6 1

5_2

6_2

Visit 6 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 6 1's page at Knotilus! Visit 6 1's page at the original Knot Atlas! 6_1 is also known as "Stevedore's Knot" (see e.g. [1]), and as the pretzel knot P(5,-1,-1).

A Kolam of a 3x3 dot array	3D depiction	Polygonal depiction
Simple square depiction	An other one	Necklace

Knot presentations

Planar diagram presentation	X1425 X7,10,8,11 X3948 X9,3,10,2 X5,12,6,1 X11,6,12,7
Gauss code	-1, 4, -3, 1, -5, 6, -2, 3, -4, 2, -6, 5
Dowker-Thistlethwaite code	4 8 12 10 2 6
Conway Notation	[42]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index $\{3,4\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-5][-3] Hyperbolic Volume 3.16396 A-Polynomial See Data:6 1/A-polynomial

[edit Notes for 6 1's three dimensional invariants] 6_1 is a ribbon knot (drawings by Yoko Mizuma): a ribbon diagram isotopy to a ribbon 6_1 has two slice disks, by Scott Carter Scott Carter notes that 6_1 bounds two distinct slice disks. He says: "this was spoken of in Fox's Example 10, 11, and 12 in a Quick Trip through Knot Theory ... BTW, the cover of Carter and Saito's Knotted Surfaces and Their Diagrams contains an illustration of such a slice disk". A picture is on the right.

Four dimensional invariants

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

[edit Notes for 6 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t+5-2t^{-1}$
Conway polynomial	$1-2z^{2}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 9, 0 }
Jones polynomial	$q^{2}-q+2-2q^{-1}+q^{-2}-q^{-3}+q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$a^{4}-z^{2}a^{2}-a^{2}-z^{2}+a^{-2}$
Kauffman polynomial (db, data sources)	$a^{3}z^{5}+az^{5}+a^{4}z^{4}+2a^{2}z^{4}+z^{4}-3a^{3}z^{3}-2az^{3}+z^{3}a^{-1}-3a^{4}z^{2}-4a^{2}z^{2}+z^{2}a^{-2}+2a^{3}z+2az+a^{4}+a^{2}-a^{-2}$
The A2 invariant	$q^{14}+q^{12}-q^{6}-q^{4}+q^{-2}+q^{-6}+q^{-8}$
The G2 invariant	$q^{66}+q^{62}-q^{60}+q^{56}-q^{54}+2q^{52}+q^{46}+q^{42}-q^{38}+q^{32}-2q^{28}+q^{26}+q^{24}-2q^{20}-2q^{18}+q^{16}-q^{14}+q^{12}-2q^{10}-q^{8}+2q^{6}-q^{4}-1+q^{-4}+q^{-10}+2q^{-14}-q^{-18}+q^{-20}+q^{-24}+q^{-28}+q^{-34}+q^{-38}$

Further Quantum Invariants

Further quantum knot invariants for 6_1.

A1 Invariants.

Weight	Invariant
1	$q^{9}-q^{3}+q^{-1}+q^{-5}$
2	$q^{26}-q^{22}-q^{16}+q^{12}+q^{8}+q^{6}+q^{-2}-q^{-4}-q^{-6}+q^{-8}+q^{-14}$
3	$q^{51}-q^{47}-q^{45}+q^{41}-q^{37}+q^{33}+q^{31}-q^{27}+q^{25}+q^{23}-q^{19}-q^{13}-q^{11}+q^{7}+q^{3}+q^{-1}+2q^{-3}+q^{-5}-q^{-7}-q^{-9}+q^{-11}+q^{-13}-q^{-15}-q^{-17}+q^{-27}$
4	$q^{84}-q^{80}-q^{78}-q^{76}+q^{74}+q^{72}+q^{70}-2q^{66}+q^{62}+q^{60}+q^{58}-q^{56}-q^{54}-q^{52}+q^{50}+2q^{48}-q^{44}-2q^{42}+q^{38}-q^{34}-2q^{32}+2q^{28}+q^{26}+q^{24}+q^{20}+2q^{18}-q^{14}-q^{12}-1-q^{-2}+2q^{-4}+q^{-6}+q^{-8}-q^{-10}-2q^{-12}+q^{-14}+2q^{-16}+3q^{-18}-2q^{-22}+q^{-28}-q^{-32}-q^{-36}+q^{-44}$
5	$q^{125}-q^{121}-q^{119}-q^{117}+q^{113}+2q^{111}+q^{109}-q^{105}-2q^{103}-q^{101}+q^{99}+2q^{97}+q^{95}-q^{91}-2q^{89}-q^{87}+2q^{83}+2q^{81}+q^{79}-q^{77}-3q^{75}-2q^{73}+2q^{69}+2q^{67}-2q^{63}-2q^{61}-q^{59}+2q^{57}+4q^{55}+2q^{53}-q^{51}-q^{49}-q^{47}+q^{45}+2q^{43}+q^{41}-2q^{39}-3q^{37}-2q^{35}+q^{31}-q^{27}-q^{25}+q^{23}+2q^{21}+q^{19}-q^{13}+q^{11}+q^{9}+2q^{5}+2q^{3}-q^{-1}-q^{-3}+q^{-7}+2q^{-9}+q^{-11}-q^{-13}-4q^{-15}-3q^{-17}+3q^{-21}+4q^{-23}+q^{-25}-2q^{-27}-4q^{-29}-q^{-31}+2q^{-33}+3q^{-35}+2q^{-37}-q^{-41}-q^{-43}+q^{-47}-q^{-55}-q^{-57}+q^{-65}$
6	$q^{174}-q^{170}-q^{168}-q^{166}+2q^{160}+2q^{158}+q^{156}-q^{152}-2q^{150}-3q^{148}+q^{144}+2q^{142}+2q^{140}+q^{138}-3q^{134}-2q^{132}-q^{130}+q^{126}+3q^{124}+3q^{122}-q^{118}-3q^{116}-3q^{114}-3q^{112}+q^{110}+4q^{108}+3q^{106}+2q^{104}-q^{102}-3q^{100}-4q^{98}-q^{96}+2q^{94}+4q^{92}+5q^{90}+2q^{88}-2q^{86}-5q^{84}-3q^{82}-q^{80}+2q^{78}+4q^{76}+2q^{74}-q^{72}-5q^{70}-4q^{68}-2q^{66}+q^{64}+4q^{62}+3q^{60}-3q^{56}-2q^{54}+2q^{50}+4q^{48}+3q^{46}-3q^{42}-q^{40}+q^{38}+q^{36}+q^{34}-q^{30}-2q^{28}+q^{24}+q^{22}-q^{18}-q^{16}-q^{14}-q^{12}-q^{10}+q^{6}+2q^{4}+2q^{2}+2-q^{-2}-2q^{-4}-q^{-8}+2q^{-10}+4q^{-12}+5q^{-14}+q^{-16}-3q^{-18}-4q^{-20}-5q^{-22}-q^{-24}+4q^{-26}+8q^{-28}+4q^{-30}-q^{-32}-5q^{-34}-8q^{-36}-4q^{-38}+q^{-40}+6q^{-42}+4q^{-44}+q^{-46}-q^{-48}-4q^{-50}-3q^{-52}+3q^{-56}+2q^{-58}+q^{-60}+q^{-62}-q^{-64}-q^{-66}+q^{-70}+q^{-76}-q^{-78}-q^{-80}-q^{-82}+q^{-90}$

A2 Invariants.

Weight	Invariant
1,0	$q^{14}+q^{12}-q^{6}-q^{4}+q^{-2}+q^{-6}+q^{-8}$
1,1	$q^{36}+2q^{32}-2q^{30}+2q^{28}-2q^{26}-2q^{22}-2q^{20}-2q^{16}+4q^{14}+q^{12}+6q^{10}+4q^{6}-2q^{4}-2-2q^{-2}-q^{-4}-2q^{-6}+2q^{-8}+2q^{-12}+2q^{-16}+q^{-20}$
2,0	$q^{36}+q^{34}+q^{32}-q^{30}-q^{28}-q^{26}-q^{24}-q^{22}-q^{20}+q^{18}+q^{16}+q^{14}+q^{12}+2q^{10}+q^{8}+q^{6}+q^{4}-q^{-2}-q^{-4}-2q^{-6}-q^{-8}+q^{-10}+q^{-12}+q^{-16}+q^{-18}+q^{-20}$
3,0	$q^{66}+q^{64}+q^{62}-2q^{58}-2q^{56}-2q^{54}+2q^{42}+2q^{40}+2q^{38}+q^{32}+2q^{30}+q^{28}-q^{24}-q^{20}-3q^{18}-3q^{16}-3q^{14}-q^{12}-q^{10}+q^{8}+2q^{6}+3q^{4}+3q^{2}+3+3q^{-2}+2q^{-4}+2q^{-6}-q^{-8}-q^{-10}+q^{-14}-3q^{-18}-3q^{-20}-q^{-22}+q^{-26}+q^{-30}+q^{-32}+q^{-34}+q^{-36}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{28}+q^{24}-q^{20}+q^{12}+2q^{10}-q^{4}-q^{2}-1-q^{-2}+q^{-4}+q^{-8}+2q^{-10}+q^{-12}+q^{-16}$
1,0,0	$q^{19}+q^{17}+q^{15}-q^{9}-q^{7}-q^{5}+q^{-3}+q^{-7}+q^{-9}+q^{-11}$

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$q^{28}+q^{24}+q^{20}-q^{12}-2q^{8}-q^{4}+q^{2}-1+q^{-2}+q^{-4}+q^{-8}+q^{-12}+q^{-16}$
1,0	$q^{46}+q^{38}-q^{34}-q^{32}+q^{20}+q^{18}+q^{16}+q^{12}-q^{6}-q^{4}-q^{-2}-q^{-4}+q^{-6}+q^{-14}+q^{-16}+q^{-18}+q^{-26}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{38}+q^{34}+q^{30}+q^{16}+q^{12}-q^{10}-q^{6}-2q^{2}-q^{-2}+q^{-6}+q^{-10}+q^{-12}+2q^{-14}+q^{-16}+q^{-18}+q^{-22}$

G2 Invariants.

Weight	Invariant
1,0	$q^{66}+q^{62}-q^{60}+q^{56}-q^{54}+2q^{52}+q^{46}+q^{42}-q^{38}+q^{32}-2q^{28}+q^{26}+q^{24}-2q^{20}-2q^{18}+q^{16}-q^{14}+q^{12}-2q^{10}-q^{8}+2q^{6}-q^{4}-1+q^{-4}+q^{-10}+2q^{-14}-q^{-18}+q^{-20}+q^{-24}+q^{-28}+q^{-34}+q^{-38}$

.

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["6 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$1-2z^{2}$

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]=

{ 9, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{3}z^{5}+az^{5}+a^{4}z^{4}+2a^{2}z^{4}+z^{4}-3a^{3}z^{3}-2az^{3}+z^{3}a^{-1}-3a^{4}z^{2}-4a^{2}z^{2}+z^{2}a^{-2}+2a^{3}z+2az+a^{4}+a^{2}-a^{-2}$

Vassiliev invariants

V2 and V3:

(-2, 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 $-8$ $8$ $32$ ${\frac {116}{3}}$ ${\frac {52}{3}}$ $-64$ $-{\frac {304}{3}}$ $-{\frac {64}{3}}$ $-24$ $-{\frac {256}{3}}$ $32$ $-{\frac {928}{3}}$ $-{\frac {416}{3}}$ $-{\frac {2791}{15}}$ ${\frac {884}{15}}$ $-{\frac {10084}{45}}$ ${\frac {343}{9}}$ $-{\frac {871}{15}}$

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

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

Integral Khovanov Homology (db, data source)

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

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, 1]]
Out[2]=	6
In[3]:=	PD[Knot[6, 1]]
Out[3]=	PD[X[1, 4, 2, 5], X[7, 10, 8, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[5, 12, 6, 1], X[11, 6, 12, 7]]
In[4]:=	GaussCode[Knot[6, 1]]
Out[4]=	GaussCode[-1, 4, -3, 1, -5, 6, -2, 3, -4, 2, -6, 5]
In[5]:=	BR[Knot[6, 1]]
Out[5]=	BR[4, {-1, -1, -2, 1, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[6, 1]][t]
Out[6]=	2 5 - - - 2 t t
In[7]:=	Conway[Knot[6, 1]][z]
Out[7]=	2 1 - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[6, 1], Knot[9, 46], Knot[11, NonAlternating, 67], Knot[11, NonAlternating, 97], Knot[11, NonAlternating, 139]}
In[9]:=	{KnotDet[Knot[6, 1]], KnotSignature[Knot[6, 1]]}
Out[9]=	{9, 0}
In[10]:=	J=Jones[Knot[6, 1]][q]
Out[10]=	-4 -3 -2 2 2 2 + q - q + q - - - q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[6, 1]}
In[12]:=	A2Invariant[Knot[6, 1]][q]
Out[12]=	-14 -12 -6 -4 2 6 8 q + q - q - q + q + q + q
In[13]:=	Kauffman[Knot[6, 1]][a, z]
Out[13]=	2 3 -2 2 4 3 z 2 2 4 2 z -a + a + a + 2 a z + 2 a z + -- - 4 a z - 3 a z + -- - 2 a a 3 3 3 4 2 4 4 4 5 3 5 2 a z - 3 a z + z + 2 a z + a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[6, 1]], Vassiliev[3][Knot[6, 1]]}
Out[14]=	{0, 1}
In[15]:=	Kh[Knot[6, 1]][q, t]
Out[15]=	1 1 1 1 1 1 5 2 - + 2 q + ----- + ----- + ----- + ---- + --- + q t + q t q 9 4 5 3 5 2 3 q t q t q t q t q t