5 2

5_1

6_1

Visit 5 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 5 2's page at Knotilus! Visit 5 2's page at the original Knot Atlas! 5_2 is also known as the 3-twist knot.

3D depiction

Simple square depiction

Lissajous curve x=cos(2t+0.2), y=cos(3t+0.7), z=cos(7t); 2 crossings can be removed

Knot presentations

Planar diagram presentation	X1425 X3849 X5,10,6,1 X9,6,10,7 X7283
Gauss code	-1, 5, -2, 1, -3, 4, -5, 2, -4, 3
Dowker-Thistlethwaite code	4 8 10 2 6
Conway Notation	[32]

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 [-8][1] Hyperbolic Volume 2.82812 A-Polynomial See Data:5 2/A-polynomial

[edit Notes for 5 2's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus ${\textrm {ConcordanceGenus}}({\textrm {Knot}}(5,2))$ Rasmussen s-Invariant -2

[edit Notes for 5 2's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t+2t^{-1}-3$
Conway polynomial	$2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 7, -2 }
Jones polynomial	$-q^{-6}+q^{-5}-q^{-4}+2q^{-3}-q^{-2}+q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$-a^{6}+a^{4}z^{2}+a^{4}+a^{2}z^{2}+a^{2}$
Kauffman polynomial (db, data sources)	$a^{7}z^{3}-2a^{7}z+a^{6}z^{4}-2a^{6}z^{2}+a^{6}+2a^{5}z^{3}-2a^{5}z+a^{4}z^{4}-a^{4}z^{2}+a^{4}+a^{3}z^{3}+a^{2}z^{2}-a^{2}$
The A2 invariant	$-q^{20}-q^{18}+q^{12}+q^{10}+q^{8}+q^{6}+q^{2}$
The G2 invariant	$q^{100}+q^{96}-q^{94}-q^{92}+q^{90}-q^{88}-q^{84}-q^{82}-q^{78}-q^{76}-q^{74}-q^{72}-q^{68}-q^{66}+q^{64}+q^{60}+q^{56}+q^{54}+2q^{50}-q^{48}+2q^{46}+q^{44}+q^{40}+q^{34}+2q^{24}+q^{20}+q^{14}+q^{10}$

Further Quantum Invariants

Further quantum knot invariants for 5_2.

A1 Invariants.

Weight	Invariant
1	$-q^{13}+q^{7}+q^{5}+q$
2	$q^{36}-q^{32}-q^{26}-q^{20}+q^{14}+q^{10}+2q^{8}+q^{2}$
3	$-q^{69}+q^{65}+q^{63}-q^{59}+q^{55}-q^{51}+q^{47}+q^{45}-q^{43}-q^{37}-2q^{35}-q^{33}-q^{31}+q^{27}+q^{21}+2q^{19}-q^{15}+q^{13}+2q^{11}+q^{9}+q^{3}$
4	$q^{112}-q^{108}-q^{106}-q^{104}+q^{102}+q^{100}+q^{98}-2q^{94}+q^{90}+q^{88}-2q^{84}-q^{82}+2q^{78}+q^{76}-q^{74}+q^{70}+2q^{68}+q^{66}-q^{64}-q^{56}-2q^{54}-q^{52}-q^{50}-q^{48}+q^{44}-q^{42}-q^{40}-q^{38}+2q^{34}-q^{30}-q^{28}+q^{26}+4q^{24}+q^{22}-q^{18}+2q^{14}+q^{12}+q^{10}+q^{4}$
5	$-q^{165}+q^{161}+q^{159}+q^{157}-q^{153}-2q^{151}-q^{149}+q^{145}+2q^{143}+q^{141}-q^{139}-2q^{137}-q^{135}+q^{133}+2q^{131}+2q^{129}-2q^{125}-3q^{123}-q^{121}+q^{119}+2q^{117}+q^{115}-2q^{113}-2q^{111}-q^{109}+q^{107}+3q^{105}+q^{103}-q^{101}-q^{99}+q^{95}+2q^{93}+q^{91}-q^{89}+q^{85}+q^{83}+q^{81}-q^{77}-q^{73}-q^{71}-q^{69}-q^{63}-2q^{61}-2q^{59}-2q^{57}+2q^{53}+q^{51}-q^{49}-2q^{47}-2q^{45}+q^{43}+3q^{41}+3q^{39}-3q^{35}-2q^{33}+3q^{29}+3q^{27}+2q^{25}-q^{21}+q^{17}+q^{15}+q^{13}+q^{11}+q^{5}$
6	$q^{228}-q^{224}-q^{222}-q^{220}+2q^{214}+2q^{212}+q^{210}-q^{206}-2q^{204}-3q^{202}+q^{198}+2q^{196}+2q^{194}+q^{192}-q^{190}-4q^{188}-2q^{186}+2q^{182}+3q^{180}+4q^{178}+q^{176}-3q^{174}-3q^{172}-3q^{170}+2q^{166}+4q^{164}+2q^{162}-2q^{160}-3q^{158}-4q^{156}-q^{154}+2q^{152}+4q^{150}+2q^{148}-q^{146}-2q^{144}-3q^{142}-q^{140}+q^{138}+3q^{136}+q^{134}-2q^{132}-2q^{130}-2q^{128}+q^{124}+2q^{122}+q^{120}-q^{118}+q^{116}+q^{114}+2q^{112}+2q^{110}+2q^{108}+q^{106}-q^{98}-q^{96}-q^{94}+q^{90}-q^{88}-q^{86}-2q^{84}-2q^{82}-2q^{80}+3q^{76}+q^{74}-2q^{70}-4q^{68}-4q^{66}-q^{64}+4q^{62}+3q^{60}+2q^{58}-2q^{56}-4q^{54}-5q^{52}-q^{50}+5q^{48}+4q^{46}+4q^{44}+q^{42}-2q^{40}-4q^{38}-2q^{36}+2q^{34}+2q^{32}+3q^{30}+2q^{28}+q^{26}-q^{24}+q^{20}+q^{16}+q^{14}+q^{12}+q^{6}$

A2 Invariants.

Weight	Invariant
0,1	$-q^{20}-q^{18}+q^{12}+q^{10}+q^{8}+q^{6}+q^{2}$
0,2	$q^{50}+q^{48}+q^{46}-q^{44}-q^{42}-q^{40}-q^{38}-q^{36}-q^{34}-q^{30}-q^{28}-q^{26}+2q^{20}+2q^{18}+2q^{16}+2q^{14}+2q^{12}+q^{10}+q^{4}$
1,0	$-q^{20}-q^{18}+q^{12}+q^{10}+q^{8}+q^{6}+q^{2}$
1,1	$q^{52}+2q^{48}-2q^{46}-2q^{42}+2q^{34}-2q^{32}-5q^{28}-4q^{24}+2q^{22}+q^{20}+2q^{18}+4q^{16}+2q^{14}+4q^{12}+2q^{8}+q^{4}$
2,0	$q^{50}+q^{48}+q^{46}-q^{44}-q^{42}-q^{40}-q^{38}-q^{36}-q^{34}-q^{30}-q^{28}-q^{26}+2q^{20}+2q^{18}+2q^{16}+2q^{14}+2q^{12}+q^{10}+q^{4}$
3,0	$-q^{90}-q^{88}-q^{86}+2q^{82}+2q^{80}+2q^{78}-q^{66}+q^{62}+2q^{60}+q^{58}-q^{54}-q^{52}-3q^{50}-4q^{48}-5q^{46}-4q^{44}-3q^{42}-2q^{40}+2q^{36}+3q^{34}+2q^{32}+3q^{30}+2q^{28}+3q^{26}+2q^{24}+q^{22}+q^{20}+2q^{18}+3q^{16}+2q^{14}+q^{6}$

A3 Invariants.

Weight	Invariant
0,0,1	$-q^{27}-q^{25}-q^{23}+q^{17}+q^{15}+q^{13}+q^{11}+q^{9}+q^{7}+q^{3}$
0,1,0	$q^{42}+q^{38}-2q^{34}-2q^{32}-2q^{30}-2q^{28}-q^{26}+q^{24}+q^{22}+2q^{20}+q^{18}+2q^{16}+q^{14}+q^{12}+2q^{10}+q^{8}+q^{4}$
1,0,0	$-q^{27}-q^{25}-q^{23}+q^{17}+q^{15}+q^{13}+q^{11}+q^{9}+q^{7}+q^{3}$
1,0,1	$q^{68}+2q^{64}-q^{60}-3q^{56}+q^{54}-q^{52}+q^{50}+3q^{48}-q^{46}+q^{44}-2q^{42}-4q^{40}-4q^{38}-4q^{36}-5q^{34}-3q^{32}-q^{30}+5q^{26}+3q^{24}+7q^{22}+6q^{20}+3q^{18}+5q^{16}+q^{14}+2q^{12}+2q^{10}+q^{6}$

A4 Invariants.

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

A5 Invariants.

Weight	Invariant
0,0,0,0,1	$-q^{41}-q^{39}-q^{37}-q^{35}-q^{33}+q^{27}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15}+q^{13}+q^{11}+q^{9}+q^{5}$
1,0,0,0,0	$-q^{41}-q^{39}-q^{37}-q^{35}-q^{33}+q^{27}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15}+q^{13}+q^{11}+q^{9}+q^{5}$

A6 Invariants.

Weight	Invariant
0,0,0,0,0,1	$-q^{48}-q^{46}-q^{44}-q^{42}-q^{40}-q^{38}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10}+q^{6}$
1,0,0,0,0,0	$-q^{48}-q^{46}-q^{44}-q^{42}-q^{40}-q^{38}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10}+q^{6}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{42}-q^{38}+q^{26}-q^{24}+q^{22}+q^{18}+q^{14}+q^{12}+q^{8}+q^{4}$
1,0	$q^{68}+q^{60}-q^{56}-q^{54}-q^{52}-q^{50}-q^{48}-q^{46}-q^{44}+q^{34}+q^{32}+q^{30}+q^{26}+q^{24}+q^{22}+q^{18}+q^{16}+q^{14}+q^{6}$

B3 Invariants.

Weight	Invariant
1,0,0	$q^{100}+q^{92}-q^{82}-q^{80}-q^{78}-q^{76}-q^{74}-q^{72}-q^{70}-q^{68}-q^{66}-q^{64}+q^{56}+q^{54}+q^{50}+q^{48}+q^{46}+q^{42}+q^{40}+q^{38}+q^{34}+q^{30}+q^{26}+q^{24}+q^{22}+q^{18}+q^{10}$

B4 Invariants.

Weight	Invariant
1,0,0,0	$q^{132}+q^{124}-q^{110}-q^{106}-q^{104}-q^{102}-q^{100}-q^{98}-q^{96}-q^{94}-q^{92}-q^{90}-q^{88}-q^{86}+q^{74}+q^{72}+q^{70}+q^{66}+q^{64}+q^{62}+q^{58}+q^{56}+q^{54}+q^{50}+q^{46}+q^{42}+q^{38}+q^{34}+q^{32}+q^{30}+q^{26}+q^{22}+q^{14}$

C3 Invariants.

Weight	Invariant
1,0,0	$-q^{58}-q^{54}-q^{48}+q^{30}+q^{26}+q^{22}+q^{20}+q^{18}+q^{16}+q^{12}+q^{10}+q^{6}$

C4 Invariants.

Weight	Invariant
1,0,0,0	$-q^{74}-q^{70}-q^{62}-q^{60}-q^{48}+q^{42}+q^{38}+q^{34}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{16}+q^{14}+q^{12}+q^{8}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{58}+q^{54}-q^{48}-2q^{46}-2q^{44}-2q^{42}-2q^{40}-2q^{38}+2q^{32}+q^{30}+2q^{28}+q^{26}+2q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+2q^{14}+q^{12}+q^{10}+q^{6}$

G2 Invariants.

Weight	Invariant
0,1	$q^{204}+q^{192}+q^{190}+q^{188}-q^{186}-q^{184}-q^{176}+q^{172}-2q^{168}-q^{166}-q^{156}+q^{154}+q^{152}+q^{142}+q^{140}+q^{138}+2q^{136}+q^{134}-q^{132}-2q^{130}-q^{122}-q^{120}-q^{118}-2q^{116}-2q^{114}-3q^{112}-2q^{110}-q^{108}-2q^{106}-2q^{104}-q^{102}-q^{100}-q^{98}-q^{96}-2q^{94}-2q^{92}+q^{88}+q^{86}+2q^{84}+q^{82}+q^{78}+2q^{74}+2q^{72}+2q^{70}+2q^{68}+3q^{66}+2q^{64}+q^{62}+q^{60}+2q^{58}+2q^{56}+q^{54}+q^{50}+3q^{48}+q^{46}+q^{36}+q^{34}+q^{32}+q^{30}+q^{18}$
1,0	$q^{100}+q^{96}-q^{94}-q^{92}+q^{90}-q^{88}-q^{84}-q^{82}-q^{78}-q^{76}-q^{74}-q^{72}-q^{68}-q^{66}+q^{64}+q^{60}+q^{56}+q^{54}+2q^{50}-q^{48}+2q^{46}+q^{44}+q^{40}+q^{34}+2q^{24}+q^{20}+q^{14}+q^{10}$

.

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 2"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$2z^{2}+1$

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]=

{ 7, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{-6}+q^{-5}-q^{-4}+2q^{-3}-q^{-2}+q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{7}z^{3}-2a^{7}z+a^{6}z^{4}-2a^{6}z^{2}+a^{6}+2a^{5}z^{3}-2a^{5}z+a^{4}z^{4}-a^{4}z^{2}+a^{4}+a^{3}z^{3}+a^{2}z^{2}-a^{2}$

Vassiliev invariants

V2 and V3:

(2, -3)

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 $8$ $-24$ $32$ ${\frac {268}{3}}$ ${\frac {44}{3}}$ $-192$ $-368$ $-64$ $-56$ ${\frac {256}{3}}$ $288$ ${\frac {2144}{3}}$ ${\frac {352}{3}}$ ${\frac {22951}{15}}$ $-{\frac {28}{5}}$ ${\frac {29764}{45}}$ ${\frac {137}{9}}$ ${\frac {1351}{15}}$

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 $s=$-2 is the signature of 5 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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[5, 2]]
Out[2]=	5
In[3]:=	PD[Knot[5, 2]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 10, 6, 1], X[9, 6, 10, 7], X[7, 2, 8, 3]]
In[4]:=	GaussCode[Knot[5, 2]]
Out[4]=	GaussCode[-1, 5, -2, 1, -3, 4, -5, 2, -4, 3]
In[5]:=	BR[Knot[5, 2]]
Out[5]=	BR[3, {-1, -1, -1, -2, 1, -2}]
In[6]:=	alex = Alexander[Knot[5, 2]][t]
Out[6]=	2 -3 + - + 2 t t
In[7]:=	Conway[Knot[5, 2]][z]
Out[7]=	2 1 + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[5, 2]}
In[9]:=	{KnotDet[Knot[5, 2]], KnotSignature[Knot[5, 2]]}
Out[9]=	{7, -2}
In[10]:=	J=Jones[Knot[5, 2]][q]
Out[10]=	-6 -5 -4 2 -2 1 -q + q - q + -- - q + - 3 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[5, 2], Knot[11, NonAlternating, 57]}
In[12]:=	A2Invariant[Knot[5, 2]][q]
Out[12]=	-20 -18 -12 -10 -8 -6 -2 -q - q + q + q + q + q + q
In[13]:=	Kauffman[Knot[5, 2]][a, z]
Out[13]=	2 4 6 5 7 2 2 4 2 6 2 3 3 -a + a + a - 2 a z - 2 a z + a z - a z - 2 a z + a z + 5 3 7 3 4 4 6 4 2 a z + a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[5, 2]], Vassiliev[3][Knot[5, 2]]}
Out[14]=	{0, -3}
In[15]:=	Kh[Knot[5, 2]][q, t]
Out[15]=	-3 1 1 1 1 1 1 1 q + - + ------ + ----- + ----- + ----- + ----- + ---- q 13 5 9 4 9 3 7 2 5 2 3 q t q t q t q t q t q t