10 141

10_140

10_142

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10 141 Quick Notes

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-]

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]=

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

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 141]]
Out[2]=	10
In[3]:=	PD[Knot[10, 141]]
Out[3]=	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[4]:=	GaussCode[Knot[10, 141]]
Out[4]=	GaussCode[-1, 10, -2, 1, 3, -5, 4, -9, -10, 2, -7, 8, 9, -3, 5, -4, -6, 7, -8, 6]
In[5]:=	BR[Knot[10, 141]]
Out[5]=	BR[3, {1, 1, 1, 1, -2, -1, -1, -1, -2, -2}]
In[6]:=	alex = Alexander[Knot[10, 141]][t]
Out[6]=	-3 3 4 2 3 5 - t + -- - - - 4 t + 3 t - t 2 t t
In[7]:=	Conway[Knot[10, 141]][z]
Out[7]=	2 4 6 1 - z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 5], Knot[10, 141]}
In[9]:=	{KnotDet[Knot[10, 141]], KnotSignature[Knot[10, 141]]}
Out[9]=	{21, 0}
In[10]:=	J=Jones[Knot[10, 141]][q]
Out[10]=	-6 2 2 3 4 3 2 3 + q - -- + -- - -- + -- - - - 2 q + q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 141]}
In[12]:=	A2Invariant[Knot[10, 141]][q]
Out[12]=	-18 -12 -10 -8 -4 2 6 q - q - q + q + q + q + q
In[13]:=	Kauffman[Knot[10, 141]][a, z]
Out[13]=	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[14]:=	{Vassiliev[2][Knot[10, 141]], Vassiliev[3][Knot[10, 141]]}
Out[14]=	{0, 1}
In[15]:=	Kh[Knot[10, 141]][q, t]
Out[15]=	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