© | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table:

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KnotPlot

This page is passe. Go here instead!    The Alternating Knot 51    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 the torus knot T(5,2). Visit 51's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 51's page at Knotilus! Acknowledgement

KnotPlot

Further views:

The VISA Interlink Logo

A shirt seen in Lisboa

A pentagonal table by Bob Mackay

PD Presentation:

X1627 X3849 X5,10,6,1 X7283 X9,4,10,5

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 8 10 2 4

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

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index Reversible 2 2 2 / 3 1

Alexander Polynomial:

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

Conway Polynomial:

1 + 3z2 + z4

Other knots with the same Alexander/Conway Polynomial:

{10132, ...}

Determinant and Signature:

{5, -4}

Jones Polynomial:

- q-7 + q-6 - q-5 + q-4 + q-2

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

{10132, ...}

A2 (sl(3)) Invariant:

- q-22 - q-20 - q-18 + q-14 + q-12 + 2q-10 + q-8 + q-6

HOMFLY-PT Polynomial:

3a4 + 4a4z2 + a4z4 - 2a6 - a6z2

Kauffman Polynomial:

3a4 - 4a4z2 + a4z4 - 2a5z + a5z3 + 2a6 - 3a6z2 + a6z4 - a7z + a7z3 + a8z2 + a9z

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

{3, -5}

Khovanov Homology: (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 51. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -3           1 j = -5           1 j = -7       1     j = -9             j = -11   1 1       j = -13             j = -15 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-19 - q-18 + q-16 - 2q-15 + q-13 - q-12 + q-10 - q-9 + q-7 + q-4
3	- q-36 + q-35 + q-31 - q-29 + q-27 - q-25 - q-21 + q-18 - q-17 + q-14 - q-13 + q-10 + q-6
4	q-58 - q-57 - q-54 + q-53 - q-52 + q-51 - q-49 + q-48 - q-47 + q-46 + q-45 - q-44 + q-43 - q-42 + q-41 - q-39 + q-38 - q-37 + q-36 - q-34 + q-33 - q-32 - q-29 + q-28 - q-27 + q-23 - q-22 + q-18 - q-17 + q-13 + q-8
5	- q-85 + q-84 + q-81 - q-79 + q-75 - q-73 - q-72 + q-69 - q-66 + q-63 - q-60 + q-58 + q-57 - q-54 + q-52 - q-48 + q-46 - q-42 + q-40 - q-39 - q-36 + q-34 - q-33 + q-28 - q-27 + q-22 - q-21 + q-16 + q-10
6	q-117 - q-116 - q-113 + 2q-110 - q-109 - q-106 + q-104 + 2q-103 - q-102 - 2q-99 + q-97 + 2q-96 - q-95 - 2q-92 + 2q-89 - q-88 - 2q-85 + q-83 + 2q-82 - q-81 - 2q-78 + q-76 + 2q-75 - q-74 - q-71 + q-69 + 2q-68 - q-67 - q-64 + 2q-61 - q-60 - q-57 + 2q-54 - q-53 - q-50 + q-47 - q-46 - q-43 + q-40 - q-39 + q-33 - q-32 + q-26 - q-25 + q-19 + q-12
7	- q-154 + q-153 + q-150 - q-147 - q-146 + q-145 + q-142 - q-141 - q-139 - q-138 + q-137 + q-136 + q-134 - q-133 - q-131 - q-130 + q-129 + q-128 + q-127 + q-126 - q-125 - q-123 - q-122 + q-121 + q-119 + q-118 - q-117 - q-115 - q-114 + q-113 + q-111 + q-110 - q-109 - q-108 - q-107 - q-106 + q-105 + q-103 + q-102 - q-101 - q-100 - q-98 + q-97 + q-95 + q-94 - q-93 - q-92 - q-90 + q-89 + q-87 + q-86 - q-85 - q-82 + q-81 + q-79 + q-78 - q-77 - q-74 + q-71 + q-70 - q-69 - q-66 + q-63 + q-62 - q-61 - q-58 + q-54 - q-53 - q-50 + q-46 - q-45 + q-38 - q-37 + q-30 - q-29 + q-22 + q-14

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]:=	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 5 3 3 a + 2 a - 2 a z - a z + a z - 4 a z - 3 a z + a z + a z + 7 3 4 4 6 4 > 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

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 51

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