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]

Length is 7, width is 4. Braid index is 4.

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} Topological 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} Concordance genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} Rasmussen s-Invariant 0

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

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 t+5-2 t^{-1} }
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1-2 z^2}
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 9, 0 }
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^2-q+2-2 q^{-1} + q^{-2} - q^{-3} + q^{-4} }
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^4-z^2 a^2-a^2-z^2+ a^{-2} }
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^3 z^5+a z^5+a^4 z^4+2 a^2 z^4+z^4-3 a^3 z^3-2 a z^3+z^3 a^{-1} -3 a^4 z^2-4 a^2 z^2+z^2 a^{-2} +2 a^3 z+2 a z+a^4+a^2- a^{-2} }
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{14}+q^{12}-q^6-q^4+ q^{-2} + q^{-6} + q^{-8} }
The G2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{66}+q^{62}-q^{60}+q^{56}-q^{54}+2 q^{52}+q^{46}+q^{42}-q^{38}+q^{32}-2 q^{28}+q^{26}+q^{24}-2 q^{20}-2 q^{18}+q^{16}-q^{14}+q^{12}-2 q^{10}-q^8+2 q^6-q^4-1+ q^{-4} + q^{-10} +2 q^{-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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^9-q^3+ q^{-1} + q^{-5} }
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{26}-q^{22}-q^{16}+q^{12}+q^8+q^6+ q^{-2} - q^{-4} - q^{-6} + q^{-8} + q^{-14} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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} +2 q^{-3} + q^{-5} - q^{-7} - q^{-9} + q^{-11} + q^{-13} - q^{-15} - q^{-17} + q^{-27} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{84}-q^{80}-q^{78}-q^{76}+q^{74}+q^{72}+q^{70}-2 q^{66}+q^{62}+q^{60}+q^{58}-q^{56}-q^{54}-q^{52}+q^{50}+2 q^{48}-q^{44}-2 q^{42}+q^{38}-q^{34}-2 q^{32}+2 q^{28}+q^{26}+q^{24}+q^{20}+2 q^{18}-q^{14}-q^{12}-1- q^{-2} +2 q^{-4} + q^{-6} + q^{-8} - q^{-10} -2 q^{-12} + q^{-14} +2 q^{-16} +3 q^{-18} -2 q^{-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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{174}-q^{170}-q^{168}-q^{166}+2 q^{160}+2 q^{158}+q^{156}-q^{152}-2 q^{150}-3 q^{148}+q^{144}+2 q^{142}+2 q^{140}+q^{138}-3 q^{134}-2 q^{132}-q^{130}+q^{126}+3 q^{124}+3 q^{122}-q^{118}-3 q^{116}-3 q^{114}-3 q^{112}+q^{110}+4 q^{108}+3 q^{106}+2 q^{104}-q^{102}-3 q^{100}-4 q^{98}-q^{96}+2 q^{94}+4 q^{92}+5 q^{90}+2 q^{88}-2 q^{86}-5 q^{84}-3 q^{82}-q^{80}+2 q^{78}+4 q^{76}+2 q^{74}-q^{72}-5 q^{70}-4 q^{68}-2 q^{66}+q^{64}+4 q^{62}+3 q^{60}-3 q^{56}-2 q^{54}+2 q^{50}+4 q^{48}+3 q^{46}-3 q^{42}-q^{40}+q^{38}+q^{36}+q^{34}-q^{30}-2 q^{28}+q^{24}+q^{22}-q^{18}-q^{16}-q^{14}-q^{12}-q^{10}+q^6+2 q^4+2 q^2+2- q^{-2} -2 q^{-4} - q^{-8} +2 q^{-10} +4 q^{-12} +5 q^{-14} + q^{-16} -3 q^{-18} -4 q^{-20} -5 q^{-22} - q^{-24} +4 q^{-26} +8 q^{-28} +4 q^{-30} - q^{-32} -5 q^{-34} -8 q^{-36} -4 q^{-38} + q^{-40} +6 q^{-42} +4 q^{-44} + q^{-46} - q^{-48} -4 q^{-50} -3 q^{-52} +3 q^{-56} +2 q^{-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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{14}+q^{12}-q^6-q^4+ q^{-2} + q^{-6} + q^{-8} }
1,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{36}+2 q^{32}-2 q^{30}+2 q^{28}-2 q^{26}-2 q^{22}-2 q^{20}-2 q^{16}+4 q^{14}+q^{12}+6 q^{10}+4 q^6-2 q^4-2-2 q^{-2} - q^{-4} -2 q^{-6} +2 q^{-8} +2 q^{-12} +2 q^{-16} + q^{-20} }
2,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}+2 q^{10}+q^8+q^6+q^4- q^{-2} - q^{-4} -2 q^{-6} - q^{-8} + q^{-10} + q^{-12} + q^{-16} + q^{-18} + q^{-20} }
3,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{66}+q^{64}+q^{62}-2 q^{58}-2 q^{56}-2 q^{54}+2 q^{42}+2 q^{40}+2 q^{38}+q^{32}+2 q^{30}+q^{28}-q^{24}-q^{20}-3 q^{18}-3 q^{16}-3 q^{14}-q^{12}-q^{10}+q^8+2 q^6+3 q^4+3 q^2+3+3 q^{-2} +2 q^{-4} +2 q^{-6} - q^{-8} - q^{-10} + q^{-14} -3 q^{-18} -3 q^{-20} - q^{-22} + q^{-26} + q^{-30} + q^{-32} + q^{-34} + q^{-36} }

A3 Invariants.

Weight	Invariant
0,1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{28}+q^{24}-q^{20}+q^{12}+2 q^{10}-q^4-q^2-1- q^{-2} + q^{-4} + q^{-8} +2 q^{-10} + q^{-12} + q^{-16} }
1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{38}+q^{36}+q^{34}+q^{32}+q^{30}-q^{28}-2 q^{26}-2 q^{24}-q^{22}-q^{20}+3 q^{16}+3 q^{14}+3 q^{12}+2 q^{10}+q^8-q^6-2 q^4-2 q^2-2-2 q^{-2} - q^{-4} + q^{-6} + q^{-10} +2 q^{-12} +2 q^{-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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{28}+q^{24}+q^{20}-q^{12}-2 q^8-q^4+q^2-1+ q^{-2} + q^{-4} + q^{-8} + q^{-12} + q^{-16} }
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{38}+q^{34}+q^{30}+q^{16}+q^{12}-q^{10}-q^6-2 q^2- q^{-2} + q^{-6} + q^{-10} + q^{-12} +2 q^{-14} + q^{-16} + q^{-18} + q^{-22} }

G2 Invariants.

Weight

Invariant

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{66}+q^{62}-q^{60}+q^{56}-q^{54}+2 q^{52}+q^{46}+q^{42}-q^{38}+q^{32}-2 q^{28}+q^{26}+q^{24}-2 q^{20}-2 q^{18}+q^{16}-q^{14}+q^{12}-2 q^{10}-q^8+2 q^6-q^4-1+ q^{-4} + q^{-10} +2 q^{-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]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 t+5-2 t^{-1} }

In[5]:=

Conway[K][z]

Out[5]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1-2 z^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]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^2-q+2-2 q^{-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]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^3 z^5+a z^5+a^4 z^4+2 a^2 z^4+z^4-3 a^3 z^3-2 a z^3+z^3 a^{-1} -3 a^4 z^2-4 a^2 z^2+z^2 a^{-2} +2 a^3 z+2 a z+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 $-8$ $8$ $32$ ${\frac {116}{3}}$ ${\frac {52}{3}}$ $-64$ $-{\frac {304}{3}}$ $-{\frac {64}{3}}$ $-24$ $-{\frac {256}{3}}$ $32$ $-{\frac {928}{3}}$ $-{\frac {416}{3}}$ $-{\frac {2791}{15}}$ ${\frac {884}{15}}$ $-{\frac {10084}{45}}$ ${\frac {343}{9}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed $j$, alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

Computer Talk

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{Include}(\textrm{ColouredJonesM.mhtml})}

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
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]=	{-2, 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

See/edit the Rolfsen_Splice_Template.