© | Dror Bar-Natan: The Knot Atlas: Torus Knots:

T(3,2) T(7,2)

TubePlot

This page is passe. Go here instead!    The 5-Crossing Torus Knot T(5,2) Also known 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. 1), as "The Pentafoil Knot" (visit Bert Jagers' pentafoil page), as the "Double Overhand Knot", and as 51. Visit T(5,2)'s page at Knotilus! Acknowledgement

Further views:

The VISA Interlink Logo

A shirt seen in Lisboa

A pentagonal table by Bob Mackay

PD Presentation:

X3948 X9,5,10,4 X5,1,6,10 X1726 X7382

Gauss Code:

{-4, 5, -1, 2, -3, 4, -5, 1, -2, 3}

Braid Representative:

Alexander Polynomial:

t-2 - t-1 + 1 - t + t2

Conway Polynomial:

1 + 3z2 + z4

Other knots with the same Alexander/Conway Polynomial:

{51, 10132, ...}

Determinant and Signature:

{5, 4}

Jones Polynomial:

q2 + q4 - q5 + q6 - q7

Other knots (up to mirrors) with the same Jones Polynomial:

{51, 10132, ...}

Coloured Jones Polynomial (in the 3-dimensional representation of sl(2); n=2):

q4 + q7 - q9 + q10 - q12 + q13 - 2q15 + q16 - q18 + q19

A2 (sl(3)) Invariant:

q6 + q8 + 2q10 + q12 + q14 - q18 - q20 - q22

Kauffman Polynomial:

a-9z + a-8z2 - a-7z + a-7z3 + 2a-6 - 3a-6z2 + a-6z4 - 2a-5z + a-5z3 + 3a-4 - 4a-4z2 + a-4z4

V2 and V3, the type 2 and 3 Vassiliev invariants:

{3, 5}

Khovanov Homology. The coefficients of the monomials t^rq^j are shown, along with their alternating sums χ (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=4 is the signature of T(5,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	TubePlot[TorusKnot[5, 2]]
Out[2]=	-Graphics-
In[3]:=	Crossings[TorusKnot[5, 2]]
Out[3]=	5
In[4]:=	PD[TorusKnot[5, 2]]
Out[4]=	PD[X[3, 9, 4, 8], X[9, 5, 10, 4], X[5, 1, 6, 10], X[1, 7, 2, 6], X[7, 3, 8, 2]]
In[5]:=	GaussCode[TorusKnot[5, 2]]
Out[5]=	GaussCode[-4, 5, -1, 2, -3, 4, -5, 1, -2, 3]
In[6]:=	BR[TorusKnot[5, 2]]
Out[6]=	BR[2, {1, 1, 1, 1, 1}]
In[7]:=	alex = Alexander[TorusKnot[5, 2]][t]
Out[7]=	-2 1 2 1 + t - - - t + t t
In[8]:=	Conway[TorusKnot[5, 2]][z]
Out[8]=	2 4 1 + 3 z + z
In[9]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=	{Knot[5, 1], Knot[10, 132]}
In[10]:=	{KnotDet[TorusKnot[5, 2]], KnotSignature[TorusKnot[5, 2]]}
Out[10]=	{5, 4}
In[11]:=	J=Jones[TorusKnot[5, 2]][q]
Out[11]=	2 4 5 6 7 q + q - q + q - q
In[12]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[12]=	{Knot[5, 1], Knot[10, 132]}
In[13]:=	ColouredJones[TorusKnot[5, 2], 2][q]
Out[13]=	4 7 9 10 12 13 15 16 18 19 q + q - q + q - q + q - 2 q + q - q + q
In[14]:=	A2Invariant[TorusKnot[5, 2]][q]
Out[14]=	6 8 10 12 14 18 20 22 q + q + 2 q + q + q - q - q - q
In[15]:=	Kauffman[TorusKnot[5, 2]][a, z]
Out[15]=	2 2 2 3 3 4 4 2 3 z z 2 z z 3 z 4 z z z z z -- + -- + -- - -- - --- + -- - ---- - ---- + -- + -- + -- + -- 6 4 9 7 5 8 6 4 7 5 6 4 a a a a a a a a a a a a
In[16]:=	{Vassiliev[2][TorusKnot[5, 2]], Vassiliev[3][TorusKnot[5, 2]]}
Out[16]=	{3, 5}
In[17]:=	Kh[TorusKnot[5, 2]][q, t]
Out[17]=	3 5 7 2 11 3 11 4 15 5 q + q + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(5,2)

T(3,2) T(7,2)