5 1

4_1

5_2

Visit 5 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 5 1's page at Knotilus! Visit 5 1's page at the original Knot Atlas! An interlaced pentagram, this is known variously as the "Cinquefoil Knot", after certain herbs and shrubs of the rose family which have 5-lobed leaves and 5-petaled flowers (see e.g. [4]), as the "Pentafoil Knot" (visit Bert Jagers' pentafoil page), as the "Double Overhand Knot", as 5_1, or finally as the torus knot T(5,2). When taken off the post the strangle knot (hitch) of practical knot tying deforms to 5_1

A kolam of a 2x3 dot array	The VISA Interlink Logo [1]	Version of the US bicentennial emblem
A pentagonal table by Bob Mackay [2]	The Utah State Parks logo	As impossible object ("Penrose" pentagram)
Folded ribbon which is single-sided (more complex version of Möbius Strip).	Non-pentagonal shape.	Pentagram of circles.
Alternate pentagram of intersecting circles.	3D-looking rendition.	Partial view of US bicentennial logo on a shirt seen in Lisboa [3]
Non-prime knot with two 5_1 configurations on a closed loop.	Knotted epitrochoid	Sum of two 5_1s, Vienna, orthodox church

This sentence was last edited by Dror. Sometime later, Scott added this sentence.

Knot presentations

Planar diagram presentation	X1627 X3849 X5,10,6,1 X7283 X9,4,10,5
Gauss code	-1, 4, -2, 5, -3, 1, -4, 2, -5, 3
Dowker-Thistlethwaite code	6 8 10 2 4
Conway Notation	[5]

Minimum Braid Representative: Length is 5, width is 2. Braid index is 2.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index 3 Nakanishi index 1 Maximal Thurston-Bennequin number [-10][3] Hyperbolic Volume Not hyperbolic A-Polynomial See Data:5 1/A-polynomial

[edit Notes for 5 1's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 2 }[/math] Topological 4 genus [math]\displaystyle{ 2 }[/math] Concordance genus [math]\displaystyle{ \textrm{ConcordanceGenus}(\textrm{Knot}(5,1)) }[/math] Rasmussen s-Invariant -4

[edit Notes for 5 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^2+ t^{-2} -t- t^{-1} +1 }[/math]
Conway polynomial	[math]\displaystyle{ z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 5, -4 }
Jones polynomial	[math]\displaystyle{ - q^{-7} + q^{-6} - q^{-5} + q^{-4} + q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^6 \left(-z^2\right)-2 a^6+a^4 z^4+4 a^4 z^2+3 a^4 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^9 z+a^8 z^2+a^7 z^3-a^7 z+a^6 z^4-3 a^6 z^2+2 a^6+a^5 z^3-2 a^5 z+a^4 z^4-4 a^4 z^2+3 a^4 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2 q^{10}+q^8+q^6 }[/math]
The G2 invariant	[math]\displaystyle{ q^{120}-q^{100}-q^{98}-q^{92}-q^{90}-q^{88}-q^{82}-q^{80}-q^{78}-q^{72}+q^{58}+q^{56}+q^{52}+2 q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+2 q^{40}+q^{38}+q^{34}+q^{32}+q^{30} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 5_1.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{15}+q^7+q^5+q^3 }[/math]
2	[math]\displaystyle{ q^{40}-q^{32}-q^{30}-q^{28}+q^{14}+q^{12}+q^{10}+q^8+q^6 }[/math]
3	[math]\displaystyle{ -q^{75}+q^{67}+q^{65}+q^{63}-q^{49}-q^{47}-q^{45}-q^{43}-q^{41}+q^{21}+q^{19}+q^{17}+q^{15}+q^{13}+q^{11}+q^9 }[/math]
4	[math]\displaystyle{ q^{120}-q^{112}-q^{110}-q^{108}+q^{94}+q^{92}+q^{90}+q^{88}+q^{86}-q^{66}-q^{64}-q^{62}-q^{60}-q^{58}-q^{56}-q^{54}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12} }[/math]
5	[math]\displaystyle{ -q^{175}+q^{167}+q^{165}+q^{163}-q^{149}-q^{147}-q^{145}-q^{143}-q^{141}+q^{121}+q^{119}+q^{117}+q^{115}+q^{113}+q^{111}+q^{109}-q^{83}-q^{81}-q^{79}-q^{77}-q^{75}-q^{73}-q^{71}-q^{69}-q^{67}+q^{35}+q^{33}+q^{31}+q^{29}+q^{27}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15} }[/math]
6	[math]\displaystyle{ q^{240}-q^{232}-q^{230}-q^{228}+q^{214}+q^{212}+q^{210}+q^{208}+q^{206}-q^{186}-q^{184}-q^{182}-q^{180}-q^{178}-q^{176}-q^{174}+q^{148}+q^{146}+q^{144}+q^{142}+q^{140}+q^{138}+q^{136}+q^{134}+q^{132}-q^{100}-q^{98}-q^{96}-q^{94}-q^{92}-q^{90}-q^{88}-q^{86}-q^{84}-q^{82}-q^{80}+q^{42}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18} }[/math]
8	[math]\displaystyle{ q^{400}-q^{392}-q^{390}-q^{388}+q^{374}+q^{372}+q^{370}+q^{368}+q^{366}-q^{346}-q^{344}-q^{342}-q^{340}-q^{338}-q^{336}-q^{334}+q^{308}+q^{306}+q^{304}+q^{302}+q^{300}+q^{298}+q^{296}+q^{294}+q^{292}-q^{260}-q^{258}-q^{256}-q^{254}-q^{252}-q^{250}-q^{248}-q^{246}-q^{244}-q^{242}-q^{240}+q^{202}+q^{200}+q^{198}+q^{196}+q^{194}+q^{192}+q^{190}+q^{188}+q^{186}+q^{184}+q^{182}+q^{180}+q^{178}-q^{134}-q^{132}-q^{130}-q^{128}-q^{126}-q^{124}-q^{122}-q^{120}-q^{118}-q^{116}-q^{114}-q^{112}-q^{110}-q^{108}-q^{106}+q^{56}+q^{54}+q^{52}+q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2 q^{10}+q^8+q^6 }[/math]
1,1	[math]\displaystyle{ q^{60}-2 q^{36}-2 q^{34}-4 q^{32}-4 q^{30}-3 q^{28}+2 q^{24}+4 q^{22}+5 q^{20}+4 q^{18}+4 q^{16}+2 q^{14}+q^{12} }[/math]
2,0	[math]\displaystyle{ q^{54}+q^{52}+2 q^{50}+q^{48}-2 q^{44}-3 q^{42}-3 q^{40}-3 q^{38}-2 q^{36}-q^{34}+q^{28}+q^{26}+2 q^{24}+2 q^{22}+3 q^{20}+2 q^{18}+2 q^{16}+q^{14}+q^{12} }[/math]
3,0	[math]\displaystyle{ -q^{96}-q^{94}-2 q^{92}-2 q^{90}-q^{88}+q^{86}+3 q^{84}+4 q^{82}+5 q^{80}+4 q^{78}+4 q^{76}+2 q^{74}+q^{72}-q^{70}-2 q^{68}-3 q^{66}-4 q^{64}-5 q^{62}-5 q^{60}-5 q^{58}-4 q^{56}-3 q^{54}-2 q^{52}-q^{50}+q^{42}+q^{40}+2 q^{38}+2 q^{36}+3 q^{34}+3 q^{32}+4 q^{30}+3 q^{28}+3 q^{26}+2 q^{24}+2 q^{22}+q^{20}+q^{18} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{50}-q^{36}-2 q^{34}-3 q^{32}-3 q^{30}-2 q^{28}-q^{26}+2 q^{24}+3 q^{22}+4 q^{20}+3 q^{18}+3 q^{16}+q^{14}+q^{12} }[/math]
1,0,0	[math]\displaystyle{ -q^{29}-q^{27}-2 q^{25}-q^{23}+q^{19}+2 q^{17}+2 q^{15}+2 q^{13}+q^{11}+q^9 }[/math]
1,0,1	[math]\displaystyle{ q^{80}+q^{58}+q^{56}+3 q^{54}+3 q^{52}+2 q^{50}-q^{48}-5 q^{46}-9 q^{44}-12 q^{42}-12 q^{40}-9 q^{38}-3 q^{36}+2 q^{34}+7 q^{32}+10 q^{30}+11 q^{28}+10 q^{26}+7 q^{24}+5 q^{22}+2 q^{20}+q^{18} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{64}+q^{62}+q^{60}+q^{58}+q^{56}-q^{50}-2 q^{48}-4 q^{46}-5 q^{44}-6 q^{42}-6 q^{40}-4 q^{38}-q^{36}+2 q^{34}+4 q^{32}+7 q^{30}+6 q^{28}+6 q^{26}+4 q^{24}+3 q^{22}+q^{20}+q^{18} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{36}-q^{34}-2 q^{32}-2 q^{30}-q^{28}+q^{24}+2 q^{22}+3 q^{20}+2 q^{18}+2 q^{16}+q^{14}+q^{12} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{50}-q^{36}-q^{32}-q^{30}+q^{26}+q^{22}+2 q^{20}+q^{18}+q^{16}+q^{14}+q^{12} }[/math]
1,0	[math]\displaystyle{ q^{80}-q^{58}-q^{56}-q^{54}-q^{52}-2 q^{50}-q^{48}-q^{46}-q^{44}+q^{38}+q^{36}+2 q^{34}+q^{32}+2 q^{30}+q^{28}+2 q^{26}+q^{24}+q^{22}+q^{18} }[/math]

B3 Invariants.

Weight	Invariant
1,0,0	[math]\displaystyle{ q^{120}-q^{86}-q^{82}-q^{80}-2 q^{78}-q^{76}-2 q^{74}-q^{72}-2 q^{70}-q^{68}-q^{66}-q^{64}+q^{58}+q^{56}+2 q^{54}+q^{52}+3 q^{50}+q^{48}+3 q^{46}+q^{44}+2 q^{42}+q^{40}+2 q^{38}+q^{34}+q^{30} }[/math]

B4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{160}-q^{114}-q^{110}-2 q^{106}-q^{104}-2 q^{102}-q^{100}-2 q^{98}-q^{96}-2 q^{94}-q^{92}-2 q^{90}-q^{88}-q^{86}+q^{78}+2 q^{74}+q^{72}+3 q^{70}+q^{68}+3 q^{66}+q^{64}+3 q^{62}+q^{60}+3 q^{58}+q^{56}+2 q^{54}+2 q^{50}+q^{46}+q^{42} }[/math]

C3 Invariants.

Weight	Invariant
1,0,0	[math]\displaystyle{ -q^{70}-q^{50}-q^{48}-q^{46}-q^{44}-q^{42}-q^{40}+q^{34}+q^{32}+2 q^{30}+2 q^{28}+2 q^{26}+q^{24}+2 q^{22}+q^{20}+q^{18} }[/math]

C4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ -q^{90}-q^{64}-q^{62}-2 q^{60}-q^{58}-q^{56}-q^{54}-q^{52}-q^{50}-q^{48}+q^{44}+2 q^{42}+2 q^{40}+2 q^{38}+2 q^{36}+2 q^{34}+2 q^{32}+2 q^{30}+2 q^{28}+q^{26}+q^{24} }[/math]

D4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{120}-q^{100}-q^{98}-3 q^{96}-3 q^{94}-q^{92}-q^{90}+2 q^{88}+6 q^{86}+9 q^{84}+11 q^{82}+14 q^{80}+11 q^{78}+9 q^{76}+3 q^{74}-4 q^{72}-12 q^{70}-18 q^{68}-24 q^{66}-27 q^{64}-27 q^{62}-24 q^{60}-17 q^{58}-11 q^{56}+7 q^{52}+14 q^{50}+19 q^{48}+22 q^{46}+19 q^{44}+19 q^{42}+14 q^{40}+10 q^{38}+6 q^{36}+4 q^{34}+q^{32}+q^{30} }[/math]
1,0,0,0	[math]\displaystyle{ q^{70}-q^{50}-q^{48}-3 q^{46}-3 q^{44}-3 q^{42}-3 q^{40}-2 q^{38}+q^{34}+3 q^{32}+4 q^{30}+4 q^{28}+4 q^{26}+3 q^{24}+2 q^{22}+q^{20}+q^{18} }[/math]

G2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{240}-q^{198}-q^{192}+q^{178}+q^{176}+q^{172}+2 q^{170}+q^{168}+q^{166}+q^{164}+q^{162}+2 q^{160}+q^{158}+q^{154}+q^{152}-q^{148}-q^{146}-q^{144}-q^{142}-2 q^{140}-3 q^{138}-2 q^{136}-2 q^{134}-3 q^{132}-4 q^{130}-4 q^{128}-3 q^{126}-3 q^{124}-4 q^{122}-4 q^{120}-3 q^{118}-2 q^{116}-2 q^{114}-3 q^{112}-2 q^{110}+q^{102}+q^{100}+2 q^{98}+3 q^{96}+2 q^{94}+2 q^{92}+4 q^{90}+3 q^{88}+3 q^{86}+4 q^{84}+3 q^{82}+3 q^{80}+4 q^{78}+2 q^{76}+2 q^{74}+3 q^{72}+2 q^{70}+q^{68}+2 q^{66}+q^{64}+q^{62}+q^{60}+q^{54} }[/math]
1,0	[math]\displaystyle{ q^{120}-q^{100}-q^{98}-q^{92}-q^{90}-q^{88}-q^{82}-q^{80}-q^{78}-q^{72}+q^{58}+q^{56}+q^{52}+2 q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+2 q^{40}+q^{38}+q^{34}+q^{32}+q^{30} }[/math]

.

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["5 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^2+ t^{-2} -t- t^{-1} +1 }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^4+3 z^2+1 }[/math]

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]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 5, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ - q^{-7} + q^{-6} - q^{-5} + q^{-4} + q^{-2} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ a^6 \left(-z^2\right)-2 a^6+a^4 z^4+4 a^4 z^2+3 a^4 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^9 z+a^8 z^2+a^7 z^3-a^7 z+a^6 z^4-3 a^6 z^2+2 a^6+a^5 z^3-2 a^5 z+a^4 z^4-4 a^4 z^2+3 a^4 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_132, ...}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {10_132, ...}

Vassiliev invariants

V2 and V3:

(3, -5)

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 [math]\displaystyle{ 12 }[/math] [math]\displaystyle{ -40 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 174 }[/math] [math]\displaystyle{ 26 }[/math] [math]\displaystyle{ -480 }[/math] [math]\displaystyle{ -\frac{2512}{3} }[/math] [math]\displaystyle{ -\frac{448}{3} }[/math] [math]\displaystyle{ -104 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 800 }[/math] [math]\displaystyle{ 2088 }[/math] [math]\displaystyle{ 312 }[/math] [math]\displaystyle{ \frac{41151}{10} }[/math] [math]\displaystyle{ \frac{2494}{15} }[/math] [math]\displaystyle{ \frac{7634}{5} }[/math] [math]\displaystyle{ \frac{43}{2} }[/math] [math]\displaystyle{ \frac{1951}{10} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-4 is the signature of 5 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 χ -3           1 1 -5           1 1 -7       1     1 -9             0 -11   1 1       0 -13             0 -15 1           -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

[math]\displaystyle{ n }[/math]	[math]\displaystyle{ J_n }[/math]
2	[math]\displaystyle{ q^{-4} + q^{-7} - q^{-9} + q^{-10} - q^{-12} + q^{-13} -2 q^{-15} + q^{-16} - q^{-18} + q^{-19} }[/math]
3	[math]\displaystyle{ q^{-6} + q^{-10} - q^{-13} + q^{-14} - q^{-17} + q^{-18} - q^{-21} - q^{-25} + q^{-27} - q^{-29} + q^{-31} + q^{-35} - q^{-36} }[/math]
4	[math]\displaystyle{ q^{-8} + q^{-13} - q^{-17} + q^{-18} - q^{-22} + q^{-23} - q^{-27} + q^{-28} - q^{-29} - q^{-32} + q^{-33} - q^{-34} + q^{-36} - q^{-37} + q^{-38} - q^{-39} + q^{-41} - q^{-42} + q^{-43} - q^{-44} + q^{-45} + q^{-46} - q^{-47} + q^{-48} - q^{-49} + q^{-51} - q^{-52} + q^{-53} - q^{-54} - q^{-57} + q^{-58} }[/math]
5	[math]\displaystyle{ q^{-10} + q^{-16} - q^{-21} + q^{-22} - q^{-27} + q^{-28} - q^{-33} + q^{-34} - q^{-36} - q^{-39} + q^{-40} - q^{-42} + q^{-46} - q^{-48} + q^{-52} - q^{-54} + q^{-57} + q^{-58} - q^{-60} + q^{-63} - q^{-66} + q^{-69} - q^{-72} - q^{-73} + q^{-75} - q^{-79} + q^{-81} + q^{-84} - q^{-85} }[/math]
6	[math]\displaystyle{ q^{-12} + q^{-19} - q^{-25} + q^{-26} - q^{-32} + q^{-33} - q^{-39} + q^{-40} - q^{-43} - q^{-46} + q^{-47} - q^{-50} - q^{-53} +2 q^{-54} - q^{-57} - q^{-60} +2 q^{-61} - q^{-64} - q^{-67} +2 q^{-68} + q^{-69} - q^{-71} - q^{-74} +2 q^{-75} + q^{-76} -2 q^{-78} - q^{-81} +2 q^{-82} + q^{-83} -2 q^{-85} - q^{-88} +2 q^{-89} -2 q^{-92} - q^{-95} +2 q^{-96} + q^{-97} -2 q^{-99} - q^{-102} +2 q^{-103} + q^{-104} - q^{-106} - q^{-109} +2 q^{-110} - q^{-113} - q^{-116} + q^{-117} }[/math]
7	[math]\displaystyle{ q^{-14} + q^{-22} - q^{-29} + q^{-30} - q^{-37} + q^{-38} - q^{-45} + q^{-46} - q^{-50} - q^{-53} + q^{-54} - q^{-58} - q^{-61} + q^{-62} + q^{-63} - q^{-66} - q^{-69} + q^{-70} + q^{-71} - q^{-74} - q^{-77} + q^{-78} + q^{-79} + q^{-81} - q^{-82} - q^{-85} + q^{-86} + q^{-87} + q^{-89} - q^{-90} - q^{-92} - q^{-93} + q^{-94} + q^{-95} + q^{-97} - q^{-98} - q^{-100} - q^{-101} + q^{-102} + q^{-103} + q^{-105} - q^{-106} - q^{-107} - q^{-108} - q^{-109} + q^{-110} + q^{-111} + q^{-113} - q^{-114} - q^{-115} - q^{-117} + q^{-118} + q^{-119} + q^{-121} - q^{-122} - q^{-123} - q^{-125} + q^{-126} + q^{-127} + q^{-128} + q^{-129} - q^{-130} - q^{-131} - q^{-133} + q^{-134} + q^{-136} + q^{-137} - q^{-138} - q^{-139} - q^{-141} + q^{-142} + q^{-145} - q^{-146} - q^{-147} + q^{-150} + q^{-153} - q^{-154} }[/math]

Computer Talk

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[5, 1]]
Out[2]=	PD[X[1, 6, 2, 7], X[3, 8, 4, 9], X[5, 10, 6, 1], X[7, 2, 8, 3], X[9, 4, 10, 5]]
In[3]:=	GaussCode[Knot[5, 1]]
Out[3]=	GaussCode[-1, 4, -2, 5, -3, 1, -4, 2, -5, 3]
In[4]:=	DTCode[Knot[5, 1]]
Out[4]=	DTCode[6, 8, 10, 2, 4]
In[5]:=	br = BR[Knot[5, 1]]
Out[5]=	BR[2, {-1, -1, -1, -1, -1}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{2, 5}
In[7]:=	BraidIndex[Knot[5, 1]]
Out[7]=	2
In[8]:=	Show[DrawMorseLink[Knot[5, 1]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[5, 1]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 2, 3, 1}
In[10]:=	alex = Alexander[Knot[5, 1]][t]
Out[10]=	-2 1 2 1 + t - - - t + t t
In[11]:=	Conway[Knot[5, 1]][z]
Out[11]=	2 4 1 + 3 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[5, 1], Knot[10, 132]}
In[13]:=	{KnotDet[Knot[5, 1]], KnotSignature[Knot[5, 1]]}
Out[13]=	{5, -4}
In[14]:=	Jones[Knot[5, 1]][q]
Out[14]=	-7 -6 -5 -4 -2 -q + q - q + q + q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[5, 1], Knot[10, 132]}
In[16]:=	A2Invariant[Knot[5, 1]][q]
Out[16]=	-22 -20 -18 -14 -12 2 -8 -6 -q - q - q + q + q + --- + q + q 10 q
In[17]:=	HOMFLYPT[Knot[5, 1]][a, z]
Out[17]=	4 6 4 2 6 2 4 4 3 a - 2 a + 4 a z - a z + a z
In[18]:=	Kauffman[Knot[5, 1]][a, z]
Out[18]=	4 6 5 7 9 4 2 6 2 8 2 3 a + 2 a - 2 a z - a z + a z - 4 a z - 3 a z + a z + 5 3 7 3 4 4 6 4 a z + a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[5, 1]], Vassiliev[3][Knot[5, 1]]}
Out[19]=	{3, -5}
In[20]:=	Kh[Knot[5, 1]][q, t]
Out[20]=	-5 -3 1 1 1 1 q + q + ------ + ------ + ------ + ----- 15 5 11 4 11 3 7 2 q t q t q t q t
In[21]:=	ColouredJones[Knot[5, 1], 2][q]
Out[21]=	-19 -18 -16 2 -13 -12 -10 -9 -7 -4 q - q + q - --- + q - q + q - q + q + q 15 q

See/edit the Rolfsen_Splice_Template.