10 67

10_66

10_68

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10 67 Quick Notes

10 67 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X7,12,8,13 X3,11,4,10 X11,3,12,2 X5,14,6,15 X13,6,14,7 X9,18,10,19 X15,20,16,1 X19,16,20,17 X17,8,18,9
Gauss code	-1, 4, -3, 1, -5, 6, -2, 10, -7, 3, -4, 2, -6, 5, -8, 9, -10, 7, -9, 8
Dowker-Thistlethwaite code	4 10 14 12 18 2 6 20 8 16
Conway Notation	[22,3,21]

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-11][-1] Hyperbolic Volume 12.4216 A-Polynomial See Data:10 67/A-polynomial

[edit Notes for 10 67's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 67's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-4t^{2}+16t-23+16t^{-1}-4t^{-2}$
Conway polynomial	$1-4z^{4}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 63, -2 }
Jones polynomial	$q-2+5q^{-1}-8q^{-2}+10q^{-3}-10q^{-4}+10q^{-5}-8q^{-6}+5q^{-7}-3q^{-8}+q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{8}-z^{4}a^{6}-2z^{4}a^{4}-2z^{2}a^{4}-z^{4}a^{2}+z^{2}+1$
Kauffman polynomial (db, data sources)	$z^{6}a^{10}-3z^{4}a^{10}+2z^{2}a^{10}+3z^{7}a^{9}-10z^{5}a^{9}+9z^{3}a^{9}-2za^{9}+3z^{8}a^{8}-7z^{6}a^{8}+2z^{4}a^{8}+z^{9}a^{7}+5z^{7}a^{7}-21z^{5}a^{7}+19z^{3}a^{7}-6za^{7}+6z^{8}a^{6}-13z^{6}a^{6}+7z^{4}a^{6}-2z^{2}a^{6}+z^{9}a^{5}+6z^{7}a^{5}-19z^{5}a^{5}+19z^{3}a^{5}-6za^{5}+3z^{8}a^{4}-2z^{6}a^{4}-z^{4}a^{4}+2z^{2}a^{4}+4z^{7}a^{3}-6z^{5}a^{3}+7z^{3}a^{3}-2za^{3}+3z^{6}a^{2}-2z^{4}a^{2}+2z^{5}a-2z^{3}a+z^{4}-2z^{2}+1$
The A2 invariant	$q^{28}-q^{26}-q^{24}+2q^{22}-2q^{20}+q^{16}-q^{14}+2q^{12}-q^{10}+q^{8}-2q^{4}+3q^{2}+q^{-4}$
The G2 invariant	$q^{142}-2q^{140}+5q^{138}-9q^{136}+9q^{134}-8q^{132}-2q^{130}+19q^{128}-34q^{126}+47q^{124}-44q^{122}+19q^{120}+18q^{118}-60q^{116}+90q^{114}-93q^{112}+61q^{110}-2q^{108}-57q^{106}+98q^{104}-99q^{102}+65q^{100}-6q^{98}-47q^{96}+72q^{94}-67q^{92}+23q^{90}+37q^{88}-76q^{86}+85q^{84}-53q^{82}-10q^{80}+71q^{78}-119q^{76}+124q^{74}-94q^{72}+23q^{70}+56q^{68}-116q^{66}+141q^{64}-111q^{62}+48q^{60}+23q^{58}-76q^{56}+93q^{54}-70q^{52}+20q^{50}+37q^{48}-62q^{46}+60q^{44}-20q^{42}-35q^{40}+73q^{38}-83q^{36}+60q^{34}-20q^{32}-30q^{30}+65q^{28}-78q^{26}+72q^{24}-41q^{22}+6q^{20}+20q^{18}-39q^{16}+42q^{14}-37q^{12}+27q^{10}-9q^{8}-2q^{6}+12q^{4}-15q^{2}+14-10q^{-2}+7q^{-4}-2q^{-6}-q^{-8}+3q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

Further Quantum Invariants

Further quantum knot invariants for 10_67.

A1 Invariants.

Weight	Invariant
1	$q^{19}-2q^{17}+2q^{15}-3q^{13}+2q^{11}+2q^{5}-3q^{3}+3q-q^{-1}+q^{-3}$
2	$q^{54}-2q^{52}-2q^{50}+7q^{48}-2q^{46}-10q^{44}+12q^{42}+5q^{40}-18q^{38}+8q^{36}+12q^{34}-18q^{32}-q^{30}+15q^{28}-7q^{26}-7q^{24}+9q^{22}+6q^{20}-11q^{18}-5q^{16}+17q^{14}-8q^{12}-13q^{10}+19q^{8}-q^{6}-12q^{4}+10q^{2}+1-5q^{-2}+3q^{-4}-q^{-8}+q^{-10}$
3	$q^{105}-2q^{103}-2q^{101}+3q^{99}+7q^{97}-2q^{95}-16q^{93}-2q^{91}+24q^{89}+15q^{87}-30q^{85}-35q^{83}+26q^{81}+57q^{79}-10q^{77}-72q^{75}-18q^{73}+78q^{71}+46q^{69}-67q^{67}-71q^{65}+48q^{63}+90q^{61}-28q^{59}-94q^{57}+2q^{55}+92q^{53}+16q^{51}-82q^{49}-33q^{47}+69q^{45}+47q^{43}-45q^{41}-61q^{39}+17q^{37}+70q^{35}+13q^{33}-74q^{31}-48q^{29}+67q^{27}+75q^{25}-47q^{23}-90q^{21}+27q^{19}+90q^{17}-6q^{15}-73q^{13}-12q^{11}+55q^{9}+16q^{7}-33q^{5}-12q^{3}+16q+9q^{-1}-8q^{-3}-3q^{-5}+4q^{-7}+q^{-9}-2q^{-11}+q^{-13}-q^{-19}+q^{-21}$
4	$q^{172}-2q^{170}-2q^{168}+3q^{166}+3q^{164}+7q^{162}-9q^{160}-16q^{158}-q^{156}+10q^{154}+42q^{152}+2q^{150}-47q^{148}-48q^{146}-19q^{144}+102q^{142}+87q^{140}-17q^{138}-124q^{136}-167q^{134}+65q^{132}+206q^{130}+175q^{128}-58q^{126}-344q^{124}-180q^{122}+128q^{120}+394q^{118}+257q^{116}-277q^{114}-444q^{112}-224q^{110}+342q^{108}+572q^{106}+74q^{104}-434q^{102}-562q^{100}+32q^{98}+608q^{96}+406q^{94}-193q^{92}-638q^{90}-253q^{88}+423q^{86}+518q^{84}+27q^{82}-520q^{80}-355q^{78}+221q^{76}+477q^{74}+153q^{72}-350q^{70}-384q^{68}+21q^{66}+402q^{64}+285q^{62}-129q^{60}-412q^{58}-261q^{56}+237q^{54}+443q^{52}+228q^{50}-328q^{48}-568q^{46}-89q^{44}+447q^{42}+590q^{40}-31q^{38}-646q^{36}-430q^{34}+181q^{32}+666q^{30}+292q^{28}-389q^{26}-488q^{24}-132q^{22}+407q^{20}+357q^{18}-73q^{16}-272q^{14}-211q^{12}+117q^{10}+193q^{8}+44q^{6}-59q^{4}-114q^{2}+4+52q^{-2}+23q^{-4}+6q^{-6}-34q^{-8}-q^{-10}+8q^{-12}+8q^{-16}-8q^{-18}+q^{-20}+q^{-22}-2q^{-24}+4q^{-26}-2q^{-28}-q^{-34}+q^{-36}$
5	$q^{255}-2q^{253}-2q^{251}+3q^{249}+3q^{247}+3q^{245}-9q^{241}-16q^{239}-q^{237}+20q^{235}+28q^{233}+20q^{231}-16q^{229}-61q^{227}-68q^{225}+q^{223}+92q^{221}+133q^{219}+78q^{217}-81q^{215}-235q^{213}-226q^{211}-2q^{209}+288q^{207}+424q^{205}+242q^{203}-218q^{201}-625q^{199}-605q^{197}-61q^{195}+661q^{193}+1013q^{191}+603q^{189}-404q^{187}-1291q^{185}-1297q^{183}-232q^{181}+1217q^{179}+1929q^{177}+1190q^{175}-641q^{173}-2252q^{171}-2248q^{169}-388q^{167}+2023q^{165}+3083q^{163}+1729q^{161}-1218q^{159}-3460q^{157}-3022q^{155}-16q^{153}+3215q^{151}+3976q^{149}+1433q^{147}-2453q^{145}-4423q^{143}-2674q^{141}+1383q^{139}+4290q^{137}+3552q^{135}-249q^{133}-3777q^{131}-3940q^{129}-670q^{127}+3027q^{125}+3893q^{123}+1304q^{121}-2285q^{119}-3584q^{117}-1591q^{115}+1651q^{113}+3157q^{111}+1679q^{109}-1195q^{107}-2755q^{105}-1692q^{103}+841q^{101}+2466q^{99}+1776q^{97}-490q^{95}-2248q^{93}-2015q^{91}-24q^{89}+2028q^{87}+2419q^{85}+764q^{83}-1652q^{81}-2864q^{79}-1751q^{77}+974q^{75}+3181q^{73}+2890q^{71}+33q^{69}-3165q^{67}-3921q^{65}-1324q^{63}+2651q^{61}+4629q^{59}+2678q^{57}-1701q^{55}-4748q^{53}-3787q^{51}+415q^{49}+4236q^{47}+4407q^{45}+873q^{43}-3228q^{41}-4365q^{39}-1845q^{37}+1936q^{35}+3761q^{33}+2345q^{31}-761q^{29}-2797q^{27}-2290q^{25}-107q^{23}+1750q^{21}+1888q^{19}+557q^{17}-894q^{15}-1320q^{13}-638q^{11}+322q^{9}+777q^{7}+535q^{5}-33q^{3}-396q-349q^{-1}-60q^{-3}+164q^{-5}+191q^{-7}+66q^{-9}-57q^{-11}-86q^{-13}-45q^{-15}+17q^{-17}+38q^{-19}+16q^{-21}-4q^{-23}-8q^{-25}-10q^{-27}-2q^{-29}+10q^{-31}+q^{-33}-4q^{-35}+2q^{-37}-2q^{-39}-2q^{-41}+4q^{-43}+q^{-45}-2q^{-47}-q^{-53}+q^{-55}$

A2 Invariants.

Weight	Invariant
1,0	$q^{28}-q^{26}-q^{24}+2q^{22}-2q^{20}+q^{16}-q^{14}+2q^{12}-q^{10}+q^{8}-2q^{4}+3q^{2}+q^{-4}$
2,0	$q^{72}-q^{70}-2q^{68}+4q^{64}+2q^{62}-7q^{60}-2q^{58}+8q^{56}+6q^{54}-8q^{52}-7q^{50}+8q^{48}+6q^{46}-10q^{44}-8q^{42}+7q^{40}+4q^{38}-4q^{36}-2q^{34}+6q^{32}+q^{30}-q^{28}+4q^{26}-4q^{24}-6q^{22}+7q^{20}+5q^{18}-11q^{16}-5q^{14}+12q^{12}+7q^{10}-11q^{8}-5q^{6}+11q^{4}+3q^{2}-5-q^{-2}+3q^{-4}+q^{-6}-q^{-8}+q^{-12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{60}-2q^{58}+q^{56}+2q^{54}-7q^{52}+5q^{50}+3q^{48}-10q^{46}+10q^{44}+4q^{42}-13q^{40}+10q^{38}+5q^{36}-13q^{34}+3q^{32}+4q^{30}-5q^{28}-4q^{26}+2q^{24}+7q^{22}-7q^{20}-2q^{18}+15q^{16}-9q^{14}-6q^{12}+15q^{10}-7q^{8}-7q^{6}+10q^{4}-q^{2}-3+4q^{-2}+q^{-4}-q^{-6}+q^{-8}$
1,0,0	$q^{37}-q^{35}-q^{31}+2q^{29}-2q^{27}+q^{25}-q^{23}+q^{21}-q^{19}+q^{17}+q^{15}-q^{13}+q^{11}-q^{9}+q^{7}-2q^{5}+3q^{3}+q^{-1}+q^{-5}$

B2 Invariants.

Weight	Invariant
0,1	$q^{60}-2q^{58}+5q^{56}-8q^{54}+11q^{52}-15q^{50}+17q^{48}-18q^{46}+16q^{44}-14q^{42}+7q^{40}-9q^{36}+19q^{34}-25q^{32}+32q^{30}-33q^{28}+34q^{26}-30q^{24}+23q^{22}-15q^{20}+6q^{18}+q^{16}-9q^{14}+14q^{12}-17q^{10}+17q^{8}-15q^{6}+14q^{4}-9q^{2}+7-4q^{-2}+3q^{-4}-q^{-6}+q^{-8}$
1,0	$q^{98}-2q^{94}-2q^{92}+3q^{90}+5q^{88}-3q^{86}-9q^{84}-2q^{82}+12q^{80}+9q^{78}-10q^{76}-15q^{74}+4q^{72}+19q^{70}+7q^{68}-17q^{66}-14q^{64}+9q^{62}+17q^{60}-q^{58}-16q^{56}-4q^{54}+11q^{52}+6q^{50}-10q^{48}-7q^{46}+8q^{44}+9q^{42}-7q^{40}-12q^{38}+4q^{36}+14q^{34}-15q^{30}-5q^{28}+15q^{26}+12q^{24}-10q^{22}-16q^{20}+3q^{18}+18q^{16}+6q^{14}-12q^{12}-12q^{10}+3q^{8}+12q^{6}+3q^{4}-5q^{2}-5+q^{-2}+4q^{-4}+2q^{-6}-q^{-8}-q^{-10}+q^{-14}$

G2 Invariants.

Weight

Invariant

1,0

$q^{142}-2q^{140}+5q^{138}-9q^{136}+9q^{134}-8q^{132}-2q^{130}+19q^{128}-34q^{126}+47q^{124}-44q^{122}+19q^{120}+18q^{118}-60q^{116}+90q^{114}-93q^{112}+61q^{110}-2q^{108}-57q^{106}+98q^{104}-99q^{102}+65q^{100}-6q^{98}-47q^{96}+72q^{94}-67q^{92}+23q^{90}+37q^{88}-76q^{86}+85q^{84}-53q^{82}-10q^{80}+71q^{78}-119q^{76}+124q^{74}-94q^{72}+23q^{70}+56q^{68}-116q^{66}+141q^{64}-111q^{62}+48q^{60}+23q^{58}-76q^{56}+93q^{54}-70q^{52}+20q^{50}+37q^{48}-62q^{46}+60q^{44}-20q^{42}-35q^{40}+73q^{38}-83q^{36}+60q^{34}-20q^{32}-30q^{30}+65q^{28}-78q^{26}+72q^{24}-41q^{22}+6q^{20}+20q^{18}-39q^{16}+42q^{14}-37q^{12}+27q^{10}-9q^{8}-2q^{6}+12q^{4}-15q^{2}+14-10q^{-2}+7q^{-4}-2q^{-6}-q^{-8}+3q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

.

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 67"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-4t^{2}+16t-23+16t^{-1}-4t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$1-4z^{4}$

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]=

{ 63, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{2}a^{8}-z^{4}a^{6}-2z^{4}a^{4}-2z^{2}a^{4}-z^{4}a^{2}+z^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{6}a^{10}-3z^{4}a^{10}+2z^{2}a^{10}+3z^{7}a^{9}-10z^{5}a^{9}+9z^{3}a^{9}-2za^{9}+3z^{8}a^{8}-7z^{6}a^{8}+2z^{4}a^{8}+z^{9}a^{7}+5z^{7}a^{7}-21z^{5}a^{7}+19z^{3}a^{7}-6za^{7}+6z^{8}a^{6}-13z^{6}a^{6}+7z^{4}a^{6}-2z^{2}a^{6}+z^{9}a^{5}+6z^{7}a^{5}-19z^{5}a^{5}+19z^{3}a^{5}-6za^{5}+3z^{8}a^{4}-2z^{6}a^{4}-z^{4}a^{4}+2z^{2}a^{4}+4z^{7}a^{3}-6z^{5}a^{3}+7z^{3}a^{3}-2za^{3}+3z^{6}a^{2}-2z^{4}a^{2}+2z^{5}a-2z^{3}a+z^{4}-2z^{2}+1$

Vassiliev invariants

V2 and V3:

(0, 0)

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 $0$ $0$ $0$ $32$ $32$ $0$ $-192$ $-64$ $-128$ $0$ $0$ $0$ $0$ $592$ $-{\frac {704}{3}}$ ${\frac {2176}{3}}$ ${\frac {304}{3}}$ $80$

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 67. 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
3											1	1
1										1		-1
-1									4	1		3
-3								5	2			-3
-5							5	3				2
-7						5	5					0
-9					5	5						0
-11				3	5							2
-13			2	5								-3
-15		1	3									2
-17		2										-2
-19	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, 67]]
Out[2]=	10
In[3]:=	PD[Knot[10, 67]]
Out[3]=	PD[X[1, 4, 2, 5], X[7, 12, 8, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], X[5, 14, 6, 15], X[13, 6, 14, 7], X[9, 18, 10, 19], X[15, 20, 16, 1], X[19, 16, 20, 17], X[17, 8, 18, 9]]
In[4]:=	GaussCode[Knot[10, 67]]
Out[4]=	GaussCode[-1, 4, -3, 1, -5, 6, -2, 10, -7, 3, -4, 2, -6, 5, -8, 9, -10, 7, -9, 8]
In[5]:=	BR[Knot[10, 67]]
Out[5]=	BR[5, {-1, -1, -1, -2, 1, -2, -3, 2, 2, 4, -3, -2, 4, -3}]
In[6]:=	alex = Alexander[Knot[10, 67]][t]
Out[6]=	4 16 2 -23 - -- + -- + 16 t - 4 t 2 t t
In[7]:=	Conway[Knot[10, 67]][z]
Out[7]=	4 1 - 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 67], Knot[10, 74], Knot[11, NonAlternating, 68]}
In[9]:=	{KnotDet[Knot[10, 67]], KnotSignature[Knot[10, 67]]}
Out[9]=	{63, -2}
In[10]:=	J=Jones[Knot[10, 67]][q]
Out[10]=	-9 3 5 8 10 10 10 8 5 -2 + q - -- + -- - -- + -- - -- + -- - -- + - + q 8 7 6 5 4 3 2 q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 67]}
In[12]:=	A2Invariant[Knot[10, 67]][q]
Out[12]=	-28 -26 -24 2 2 -16 -14 2 -10 -8 2 q - q - q + --- - --- + q - q + --- - q + q - -- + 22 20 12 4 q q q q 3 4 -- + q 2 q
In[13]:=	Kauffman[Knot[10, 67]][a, z]
Out[13]=	3 5 7 9 2 4 2 6 2 1 - 2 a z - 6 a z - 6 a z - 2 a z - 2 z + 2 a z - 2 a z + 10 2 3 3 3 5 3 7 3 9 3 4 2 a z - 2 a z + 7 a z + 19 a z + 19 a z + 9 a z + z - 2 4 4 4 6 4 8 4 10 4 5 3 5 2 a z - a z + 7 a z + 2 a z - 3 a z + 2 a z - 6 a z - 5 5 7 5 9 5 2 6 4 6 6 6 19 a z - 21 a z - 10 a z + 3 a z - 2 a z - 13 a z - 8 6 10 6 3 7 5 7 7 7 9 7 4 8 7 a z + a z + 4 a z + 6 a z + 5 a z + 3 a z + 3 a z + 6 8 8 8 5 9 7 9 6 a z + 3 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[10, 67]], Vassiliev[3][Knot[10, 67]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 67]][q, t]
Out[15]=	2 4 1 2 1 3 2 5 3 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 5 5 5 5 5 5 3 5 t ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + - + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 3 2 q t + q t