6 2

6_1

6_3

Visit 6 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 6 2's page at Knotilus! Visit 6 2's page at the original Knot Atlas!

Dror likes to call 6_2 "The Miller Institute Knot", as it is the logo of the Miller Institute for Basic Research. The bowline knot of practical knot tying deforms to 6_2.

The Miller Institute Mug [1]

Simple square depiction

3D depiction

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 X7,12,8,1 X11,6,12,7
Gauss code	-1, 4, -3, 1, -2, 6, -5, 3, -4, 2, -6, 5
Dowker-Thistlethwaite code	4 8 10 12 2 6
Conway Notation	[312]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index $\{3,4\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-1] Hyperbolic Volume 4.40083 A-Polynomial See Data:6 2/A-polynomial

[edit Notes for 6 2's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 6 2's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{2}+3t-3+3t^{-1}-t^{-2}$
Conway polynomial	$-z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 11, -2 }
Jones polynomial	$q-1+2q^{-1}-2q^{-2}+2q^{-3}-2q^{-4}+q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{4}+a^{4}-z^{4}a^{2}-3z^{2}a^{2}-2a^{2}+z^{2}+2$
Kauffman polynomial (db, data sources)	$z^{2}a^{6}+2z^{3}a^{5}-za^{5}+2z^{4}a^{4}-2z^{2}a^{4}+a^{4}+z^{5}a^{3}-za^{3}+3z^{4}a^{2}-6z^{2}a^{2}+2a^{2}+z^{5}a-2z^{3}a+z^{4}-3z^{2}+2$
The A2 invariant	$q^{16}-q^{8}-q^{4}+q^{2}+1+q^{-2}+q^{-4}$
The G2 invariant	$q^{86}-q^{84}+q^{82}-q^{80}-q^{78}-q^{74}+3q^{72}-2q^{70}+q^{68}+2q^{62}-q^{60}+q^{58}+q^{56}+q^{52}+3q^{46}-3q^{44}+q^{42}-q^{40}-q^{38}+2q^{36}-4q^{34}+q^{32}-2q^{30}-3q^{24}+q^{22}-q^{20}-q^{14}+q^{12}+q^{10}+2q^{6}-q^{4}+2q^{2}+1-q^{-2}+3q^{-4}-q^{-6}+2q^{-8}-q^{-12}+2q^{-14}+q^{-18}$

Further Quantum Invariants

Further quantum knot invariants for 6_2.

A1 Invariants.

Weight	Invariant
1	$q^{11}-q^{9}+q+q^{-3}$
2	$q^{30}-q^{28}-q^{26}+2q^{24}-q^{22}-q^{20}+q^{18}+q^{10}-q^{6}+q^{4}+q^{2}-1+q^{-2}+q^{-4}-q^{-6}+q^{-10}$
3	$q^{57}-q^{55}-q^{53}+q^{51}+q^{49}-q^{47}-3q^{45}+2q^{43}+3q^{41}-q^{39}-2q^{37}+q^{35}+2q^{33}-q^{29}-q^{27}-2q^{19}-q^{17}+2q^{15}+2q^{13}-q^{11}+2q^{7}+2q^{5}-q^{3}-2q+q^{-1}+2q^{-3}-2q^{-7}+2q^{-11}+q^{-13}-q^{-15}-q^{-17}+q^{-21}$
4	$q^{92}-q^{90}-q^{88}+q^{86}+q^{82}-3q^{80}-q^{78}+3q^{76}+2q^{74}+3q^{72}-5q^{70}-4q^{68}+3q^{66}+4q^{64}+3q^{62}-5q^{60}-5q^{58}+2q^{54}+3q^{52}-q^{50}-2q^{48}+q^{44}+2q^{42}+2q^{40}+q^{38}-2q^{36}-2q^{34}+2q^{30}+q^{28}-3q^{26}-3q^{24}-q^{22}+4q^{20}+2q^{18}-3q^{16}-2q^{14}+5q^{10}+3q^{8}-2q^{6}-2q^{4}-2q^{2}+4+4q^{-2}-2q^{-6}-4q^{-8}+q^{-10}+3q^{-12}+2q^{-14}-4q^{-18}-q^{-20}+q^{-22}+2q^{-24}+2q^{-26}-q^{-28}-q^{-30}-q^{-32}+q^{-36}$
5	$q^{135}-q^{133}-q^{131}+q^{129}-q^{123}-q^{121}+3q^{117}+4q^{115}-5q^{111}-5q^{109}+4q^{105}+8q^{103}+3q^{101}-8q^{99}-10q^{97}-5q^{95}+5q^{93}+11q^{91}+7q^{89}-2q^{87}-9q^{85}-6q^{83}+q^{81}+6q^{79}+6q^{77}+q^{75}-4q^{73}-5q^{71}-2q^{69}+q^{67}+2q^{65}+3q^{63}+q^{61}-q^{59}-2q^{57}-3q^{55}-q^{53}+3q^{51}+4q^{49}+q^{47}-3q^{45}-5q^{43}-q^{41}+5q^{39}+5q^{37}+q^{35}-4q^{33}-6q^{31}-q^{29}+5q^{27}+6q^{25}-6q^{21}-7q^{19}-2q^{17}+7q^{15}+8q^{13}+3q^{11}-4q^{9}-7q^{7}-3q^{5}+3q^{3}+8q+6q^{-1}-q^{-3}-6q^{-5}-6q^{-7}-2q^{-9}+4q^{-11}+7q^{-13}+4q^{-15}-q^{-17}-5q^{-19}-5q^{-21}-q^{-23}+3q^{-25}+5q^{-27}+3q^{-29}-q^{-31}-4q^{-33}-3q^{-35}-q^{-37}+q^{-39}+3q^{-41}+2q^{-43}-q^{-47}-q^{-49}-q^{-51}+q^{-55}$
6	$q^{186}-q^{184}-q^{182}+q^{180}-2q^{174}+q^{172}+5q^{166}+2q^{164}-2q^{162}-6q^{160}-3q^{158}-3q^{156}+3q^{154}+12q^{152}+7q^{150}-3q^{148}-14q^{146}-10q^{144}-8q^{142}+4q^{140}+21q^{138}+17q^{136}+5q^{134}-14q^{132}-18q^{130}-17q^{128}-3q^{126}+18q^{124}+20q^{122}+12q^{120}-5q^{118}-13q^{116}-17q^{114}-9q^{112}+7q^{110}+13q^{108}+12q^{106}+2q^{104}-4q^{102}-10q^{100}-9q^{98}-2q^{96}+3q^{94}+6q^{92}+4q^{90}+3q^{88}-q^{86}-3q^{84}-4q^{82}-3q^{80}+4q^{76}+6q^{74}+4q^{72}-6q^{68}-6q^{66}-3q^{64}+5q^{62}+8q^{60}+5q^{58}-q^{56}-11q^{54}-8q^{52}-q^{50}+8q^{48}+9q^{46}+4q^{44}-4q^{42}-14q^{40}-9q^{38}+q^{36}+12q^{34}+12q^{32}+7q^{30}-4q^{28}-16q^{26}-12q^{24}-q^{22}+12q^{20}+14q^{18}+11q^{16}-15q^{12}-15q^{10}-7q^{8}+7q^{6}+12q^{4}+15q^{2}+7-8q^{-2}-13q^{-4}-12q^{-6}-2q^{-8}+4q^{-10}+14q^{-12}+13q^{-14}+2q^{-16}-5q^{-18}-11q^{-20}-9q^{-22}-6q^{-24}+5q^{-26}+10q^{-28}+8q^{-30}+4q^{-32}-2q^{-34}-6q^{-36}-10q^{-38}-3q^{-40}+2q^{-42}+5q^{-44}+6q^{-46}+4q^{-48}+q^{-50}-5q^{-52}-4q^{-54}-3q^{-56}-q^{-58}+q^{-60}+3q^{-62}+3q^{-64}-q^{-70}-q^{-72}-q^{-74}+q^{-78}$

A2 Invariants.

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

A3 Invariants.

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

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{46}-q^{42}-q^{36}-q^{34}+q^{32}+q^{30}+q^{28}+2q^{26}+3q^{24}+q^{22}+q^{18}-2q^{16}-3q^{14}-2q^{12}-2q^{10}-3q^{8}-q^{6}+q^{4}+2q^{2}+2+3q^{-2}+3q^{-4}+2q^{-6}+q^{-8}+q^{-10}$
1,0,0,0	$q^{26}+q^{22}+q^{20}-q^{14}-q^{12}-2q^{10}-q^{8}-q^{6}+q^{4}+q^{2}+2+2q^{-2}+q^{-4}+q^{-6}$

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight	Invariant
1,0	$q^{86}-q^{84}+q^{82}-q^{80}-q^{78}-q^{74}+3q^{72}-2q^{70}+q^{68}+2q^{62}-q^{60}+q^{58}+q^{56}+q^{52}+3q^{46}-3q^{44}+q^{42}-q^{40}-q^{38}+2q^{36}-4q^{34}+q^{32}-2q^{30}-3q^{24}+q^{22}-q^{20}-q^{14}+q^{12}+q^{10}+2q^{6}-q^{4}+2q^{2}+1-q^{-2}+3q^{-4}-q^{-6}+2q^{-8}-q^{-12}+2q^{-14}+q^{-18}$

.

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["6 2"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 11, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{2}a^{4}+a^{4}-z^{4}a^{2}-3z^{2}a^{2}-2a^{2}+z^{2}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

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 $-4$ $8$ $8$ ${\frac {34}{3}}$ ${\frac {38}{3}}$ $-32$ $-{\frac {208}{3}}$ $-{\frac {64}{3}}$ $-24$ $-{\frac {32}{3}}$ $32$ $-{\frac {136}{3}}$ $-{\frac {152}{3}}$ ${\frac {2129}{30}}$ ${\frac {662}{15}}$ $-{\frac {1862}{45}}$ ${\frac {463}{18}}$ $-{\frac {751}{30}}$

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

\   r    \    j   \	-4	-3	-2	-1	0	1	2	χ
3							1	1
1								0
-1					2	1		1
-3				1	1			0
-5			1	1				0
-7		1	1					0
-9		1						-1
-11	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[6, 2]]
Out[2]=	6
In[3]:=	PD[Knot[6, 2]]
Out[3]=	PD[X[1, 4, 2, 5], X[5, 10, 6, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[7, 12, 8, 1], X[11, 6, 12, 7]]
In[4]:=	GaussCode[Knot[6, 2]]
Out[4]=	GaussCode[-1, 4, -3, 1, -2, 6, -5, 3, -4, 2, -6, 5]
In[5]:=	BR[Knot[6, 2]]
Out[5]=	BR[3, {-1, -1, -1, 2, -1, 2}]
In[6]:=	alex = Alexander[Knot[6, 2]][t]
Out[6]=	-2 3 2 -3 - t + - + 3 t - t t
In[7]:=	Conway[Knot[6, 2]][z]
Out[7]=	2 4 1 - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[6, 2]}
In[9]:=	{KnotDet[Knot[6, 2]], KnotSignature[Knot[6, 2]]}
Out[9]=	{11, -2}
In[10]:=	J=Jones[Knot[6, 2]][q]
Out[10]=	-5 2 2 2 2 -1 + q - -- + -- - -- + - + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[6, 2]}
In[12]:=	A2Invariant[Knot[6, 2]][q]
Out[12]=	-16 -8 -4 -2 2 4 1 + q - q - q + q + q + q
In[13]:=	Kauffman[Knot[6, 2]][a, z]
Out[13]=	2 4 3 5 2 2 2 4 2 6 2 2 + 2 a + a - a z - a z - 3 z - 6 a z - 2 a z + a z - 3 5 3 4 2 4 4 4 5 3 5 2 a z + 2 a z + z + 3 a z + 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[6, 2]], Vassiliev[3][Knot[6, 2]]}
Out[14]=	{0, 1}
In[15]:=	Kh[Knot[6, 2]][q, t]
Out[15]=	-3 2 1 1 1 1 1 1 1 t q + - + ------ + ----- + ----- + ----- + ----- + ---- + ---- + - + q 11 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 3 2 q t