^{138}+16q^{136}+3q^{134}+8q^{132}-44q^{130}-23q^{128}+61q^{126}+49q^{124}+30q^{122}-150q^{120}-128q^{118}+124q^{116}+220q^{114}+181q^{112}-321q^{110}-456q^{108}+43q^{106}+518q^{104}+655q^{102}-331q^{100}-994q^{98}-452q^{96}+641q^{94}+1412q^{92}+153q^{90}-1329q^{88}-1285q^{86}+208q^{84}+1910q^{82}+1010q^{80}-1006q^{78}-1852q^{76}-622q^{74}+1662q^{72}+1604q^{70}-189q^{68}-1684q^{66}-1238q^{64}+821q^{62}+1548q^{60}+591q^{58}-975q^{56}-1357q^{54}-113q^{52}+1058q^{50}+1082q^{48}-144q^{46}-1164q^{44}-926q^{42}+446q^{40}+1384q^{38}+658q^{36}-833q^{34}-1601q^{32}-244q^{30}+1456q^{28}+1405q^{26}-260q^{24}-1958q^{22}-1006q^{20}+1069q^{18}+1834q^{16}+551q^{14}-1675q^{12}-1521q^{10}+235q^{8}+1602q^{6}+1187q^{4}-828q^{2}-1389-510q^{-2}+814q^{-4}+1180q^{-6}-20q^{-8}-735q^{-10}-671q^{-12}+96q^{-14}+673q^{-16}+253q^{-18}-152q^{-20}-380q^{-22}-146q^{-24}+210q^{-26}+150q^{-28}+47q^{-30}-110q^{-32}-91q^{-34}+32q^{-36}+33q^{-38}+33q^{-40}-14q^{-42}-23q^{-44}+2q^{-46}+2q^{-48}+7q^{-50}-q^{-52}-3q^{-54}+q^{-56}$
5	$-q^{225}+2q^{223}+q^{221}-3q^{219}-q^{213}+5q^{211}+3q^{209}-12q^{207}-6q^{205}+10q^{203}+16q^{201}+17q^{199}-10q^{197}-55q^{195}-55q^{193}+27q^{191}+123q^{189}+126q^{187}-14q^{185}-229q^{183}-306q^{181}-61q^{179}+398q^{177}+625q^{175}+264q^{173}-543q^{171}-1137q^{169}-763q^{167}+587q^{165}+1857q^{163}+1653q^{161}-320q^{159}-2637q^{157}-3055q^{155}-535q^{153}+3253q^{151}+4911q^{149}+2191q^{147}-3332q^{145}-6933q^{143}-4701q^{141}+2443q^{139}+8665q^{137}+7895q^{135}-411q^{133}-9580q^{131}-11166q^{129}-2720q^{127}+9170q^{125}+13925q^{123}+6507q^{121}-7380q^{119}-15506q^{117}-10190q^{115}+4392q^{113}+15509q^{111}+13140q^{109}-786q^{107}-14023q^{105}-14803q^{103}-2703q^{101}+11307q^{99}+15010q^{97}+5640q^{95}-8001q^{93}-13975q^{91}-7660q^{89}+4641q^{87}+12053q^{85}+8802q^{83}-1525q^{81}-9779q^{79}-9322q^{77}-1139q^{75}+7490q^{73}+9474q^{71}+3499q^{69}-5335q^{67}-9620q^{65}-5685q^{63}+3310q^{61}+9789q^{59}+7910q^{57}-1225q^{55}-9931q^{53}-10218q^{51}-1116q^{49}+9775q^{47}+12500q^{45}+3829q^{43}-9018q^{41}-14397q^{39}-6894q^{37}+7358q^{35}+15572q^{33}+9970q^{31}-4761q^{29}-15524q^{27}-12623q^{25}+1377q^{23}+14091q^{21}+14345q^{19}+2231q^{17}-11319q^{15}-14647q^{13}-5480q^{11}+7621q^{9}+13462q^{7}+7753q^{5}-3742q^{3}-10984q-8606q^{-1}+308q^{-3}+7784q^{-5}+8132q^{-7}+2075q^{-9}-4603q^{-11}-6592q^{-13}-3225q^{-15}+1961q^{-17}+4625q^{-19}+3290q^{-21}-200q^{-23}-2761q^{-25}-2676q^{-27}-646q^{-29}+1331q^{-31}+1794q^{-33}+867q^{-35}-438q^{-37}-1058q^{-39}-702q^{-41}+39q^{-43}+494q^{-45}+450q^{-47}+117q^{-49}-203q^{-51}-254q^{-53}-90q^{-55}+60q^{-57}+107q^{-59}+60q^{-61}-5q^{-63}-47q^{-65}-33q^{-67}+5q^{-69}+15q^{-71}+8q^{-73}+2q^{-75}-2q^{-77}-7q^{-79}+q^{-81}+3q^{-83}-q^{-85}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{26}-q^{24}+2q^{22}+3q^{16}-3q^{14}-q^{10}-q^{8}+3q^{6}-2q^{4}+4q^{2}-q^{-2}+2q^{-4}-q^{-6}$
2,0	$q^{66}+q^{64}-q^{62}-4q^{60}-q^{58}+5q^{56}+q^{54}-6q^{52}+13q^{48}+3q^{46}-16q^{44}-5q^{42}+15q^{40}-q^{38}-19q^{36}+q^{34}+17q^{32}-11q^{28}+9q^{26}+5q^{24}-10q^{22}+4q^{20}+6q^{18}-11q^{16}-5q^{14}+16q^{12}+q^{10}-18q^{8}+3q^{6}+21q^{4}-4q^{2}-16+8q^{-2}+12q^{-4}-5q^{-6}-7q^{-8}+2q^{-10}+5q^{-12}-2q^{-14}-2q^{-16}+q^{-18}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{54}-2q^{52}+q^{50}+4q^{48}-9q^{46}+2q^{44}+12q^{42}-19q^{40}+2q^{38}+24q^{36}-25q^{34}-q^{32}+27q^{30}-20q^{28}-6q^{26}+17q^{24}-4q^{22}-9q^{20}-q^{18}+13q^{16}-5q^{14}-18q^{12}+23q^{10}+4q^{8}-27q^{6}+23q^{4}+8q^{2}-22+15q^{-2}+5q^{-4}-12q^{-6}+6q^{-8}+q^{-10}-3q^{-12}+q^{-14}$
1,0,0	$-q^{35}-q^{33}-q^{31}+2q^{29}+3q^{25}+3q^{21}-4q^{19}-3q^{15}-q^{13}+q^{9}+3q^{7}-q^{5}+4q^{3}-q+2q^{-1}-2q^{-3}+2q^{-5}-q^{-7}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{54}+2q^{52}-5q^{50}+8q^{48}-13q^{46}+18q^{44}-24q^{42}+29q^{40}-30q^{38}+30q^{36}-23q^{34}+15q^{32}-q^{30}-14q^{28}+30q^{26}-45q^{24}+54q^{22}-61q^{20}+59q^{18}-55q^{16}+43q^{14}-28q^{12}+13q^{10}+4q^{8}-15q^{6}+27q^{4}-30q^{2}+32-29q^{-2}+25q^{-4}-18q^{-6}+12q^{-8}-7q^{-10}+3q^{-12}-q^{-14}$
1,0	$q^{88}-2q^{84}-2q^{82}+3q^{80}+6q^{78}-q^{76}-11q^{74}-7q^{72}+11q^{70}+18q^{68}-3q^{66}-26q^{64}-12q^{62}+23q^{60}+28q^{58}-10q^{56}-34q^{54}-7q^{52}+30q^{50}+19q^{48}-20q^{46}-24q^{44}+10q^{42}+24q^{40}-3q^{38}-22q^{36}-2q^{34}+21q^{32}+6q^{30}-19q^{28}-11q^{26}+18q^{24}+16q^{22}-16q^{20}-23q^{18}+10q^{16}+31q^{14}+2q^{12}-32q^{10}-18q^{8}+26q^{6}+30q^{4}-8q^{2}-30-7q^{-2}+22q^{-4}+17q^{-6}-9q^{-8}-15q^{-10}-q^{-12}+9q^{-14}+4q^{-16}-3q^{-18}-3q^{-20}+q^{-24}$

G2 Invariants.

Weight

Invariant

1,0

$q^{128}-2q^{126}+5q^{124}-8q^{122}+8q^{120}-7q^{118}-2q^{116}+16q^{114}-31q^{112}+46q^{110}-52q^{108}+38q^{106}-8q^{104}-43q^{102}+98q^{100}-138q^{98}+143q^{96}-104q^{94}+19q^{92}+89q^{90}-183q^{88}+235q^{86}-206q^{84}+111q^{82}+22q^{80}-145q^{78}+206q^{76}-180q^{74}+88q^{72}+41q^{70}-135q^{68}+154q^{66}-84q^{64}-50q^{62}+184q^{60}-258q^{58}+227q^{56}-101q^{54}-87q^{52}+259q^{50}-356q^{48}+339q^{46}-215q^{44}+20q^{42}+165q^{40}-285q^{38}+298q^{36}-206q^{34}+58q^{32}+88q^{30}-170q^{28}+162q^{26}-69q^{24}-56q^{22}+161q^{20}-188q^{18}+126q^{16}+2q^{14}-141q^{12}+238q^{10}-247q^{8}+177q^{6}-51q^{4}-82q^{2}+173-199q^{-2}+165q^{-4}-88q^{-6}+7q^{-8}+53q^{-10}-83q^{-12}+78q^{-14}-52q^{-16}+26q^{-18}-q^{-20}-12q^{-22}+14q^{-24}-13q^{-26}+7q^{-28}-3q^{-30}+q^{-32}$

.

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 73"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-7t^{2}+20t-27+20t^{-1}-7t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

In[6]:=

Alexander[K, 2][t]

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

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 83, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{2}+4q-7+11q^{-1}-13q^{-2}+14q^{-3}-13q^{-4}+10q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-6z^{4}a^{8}+4z^{2}a^{8}-a^{8}+4z^{7}a^{7}-5z^{5}a^{7}+z^{3}a^{7}+3z^{8}a^{6}+3z^{6}a^{6}-14z^{4}a^{6}+12z^{2}a^{6}-3a^{6}+z^{9}a^{5}+10z^{7}a^{5}-21z^{5}a^{5}+14z^{3}a^{5}-3za^{5}+7z^{8}a^{4}-2z^{6}a^{4}-17z^{4}a^{4}+17z^{2}a^{4}-4a^{4}+z^{9}a^{3}+12z^{7}a^{3}-26z^{5}a^{3}+16z^{3}a^{3}-3za^{3}+4z^{8}a^{2}+2z^{6}a^{2}-16z^{4}a^{2}+12z^{2}a^{2}-3a^{2}+6z^{7}a-10z^{5}a+4z^{3}a-za+4z^{6}-7z^{4}+3z^{2}+z^{5}a^{-1}-z^{3}a^{-1}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {10_83, ...}

Vassiliev invariants

V2 and V3:

(1, -2)

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 $4$ $-16$ $8$ ${\frac {206}{3}}$ ${\frac {34}{3}}$ $-64$ $-{\frac {832}{3}}$ $-{\frac {160}{3}}$ $-48$ ${\frac {32}{3}}$ $128$ ${\frac {824}{3}}$ ${\frac {136}{3}}$ ${\frac {31231}{30}}$ $-{\frac {1582}{15}}$ ${\frac {26702}{45}}$ ${\frac {257}{18}}$ ${\frac {2431}{30}}$

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 $s=$-2 is the signature of 10 73. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{7}-4q^{6}+2q^{5}+13q^{4}-24q^{3}-q^{2}+52q-57-24q^{-1}+113q^{-2}-83q^{-3}-64q^{-4}+166q^{-5}-86q^{-6}-100q^{-7}+182q^{-8}-66q^{-9}-111q^{-10}+151q^{-11}-32q^{-12}-90q^{-13}+90q^{-14}-6q^{-15}-50q^{-16}+36q^{-17}+2q^{-18}-17q^{-19}+9q^{-20}+q^{-21}-3q^{-22}+q^{-23}$
3	$-q^{15}+4q^{14}-2q^{13}-8q^{12}+23q^{10}+7q^{9}-54q^{8}-20q^{7}+90q^{6}+66q^{5}-144q^{4}-136q^{3}+188q^{2}+252q-229-384q^{-1}+223q^{-2}+558q^{-3}-206q^{-4}-711q^{-5}+136q^{-6}+869q^{-7}-57q^{-8}-981q^{-9}-52q^{-10}+1068q^{-11}+152q^{-12}-1098q^{-13}-257q^{-14}+1086q^{-15}+344q^{-16}-1019q^{-17}-418q^{-18}+911q^{-19}+464q^{-20}-767q^{-21}-476q^{-22}+600q^{-23}+454q^{-24}-431q^{-25}-405q^{-26}+288q^{-27}+324q^{-28}-167q^{-29}-242q^{-30}+87q^{-31}+164q^{-32}-40q^{-33}-100q^{-34}+15q^{-35}+56q^{-36}-5q^{-37}-28q^{-38}+q^{-39}+14q^{-40}-2q^{-41}-4q^{-42}-q^{-43}+3q^{-44}-q^{-45}$
4	$q^{26}-4q^{25}+2q^{24}+8q^{23}-5q^{22}+q^{21}-29q^{20}+11q^{19}+55q^{18}-5q^{17}-152q^{15}-8q^{14}+212q^{13}+98q^{12}+60q^{11}-508q^{10}-242q^{9}+440q^{8}+503q^{7}+480q^{6}-1085q^{5}-1009q^{4}+376q^{3}+1218q^{2}+1680q-1451-2333q^{-1}-503q^{-2}+1779q^{-3}+3695q^{-4}-1036q^{-5}-3700q^{-6}-2259q^{-7}+1625q^{-8}+5921q^{-9}+247q^{-10}-4465q^{-11}-4334q^{-12}+673q^{-13}+7619q^{-14}+1912q^{-15}-4414q^{-16}-6034q^{-17}-684q^{-18}+8387q^{-19}+3403q^{-20}-3688q^{-21}-6972q^{-22}-2056q^{-23}+8149q^{-24}+4423q^{-25}-2462q^{-26}-6996q^{-27}-3227q^{-28}+6905q^{-29}+4805q^{-30}-896q^{-31}-6039q^{-32}-3954q^{-33}+4846q^{-34}+4359q^{-35}+599q^{-36}-4246q^{-37}-3896q^{-38}+2562q^{-39}+3129q^{-40}+1445q^{-41}-2230q^{-42}-2996q^{-43}+860q^{-44}+1636q^{-45}+1401q^{-46}-748q^{-47}-1737q^{-48}+89q^{-49}+543q^{-50}+859q^{-51}-85q^{-52}-751q^{-53}-48q^{-54}+68q^{-55}+360q^{-56}+50q^{-57}-249q^{-58}-9q^{-59}-28q^{-60}+108q^{-61}+28q^{-62}-69q^{-63}+10q^{-64}-16q^{-65}+24q^{-66}+7q^{-67}-17q^{-68}+5q^{-69}-3q^{-70}+4q^{-71}+q^{-72}-3q^{-73}+q^{-74}$
5	$-q^{40}+4q^{39}-2q^{38}-8q^{37}+5q^{36}+4q^{35}+5q^{34}+11q^{33}-12q^{32}-46q^{31}-9q^{30}+46q^{29}+70q^{28}+58q^{27}-59q^{26}-196q^{25}-173q^{24}+97q^{23}+390q^{22}+391q^{21}-15q^{20}-651q^{19}-914q^{18}-259q^{17}+1010q^{16}+1696q^{15}+912q^{14}-1114q^{13}-2891q^{12}-2289q^{11}+925q^{10}+4257q^{9}+4402q^{8}+221q^{7}-5555q^{6}-7475q^{5}-2442q^{4}+6246q^{3}+11080q^{2}+6278q-5903-14951q^{-1}-11356q^{-2}+3868q^{-3}+18322q^{-4}+17773q^{-5}-194q^{-6}-20792q^{-7}-24457q^{-8}-5299q^{-9}+21650q^{-10}+31323q^{-11}+11920q^{-12}-21046q^{-13}-37171q^{-14}-19299q^{-15}+18749q^{-16}+42086q^{-17}+26651q^{-18}-15444q^{-19}-45385q^{-20}-33551q^{-21}+11246q^{-22}+47465q^{-23}+39498q^{-24}-6773q^{-25}-48110q^{-26}-44442q^{-27}+2144q^{-28}+47752q^{-29}+48204q^{-30}+2362q^{-31}-46231q^{-32}-50921q^{-33}-6851q^{-34}+43817q^{-35}+52489q^{-36}+11196q^{-37}-40256q^{-38}-52905q^{-39}-15480q^{-40}+35634q^{-41}+52032q^{-42}+19450q^{-43}-29929q^{-44}-49654q^{-45}-22892q^{-46}+23333q^{-47}+45717q^{-48}+25424q^{-49}-16288q^{-50}-40284q^{-51}-26595q^{-52}+9323q^{-53}+33617q^{-54}+26204q^{-55}-3051q^{-56}-26358q^{-57}-24226q^{-58}-1794q^{-59}+19035q^{-60}+20888q^{-61}+5075q^{-62}-12471q^{-63}-16808q^{-64}-6549q^{-65}+7145q^{-66}+12442q^{-67}+6661q^{-68}-3302q^{-69}-8502q^{-70}-5779q^{-71}+923q^{-72}+5298q^{-73}+4429q^{-74}+299q^{-75}-2979q^{-76}-3059q^{-77}-735q^{-78}+1510q^{-79}+1909q^{-80}+717q^{-81}-662q^{-82}-1086q^{-83}-531q^{-84}+240q^{-85}+559q^{-86}+343q^{-87}-68q^{-88}-279q^{-89}-170q^{-90}+13q^{-91}+100q^{-92}+98q^{-93}+9q^{-94}-64q^{-95}-30q^{-96}+10q^{-97}+4q^{-98}+16q^{-99}+9q^{-100}-19q^{-101}-3q^{-102}+9q^{-103}-2q^{-104}+3q^{-106}-4q^{-107}-q^{-108}+3q^{-109}-q^{-110}$
6	$q^{57}-4q^{56}+2q^{55}+8q^{54}-5q^{53}-4q^{52}-10q^{51}+13q^{50}-10q^{49}+3q^{48}+60q^{47}-21q^{46}-40q^{45}-80q^{44}+22q^{43}-5q^{42}+61q^{41}+286q^{40}+12q^{39}-178q^{38}-448q^{37}-157q^{36}-135q^{35}+324q^{34}+1215q^{33}+621q^{32}-195q^{31}-1615q^{30}-1455q^{29}-1478q^{28}+365q^{27}+3740q^{26}+3768q^{25}+1917q^{24}-2962q^{23}-5348q^{22}-7543q^{21}-3334q^{20}+6610q^{19}+12056q^{18}+11873q^{17}+1241q^{16}-9581q^{15}-22247q^{14}-19489q^{13}+803q^{12}+21987q^{11}+34991q^{10}+23437q^{9}-1149q^{8}-40193q^{7}-54756q^{6}-30334q^{5}+16101q^{4}+63341q^{3}+71447q^{2}+40497q-39205-98262q^{-1}-95029q^{-2}-29381q^{-3}+70285q^{-4}+131292q^{-5}+122734q^{-6}+6659q^{-7}-119636q^{-8}-175737q^{-9}-120110q^{-10}+29122q^{-11}+169919q^{-12}+224103q^{-13}+100467q^{-14}-92576q^{-15}-237228q^{-16}-231371q^{-17}-60200q^{-18}+162216q^{-19}+307836q^{-20}+215010q^{-21}-19462q^{-22}-255369q^{-23}-326396q^{-24}-169328q^{-25}+111873q^{-26}+350715q^{-27}+314825q^{-28}+72448q^{-29}-233442q^{-30}-383771q^{-31}-265787q^{-32}+42953q^{-33}+354768q^{-34}+381039q^{-35}+155815q^{-36}-190614q^{-37}-404833q^{-38}-334412q^{-39}-23710q^{-40}+334111q^{-41}+414974q^{-42}+220484q^{-43}-140794q^{-44}-399768q^{-45}-377429q^{-46}-83357q^{-47}+296595q^{-48}+423848q^{-49}+270326q^{-50}-83847q^{-51}-371723q^{-52}-399730q^{-53}-141303q^{-54}+238295q^{-55}+406706q^{-56}+308004q^{-57}-13867q^{-58}-313815q^{-59}-396214q^{-60}-197255q^{-61}+153731q^{-62}+353656q^{-63}+323626q^{-64}+64742q^{-65}-221298q^{-66}-353478q^{-67}-235511q^{-68}+52358q^{-69}+260334q^{-70}+299383q^{-71}+129465q^{-72}-108265q^{-73}-266947q^{-74}-233351q^{-75}-36508q^{-76}+144548q^{-77}+229746q^{-78}+152717q^{-79}-8930q^{-80}-157047q^{-81}-184327q^{-82}-81848q^{-83}+43312q^{-84}+136431q^{-85}+127673q^{-86}+45298q^{-87}-61599q^{-88}-110955q^{-89}-77876q^{-90}-13719q^{-91}+56081q^{-92}+77222q^{-93}+51529q^{-94}-7043q^{-95}-47644q^{-96}-47814q^{-97}-26772q^{-98}+10952q^{-99}+32755q^{-100}+33170q^{-101}+9375q^{-102}-12557q^{-103}-19668q^{-104}-18099q^{-105}-3439q^{-106}+8773q^{-107}+14550q^{-108}+7515q^{-109}-718q^{-110}-4963q^{-111}-7712q^{-112}-3764q^{-113}+824q^{-114}+4639q^{-115}+3006q^{-116}+889q^{-117}-360q^{-118}-2299q^{-119}-1624q^{-120}-432q^{-121}+1179q^{-122}+725q^{-123}+367q^{-124}+299q^{-125}-504q^{-126}-453q^{-127}-249q^{-128}+291q^{-129}+88q^{-130}+43q^{-131}+162q^{-132}-85q^{-133}-89q^{-134}-77q^{-135}+87q^{-136}-6q^{-137}-20q^{-138}+48q^{-139}-14q^{-140}-10q^{-141}-19q^{-142}+26q^{-143}-2q^{-144}-13q^{-145}+10q^{-146}-3q^{-147}-3q^{-149}+4q^{-150}+q^{-151}-3q^{-152}+q^{-153}$
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, 73]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], X[16, 14, 17, 13], X[14, 7, 15, 8], X[6, 15, 7, 16], X[20, 17, 1, 18], X[18, 11, 19, 12], X[12, 19, 13, 20]]
In[3]:=	GaussCode[Knot[10, 73]]
Out[3]=	GaussCode[1, -4, 3, -1, 2, -7, 6, -3, 4, -2, 9, -10, 5, -6, 7, -5, 8, -9, 10, -8]
In[4]:=	DTCode[Knot[10, 73]]
Out[4]=	DTCode[4, 8, 10, 14, 2, 18, 16, 6, 20, 12]
In[5]:=	br = BR[Knot[10, 73]]
Out[5]=	BR[5, {-1, -1, -2, 1, -2, -1, 3, -2, 3, -4, 3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 73]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 73]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 73]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 73]][t]
Out[10]=	-3 7 20 2 3 -27 + t - -- + -- + 20 t - 7 t + t 2 t t
In[11]:=	Conway[Knot[10, 73]][z]
Out[11]=	2 4 6 1 + z - z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 73]}
In[13]:=	{KnotDet[Knot[10, 73]], KnotSignature[Knot[10, 73]]}
Out[13]=	{83, -2}
In[14]:=	Jones[Knot[10, 73]][q]
Out[14]=	-8 3 6 10 13 14 13 11 2 -7 - q + -- - -- + -- - -- + -- - -- + -- + 4 q - q 7 6 5 4 3 2 q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 73], Knot[10, 83]}
In[16]:=	A2Invariant[Knot[10, 73]][q]
Out[16]=	-26 -24 2 3 3 -10 -8 3 2 4 2 -q - q + --- + --- - --- - q - q + -- - -- + -- - q + 22 16 14 6 4 2 q q q q q q 4 6 2 q - q
In[17]:=	HOMFLYPT[Knot[10, 73]][a, z]
Out[17]=	2 4 6 8 2 2 2 4 2 6 2 4 3 a - 4 a + 3 a - a - z + 5 a z - 6 a z + 3 a z - z + 2 4 4 4 2 6 3 a z - 3 a z + a z
In[18]:=	Kauffman[Knot[10, 73]][a, z]
Out[18]=	2 4 6 8 3 5 9 2 -3 a - 4 a - 3 a - a - a z - 3 a z - 3 a z + a z + 3 z + 3 2 2 4 2 6 2 8 2 z 3 3 3 12 a z + 17 a z + 12 a z + 4 a z - -- + 4 a z + 16 a z + a 5 3 7 3 9 3 4 2 4 4 4 6 4 14 a z + a z - 2 a z - 7 z - 16 a z - 17 a z - 14 a z - 5 8 4 z 5 3 5 5 5 7 5 9 5 6 a z + -- - 10 a z - 26 a z - 21 a z - 5 a z + a z + a 6 2 6 4 6 6 6 8 6 7 3 7 4 z + 2 a z - 2 a z + 3 a z + 3 a z + 6 a z + 12 a z + 5 7 7 7 2 8 4 8 6 8 3 9 5 9 10 a z + 4 a z + 4 a z + 7 a z + 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 73]], Vassiliev[3][Knot[10, 73]]}
Out[19]=	{1, -2}
In[20]:=	Kh[Knot[10, 73]][q, t]
Out[20]=	5 7 1 2 1 4 2 6 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 q q t q t q t q t q t q t q t 7 6 7 7 6 7 3 t 2 ----- + ----- + ----- + ----- + ---- + ---- + --- + 4 q t + q t + 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t 3 2 5 3 3 q t + q t
In[21]:=	ColouredJones[Knot[10, 73], 2][q]
Out[21]=	-23 3 -21 9 17 2 36 50 6 90 -57 + q - --- + q + --- - --- + --- + --- - --- - --- + --- - 22 20 19 18 17 16 15 14 q q q q q q q q 90 32 151 111 66 182 100 86 166 64 83 113 --- - --- + --- - --- - -- + --- - --- - -- + --- - -- - -- + --- - 13 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q 24 2 3 4 5 6 7 -- + 52 q - q - 24 q + 13 q + 2 q - 4 q + q q

See/edit the Rolfsen_Splice_Template. Back to the top.

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