10 141

10_140

10_142

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10 141 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X14,6,15,5 X16,8,17,7 X6,16,7,15 X17,20,18,1 X11,18,12,19 X19,12,20,13 X8,14,9,13 X9,2,10,3
Gauss code	-1, 10, -2, 1, 3, -5, 4, -9, -10, 2, -7, 8, 9, -3, 5, -4, -6, 7, -8, 6
Dowker-Thistlethwaite code	4 10 -14 -16 2 18 -8 -6 20 12
Conway Notation	[4,21,21-]

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 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-3] Hyperbolic Volume 7.93647 A-Polynomial See Data:10 141/A-polynomial

[edit Notes for 10 141's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for 10 141's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+3t^{2}-4t+5-4t^{-1}+3t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-3z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 21, 0 }
Jones polynomial	$q^{2}-2q+3-3q^{-1}+4q^{-2}-3q^{-3}+2q^{-4}-2q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{6}+a^{4}z^{4}-5a^{2}z^{4}+z^{4}+3a^{4}z^{2}-7a^{2}z^{2}+3z^{2}+a^{4}-2a^{2}+2$
Kauffman polynomial (db, data sources)	$a^{4}z^{8}+a^{2}z^{8}+2a^{5}z^{7}+3a^{3}z^{7}+az^{7}+a^{6}z^{6}-3a^{4}z^{6}-4a^{2}z^{6}-9a^{5}z^{5}-12a^{3}z^{5}-3az^{5}-4a^{6}z^{4}+a^{4}z^{4}+8a^{2}z^{4}+3z^{4}+10a^{5}z^{3}+13a^{3}z^{3}+5az^{3}+2z^{3}a^{-1}+3a^{6}z^{2}-a^{4}z^{2}-9a^{2}z^{2}+z^{2}a^{-2}-4z^{2}-2a^{5}z-4a^{3}z-3az-za^{-1}+a^{4}+2a^{2}+2$
The A2 invariant	$q^{18}-q^{12}-q^{10}+q^{8}+q^{4}+q^{-2}+q^{-6}$
The G2 invariant	$q^{94}-q^{92}+2q^{90}-3q^{88}+q^{86}-q^{84}-2q^{82}+6q^{80}-7q^{78}+6q^{76}-2q^{74}-3q^{72}+6q^{70}-5q^{68}+4q^{66}-3q^{62}+7q^{60}-q^{58}-2q^{56}+4q^{54}-8q^{52}+7q^{50}-7q^{46}+3q^{44}-4q^{42}+9q^{40}-4q^{38}-2q^{36}-q^{34}-3q^{32}+7q^{30}-6q^{28}-q^{26}+q^{22}+5q^{20}-3q^{18}-3q^{16}+5q^{14}-5q^{12}+4q^{10}-6q^{6}+8q^{4}-3q^{2}+2+2q^{-2}-3q^{-4}+2q^{-6}-q^{-8}+q^{-10}+2q^{-12}+2q^{-18}+2q^{-24}-2q^{-26}+q^{-28}-q^{-30}-q^{-32}+q^{-34}-q^{-36}+q^{-38}$

Further Quantum Invariants

Further quantum knot invariants for 10_141.

A1 Invariants.

Weight	Invariant
1	$q^{13}-q^{11}-q^{7}+q^{5}+q^{3}+q^{-1}-q^{-3}+q^{-5}$
2	$q^{38}-q^{36}-2q^{34}+2q^{32}+q^{30}-2q^{28}+q^{26}+3q^{24}-q^{22}-2q^{20}-q^{16}-2q^{14}+q^{12}+2q^{10}-q^{8}+q^{6}+3q^{4}+q^{2}-2+q^{-2}+q^{-4}-2q^{-6}+q^{-14}$
3	$q^{75}-q^{73}-2q^{71}+3q^{67}+3q^{65}-3q^{63}-4q^{61}+q^{59}+5q^{57}+2q^{55}-5q^{53}-6q^{51}+q^{49}+5q^{47}+3q^{45}-3q^{43}-4q^{41}+3q^{39}+7q^{37}+2q^{35}-7q^{33}-4q^{31}+4q^{29}+3q^{27}-6q^{25}-3q^{23}+4q^{21}+2q^{19}-4q^{17}-2q^{15}+4q^{13}+4q^{11}-3q^{7}-2q^{5}+4q^{3}+7q-2q^{-1}-8q^{-3}-2q^{-5}+8q^{-7}+5q^{-9}-5q^{-11}-5q^{-13}+q^{-15}+5q^{-17}+q^{-19}-3q^{-21}-2q^{-23}+2q^{-27}$
5	$q^{185}-q^{183}-2q^{181}+q^{177}+3q^{175}+3q^{173}+q^{171}-5q^{169}-7q^{167}-4q^{165}+q^{163}+8q^{161}+12q^{159}+7q^{157}-8q^{155}-15q^{153}-14q^{151}-5q^{149}+12q^{147}+23q^{145}+19q^{143}+q^{141}-16q^{139}-27q^{137}-23q^{135}-q^{133}+20q^{131}+30q^{129}+22q^{127}+3q^{125}-22q^{123}-37q^{121}-30q^{119}-3q^{117}+31q^{115}+50q^{113}+40q^{111}-2q^{109}-47q^{107}-65q^{105}-36q^{103}+26q^{101}+71q^{99}+68q^{97}+12q^{95}-64q^{93}-91q^{91}-48q^{89}+33q^{87}+88q^{85}+76q^{83}-77q^{79}-87q^{77}-23q^{75}+57q^{73}+87q^{71}+46q^{69}-34q^{67}-79q^{65}-50q^{63}+19q^{61}+64q^{59}+51q^{57}-7q^{55}-52q^{53}-43q^{51}+4q^{49}+35q^{47}+29q^{45}-7q^{43}-30q^{41}-18q^{39}+11q^{37}+24q^{35}+11q^{33}-15q^{31}-24q^{29}-16q^{27}+9q^{25}+31q^{23}+30q^{21}+9q^{19}-25q^{17}-45q^{15}-36q^{13}+9q^{11}+59q^{9}+71q^{7}+22q^{5}-55q^{3}-96q-63q^{-1}+29q^{-3}+110q^{-5}+101q^{-7}+6q^{-9}-97q^{-11}-115q^{-13}-43q^{-15}+62q^{-17}+111q^{-19}+63q^{-21}-28q^{-23}-82q^{-25}-61q^{-27}-2q^{-29}+45q^{-31}+48q^{-33}+13q^{-35}-19q^{-37}-25q^{-39}-11q^{-41}+4q^{-43}+9q^{-45}+7q^{-47}-2q^{-49}-3q^{-51}+q^{-53}+3q^{-55}+q^{-57}-q^{-59}-3q^{-61}-3q^{-63}+q^{-65}+q^{-67}+q^{-69}$
6	$q^{258}-q^{256}-2q^{254}+q^{250}+3q^{248}+q^{246}+3q^{244}-q^{242}-7q^{240}-6q^{238}-4q^{236}+2q^{234}+5q^{232}+15q^{230}+11q^{228}+q^{226}-10q^{224}-19q^{222}-18q^{220}-15q^{218}+12q^{216}+27q^{214}+31q^{212}+22q^{210}-24q^{206}-48q^{204}-39q^{202}-17q^{200}+17q^{198}+43q^{196}+56q^{194}+45q^{192}+4q^{190}-32q^{188}-61q^{186}-61q^{184}-41q^{182}+q^{180}+51q^{178}+80q^{176}+79q^{174}+43q^{172}-14q^{170}-82q^{168}-121q^{166}-102q^{164}-32q^{162}+62q^{160}+142q^{158}+165q^{156}+94q^{154}-39q^{152}-158q^{150}-211q^{148}-158q^{146}-6q^{144}+169q^{142}+259q^{140}+209q^{138}+49q^{136}-157q^{134}-289q^{132}-257q^{130}-74q^{128}+154q^{126}+300q^{124}+289q^{122}+95q^{120}-158q^{118}-319q^{116}-293q^{114}-97q^{112}+154q^{110}+327q^{108}+286q^{106}+68q^{104}-183q^{102}-312q^{100}-249q^{98}-34q^{96}+211q^{94}+301q^{92}+189q^{90}-35q^{88}-217q^{86}-249q^{84}-119q^{82}+97q^{80}+220q^{78}+179q^{76}+28q^{74}-122q^{72}-174q^{70}-104q^{68}+37q^{66}+126q^{64}+106q^{62}+19q^{60}-64q^{58}-87q^{56}-42q^{54}+28q^{52}+59q^{50}+33q^{48}-19q^{46}-53q^{44}-41q^{42}+6q^{40}+53q^{38}+67q^{36}+37q^{34}-23q^{32}-76q^{30}-91q^{28}-52q^{26}+22q^{24}+106q^{22}+147q^{20}+102q^{18}-17q^{16}-150q^{14}-215q^{12}-166q^{10}+10q^{8}+212q^{6}+306q^{4}+218q^{2}-19-267q^{-2}-389q^{-4}-259q^{-6}+44q^{-8}+329q^{-10}+419q^{-12}+255q^{-14}-60q^{-16}-354q^{-18}-407q^{-20}-213q^{-22}+91q^{-24}+309q^{-26}+335q^{-28}+170q^{-30}-90q^{-32}-241q^{-34}-237q^{-36}-104q^{-38}+50q^{-40}+152q^{-42}+156q^{-44}+63q^{-46}-23q^{-48}-77q^{-50}-78q^{-52}-50q^{-54}+35q^{-58}+33q^{-60}+28q^{-62}+9q^{-64}-4q^{-66}-18q^{-68}-15q^{-70}-7q^{-72}-3q^{-74}+7q^{-76}+8q^{-78}+9q^{-80}+q^{-82}-2q^{-84}-4q^{-86}-5q^{-88}-q^{-90}+2q^{-94}+q^{-96}$

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	$q^{40}-q^{38}-2q^{32}+q^{30}+q^{26}+2q^{24}+q^{22}+q^{18}-2q^{14}-2q^{10}-q^{8}-2q^{6}+2q^{2}+1+3q^{-2}+3q^{-4}+q^{-10}-q^{-12}-q^{-14}+q^{-16}$
1,0,0	$q^{23}+q^{19}-q^{17}-q^{13}+q+2q^{-3}+q^{-7}$
1,0,1	$q^{66}-2q^{64}+3q^{62}-3q^{60}+q^{58}+q^{56}-6q^{54}+8q^{52}-7q^{50}+9q^{48}-4q^{46}+4q^{44}-5q^{40}+8q^{38}-15q^{36}+12q^{34}-18q^{32}+13q^{30}-8q^{28}+3q^{26}+11q^{24}-9q^{22}+17q^{20}-13q^{18}+16q^{16}-17q^{14}+8q^{12}-7q^{10}-7q^{8}+8q^{6}-8q^{4}+15q^{2}-5+12q^{-2}-2q^{-4}+q^{-6}-2q^{-8}-3q^{-10}-3q^{-14}+7q^{-16}-2q^{-18}+3q^{-20}+q^{-22}-2q^{-24}+q^{-26}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{50}-q^{42}-q^{40}-q^{38}-2q^{36}-q^{34}+2q^{32}+2q^{30}+2q^{28}+4q^{26}+6q^{24}+3q^{22}-q^{20}-3q^{16}-8q^{14}-6q^{12}-4q^{10}-4q^{8}+5q^{4}+5q^{2}+4+5q^{-2}+4q^{-4}-2q^{-8}+q^{-10}-q^{-14}+q^{-18}$
1,0,0,0	$q^{28}+q^{24}-q^{16}-q^{12}-q^{8}+q^{2}+1+q^{-2}+2q^{-4}+q^{-8}$

B2 Invariants.

Weight	Invariant
0,1	$q^{40}-q^{38}+2q^{36}-2q^{34}+2q^{32}-3q^{30}+2q^{28}-q^{26}+q^{22}-2q^{20}+3q^{18}-4q^{16}+4q^{14}-4q^{12}+4q^{10}-3q^{8}+2q^{6}+1-q^{-2}+3q^{-4}-2q^{-6}+2q^{-8}-q^{-10}+q^{-12}-q^{-14}+q^{-16}$
1,0	$q^{66}-q^{62}-q^{60}+q^{58}+q^{56}-2q^{54}-2q^{52}+q^{50}+3q^{48}-2q^{44}+3q^{40}+2q^{38}-q^{36}-2q^{34}+q^{30}-q^{26}-q^{24}+q^{22}+q^{20}-q^{18}-2q^{16}+q^{14}+2q^{12}-q^{10}-3q^{8}+2q^{4}+q^{2}-1+2q^{-4}+3q^{-6}-q^{-10}+q^{-14}+q^{-16}-q^{-20}-q^{-22}+q^{-26}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{54}-q^{52}+q^{50}-2q^{48}+2q^{46}-3q^{44}+q^{42}-2q^{40}+2q^{38}+q^{34}+2q^{32}+q^{30}+4q^{28}-q^{26}+3q^{24}-3q^{22}+2q^{20}-5q^{18}+q^{16}-6q^{14}-4q^{10}+q^{4}+3q^{2}+2+4q^{-2}+q^{-4}+4q^{-6}-q^{-8}+q^{-10}-q^{-12}+q^{-14}-q^{-16}-q^{-20}+q^{-22}$

G2 Invariants.

Weight

Invariant

1,0

$q^{94}-q^{92}+2q^{90}-3q^{88}+q^{86}-q^{84}-2q^{82}+6q^{80}-7q^{78}+6q^{76}-2q^{74}-3q^{72}+6q^{70}-5q^{68}+4q^{66}-3q^{62}+7q^{60}-q^{58}-2q^{56}+4q^{54}-8q^{52}+7q^{50}-7q^{46}+3q^{44}-4q^{42}+9q^{40}-4q^{38}-2q^{36}-q^{34}-3q^{32}+7q^{30}-6q^{28}-q^{26}+q^{22}+5q^{20}-3q^{18}-3q^{16}+5q^{14}-5q^{12}+4q^{10}-6q^{6}+8q^{4}-3q^{2}+2+2q^{-2}-3q^{-4}+2q^{-6}-q^{-8}+q^{-10}+2q^{-12}+2q^{-18}+2q^{-24}-2q^{-26}+q^{-28}-q^{-30}-q^{-32}+q^{-34}-q^{-36}+q^{-38}$

.

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["10 141"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+3t^{2}-4t+5-4t^{-1}+3t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 21, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{2}-2q+3-3q^{-1}+4q^{-2}-3q^{-3}+2q^{-4}-2q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{4}z^{8}+a^{2}z^{8}+2a^{5}z^{7}+3a^{3}z^{7}+az^{7}+a^{6}z^{6}-3a^{4}z^{6}-4a^{2}z^{6}-9a^{5}z^{5}-12a^{3}z^{5}-3az^{5}-4a^{6}z^{4}+a^{4}z^{4}+8a^{2}z^{4}+3z^{4}+10a^{5}z^{3}+13a^{3}z^{3}+5az^{3}+2z^{3}a^{-1}+3a^{6}z^{2}-a^{4}z^{2}-9a^{2}z^{2}+z^{2}a^{-2}-4z^{2}-2a^{5}z-4a^{3}z-3az-za^{-1}+a^{4}+2a^{2}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_5, ...}

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 $-4$ $8$ $8$ ${\frac {82}{3}}$ ${\frac {86}{3}}$ $-32$ $-{\frac {304}{3}}$ $-{\frac {160}{3}}$ $-24$ $-{\frac {32}{3}}$ $32$ $-{\frac {328}{3}}$ $-{\frac {344}{3}}$ ${\frac {1409}{30}}$ ${\frac {1154}{5}}$ $-{\frac {12902}{45}}$ ${\frac {1087}{18}}$ $-{\frac {2911}{30}}$

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=$0 is the signature of 10 141. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{6}-q^{5}+q^{3}-3q^{2}+3q+1-6q^{-1}+6q^{-2}+3q^{-3}-8q^{-4}+4q^{-5}+6q^{-6}-9q^{-7}+q^{-8}+7q^{-9}-8q^{-10}-q^{-11}+8q^{-12}-4q^{-13}-3q^{-14}+5q^{-15}-q^{-16}-2q^{-17}+q^{-18}$
3	$2q^{12}-2q^{11}-2q^{10}-q^{9}+6q^{8}+2q^{7}-6q^{6}-7q^{5}+6q^{4}+12q^{3}-3q^{2}-17q+18q^{-1}+6q^{-2}-20q^{-3}-6q^{-4}+17q^{-5}+9q^{-6}-16q^{-7}-6q^{-8}+11q^{-9}+7q^{-10}-10q^{-11}-4q^{-12}+4q^{-13}+4q^{-14}-q^{-15}-3q^{-16}-4q^{-17}+q^{-18}+8q^{-19}+2q^{-20}-8q^{-21}-6q^{-22}+9q^{-23}+8q^{-24}-6q^{-25}-10q^{-26}+2q^{-27}+9q^{-28}+q^{-29}-7q^{-30}-2q^{-31}+4q^{-32}+2q^{-33}-q^{-34}-2q^{-35}+q^{-36}$
4	$q^{21}+q^{20}-4q^{19}-q^{18}+q^{17}+6q^{16}+5q^{15}-13q^{14}-7q^{13}-q^{12}+21q^{11}+21q^{10}-23q^{9}-29q^{8}-20q^{7}+40q^{6}+57q^{5}-16q^{4}-56q^{3}-57q^{2}+43q+91+7q^{-1}-63q^{-2}-87q^{-3}+28q^{-4}+104q^{-5}+26q^{-6}-54q^{-7}-95q^{-8}+16q^{-9}+100q^{-10}+25q^{-11}-42q^{-12}-87q^{-13}+8q^{-14}+89q^{-15}+21q^{-16}-30q^{-17}-75q^{-18}-4q^{-19}+75q^{-20}+20q^{-21}-12q^{-22}-57q^{-23}-19q^{-24}+51q^{-25}+15q^{-26}+9q^{-27}-30q^{-28}-24q^{-29}+23q^{-30}-q^{-31}+18q^{-32}-3q^{-33}-14q^{-34}+9q^{-35}-21q^{-36}+8q^{-37}+8q^{-38}+3q^{-39}+16q^{-40}-25q^{-41}-6q^{-42}+5q^{-44}+24q^{-45}-10q^{-46}-7q^{-47}-9q^{-48}-4q^{-49}+17q^{-50}-5q^{-53}-6q^{-54}+5q^{-55}+q^{-56}+2q^{-57}-q^{-58}-2q^{-59}+q^{-60}$
5	$q^{32}-4q^{29}+2q^{27}+3q^{26}+2q^{25}-2q^{24}-8q^{23}+q^{22}+11q^{21}+5q^{20}-3q^{19}-17q^{18}-22q^{17}+7q^{16}+43q^{15}+40q^{14}-6q^{13}-64q^{12}-81q^{11}-14q^{10}+97q^{9}+131q^{8}+42q^{7}-113q^{6}-186q^{5}-86q^{4}+115q^{3}+234q^{2}+137q-104-267q^{-1}-178q^{-2}+82q^{-3}+275q^{-4}+214q^{-5}-55q^{-6}-279q^{-7}-228q^{-8}+37q^{-9}+266q^{-10}+234q^{-11}-21q^{-12}-258q^{-13}-227q^{-14}+15q^{-15}+241q^{-16}+226q^{-17}-12q^{-18}-232q^{-19}-214q^{-20}+2q^{-21}+212q^{-22}+214q^{-23}+11q^{-24}-196q^{-25}-208q^{-26}-29q^{-27}+165q^{-28}+205q^{-29}+56q^{-30}-138q^{-31}-195q^{-32}-74q^{-33}+96q^{-34}+176q^{-35}+101q^{-36}-58q^{-37}-154q^{-38}-104q^{-39}+16q^{-40}+112q^{-41}+111q^{-42}+19q^{-43}-78q^{-44}-92q^{-45}-39q^{-46}+31q^{-47}+68q^{-48}+46q^{-49}-2q^{-50}-36q^{-51}-36q^{-52}-14q^{-53}+6q^{-54}+17q^{-55}+16q^{-56}+9q^{-57}+6q^{-58}-4q^{-59}-13q^{-60}-17q^{-61}-11q^{-62}+2q^{-63}+21q^{-64}+21q^{-65}+6q^{-66}-9q^{-67}-21q^{-68}-19q^{-69}-q^{-70}+17q^{-71}+17q^{-72}+8q^{-73}-3q^{-74}-15q^{-75}-12q^{-76}+8q^{-78}+7q^{-79}+4q^{-80}-7q^{-82}-4q^{-83}+q^{-84}+2q^{-85}+q^{-86}+2q^{-87}-q^{-88}-2q^{-89}+q^{-90}$
6	$q^{45}+q^{44}-2q^{43}-q^{42}-4q^{41}+q^{40}+2q^{39}+4q^{38}+9q^{37}-3q^{36}-2q^{35}-14q^{34}-3q^{33}-6q^{32}+q^{31}+23q^{30}+10q^{29}+17q^{28}-9q^{27}-q^{26}-41q^{25}-49q^{24}-5q^{23}+11q^{22}+71q^{21}+77q^{20}+92q^{19}-45q^{18}-151q^{17}-159q^{16}-122q^{15}+67q^{14}+228q^{13}+352q^{12}+120q^{11}-177q^{10}-377q^{9}-426q^{8}-127q^{7}+281q^{6}+646q^{5}+435q^{4}-13q^{3}-467q^{2}-711q-430+151q^{-1}+768q^{-2}+683q^{-3}+224q^{-4}-379q^{-5}-805q^{-6}-632q^{-7}-25q^{-8}+719q^{-9}+748q^{-10}+357q^{-11}-260q^{-12}-760q^{-13}-673q^{-14}-109q^{-15}+645q^{-16}+709q^{-17}+372q^{-18}-207q^{-19}-700q^{-20}-643q^{-21}-123q^{-22}+598q^{-23}+662q^{-24}+360q^{-25}-173q^{-26}-648q^{-27}-617q^{-28}-154q^{-29}+528q^{-30}+617q^{-31}+383q^{-32}-90q^{-33}-561q^{-34}-597q^{-35}-243q^{-36}+387q^{-37}+547q^{-38}+435q^{-39}+60q^{-40}-410q^{-41}-556q^{-42}-366q^{-43}+171q^{-44}+417q^{-45}+467q^{-46}+242q^{-47}-186q^{-48}-444q^{-49}-456q^{-50}-74q^{-51}+202q^{-52}+404q^{-53}+371q^{-54}+65q^{-55}-226q^{-56}-415q^{-57}-247q^{-58}-49q^{-59}+208q^{-60}+345q^{-61}+226q^{-62}+27q^{-63}-221q^{-64}-236q^{-65}-195q^{-66}-20q^{-67}+162q^{-68}+194q^{-69}+159q^{-70}-15q^{-71}-76q^{-72}-145q^{-73}-110q^{-74}-13q^{-75}+42q^{-76}+106q^{-77}+38q^{-78}+43q^{-79}-12q^{-80}-39q^{-81}-36q^{-82}-38q^{-83}+12q^{-84}-32q^{-85}+24q^{-86}+27q^{-87}+29q^{-88}+21q^{-89}-2q^{-90}+13q^{-91}-61q^{-92}-26q^{-93}-15q^{-94}+9q^{-95}+21q^{-96}+27q^{-97}+49q^{-98}-20q^{-99}-15q^{-100}-28q^{-101}-17q^{-102}-13q^{-103}+5q^{-104}+40q^{-105}+4q^{-106}+9q^{-107}-6q^{-108}-8q^{-109}-17q^{-110}-10q^{-111}+13q^{-112}+q^{-113}+8q^{-114}+3q^{-115}+3q^{-116}-7q^{-117}-6q^{-118}+3q^{-119}-2q^{-120}+2q^{-121}+q^{-122}+2q^{-123}-q^{-124}-2q^{-125}+q^{-126}$
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, 141]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[14, 6, 15, 5], X[16, 8, 17, 7], X[6, 16, 7, 15], X[17, 20, 18, 1], X[11, 18, 12, 19], X[19, 12, 20, 13], X[8, 14, 9, 13], X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[10, 141]]
Out[3]=	GaussCode[-1, 10, -2, 1, 3, -5, 4, -9, -10, 2, -7, 8, 9, -3, 5, -4, -6, 7, -8, 6]
In[4]:=	DTCode[Knot[10, 141]]
Out[4]=	DTCode[4, 10, -14, -16, 2, 18, -8, -6, 20, 12]
In[5]:=	br = BR[Knot[10, 141]]
Out[5]=	BR[3, {1, 1, 1, 1, -2, -1, -1, -1, -2, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 141]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 141]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 141]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 141]][t]
Out[10]=	-3 3 4 2 3 5 - t + -- - - - 4 t + 3 t - t 2 t t
In[11]:=	Conway[Knot[10, 141]][z]
Out[11]=	2 4 6 1 - z - 3 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 5], Knot[10, 141]}
In[13]:=	{KnotDet[Knot[10, 141]], KnotSignature[Knot[10, 141]]}
Out[13]=	{21, 0}
In[14]:=	Jones[Knot[10, 141]][q]
Out[14]=	-6 2 2 3 4 3 2 3 + q - -- + -- - -- + -- - - - 2 q + q 5 4 3 2 q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 141]}
In[16]:=	A2Invariant[Knot[10, 141]][q]
Out[16]=	-18 -12 -10 -8 -4 2 6 q - q - q + q + q + q + q
In[17]:=	HOMFLYPT[Knot[10, 141]][a, z]
Out[17]=	2 4 2 2 2 4 2 4 2 4 4 4 2 6 2 - 2 a + a + 3 z - 7 a z + 3 a z + z - 5 a z + a z - a z
In[18]:=	Kauffman[Knot[10, 141]][a, z]
Out[18]=	2 2 4 z 3 5 2 z 2 2 2 + 2 a + a - - - 3 a z - 4 a z - 2 a z - 4 z + -- - 9 a z - a 2 a 3 4 2 6 2 2 z 3 3 3 5 3 4 a z + 3 a z + ---- + 5 a z + 13 a z + 10 a z + 3 z + a 2 4 4 4 6 4 5 3 5 5 5 2 6 8 a z + a z - 4 a z - 3 a z - 12 a z - 9 a z - 4 a z - 4 6 6 6 7 3 7 5 7 2 8 4 8 3 a z + a z + a z + 3 a z + 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 141]], Vassiliev[3][Knot[10, 141]]}
Out[19]=	{-1, 1}
In[20]:=	Kh[Knot[10, 141]][q, t]
Out[20]=	2 1 1 1 1 1 2 1 - + 2 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 q t q t q t q t q t q t q t 2 2 1 2 3 5 2 ----- + ----- + ---- + --- + q t + q t + q t 5 2 3 2 3 q t q t q t q t
In[21]:=	ColouredJones[Knot[10, 141], 2][q]
Out[21]=	-18 2 -16 5 3 4 8 -11 8 7 -8 1 + q - --- - q + --- - --- - --- + --- - q - --- + -- + q - 17 15 14 13 12 10 9 q q q q q q q 9 6 4 8 3 6 6 2 3 5 6 -- + -- + -- - -- + -- + -- - - + 3 q - 3 q + q - q + q 7 6 5 4 3 2 q q q q q q q

See/edit the Rolfsen_Splice_Template. Back to the top.

10_140 10_142