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KnotPlot

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KnotPlot

PD Presentation:

X1627 X11,16,12,17 X5,13,6,12 X3,15,4,14 X13,5,14,4 X15,3,16,2 X7,20,8,1 X9,18,10,19 X17,10,18,11 X19,8,20,9

Gauss Code:

{-1, 6, -4, 5, -3, 1, -7, 10, -8, 9, -2, 3, -5, 4, -6, 2, -9, 8, -10, 7}

DT (Dowker-Thistlethwaite) Code:

6 14 12 20 18 16 4 2 10 8

Minimum Braid Representative: Length is 13, width is 6 Braid index is 6

A Morse Link Presentation:

3D Invariants:

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

Alexander Polynomial:

- 6t-1 + 13 - 6t

Conway Polynomial:

1 - 6z2

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{25, 0}

Jones Polynomial:

q-6 - q-5 + 2q-4 - 3q-3 + 3q-2 - 4q-1 + 4 - 3q + 2q2 - q3 + q4

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

{...}

A2 (sl(3)) Invariant:

q-20 + q-18 + q-14 - q-10 - q-6 - q-4 + q2 - q4 + q8 + q12 + q14

HOMFLY-PT Polynomial:

a-4 - a-2z2 - 2z2 - a2 - 2a2z2 - a4z2 + a6

Kauffman Polynomial:

a-4 - 3a-4z2 + a-4z4 - 2a-3z3 + a-3z5 + a-2z2 - 2a-2z4 + a-2z6 + 4a-1z3 - 3a-1z5 + a-1z7 + 6z4 - 4z6 + z8 + 6az - 15az3 + 15az5 - 6az7 + az9 + a2 - 12a2z2 + 18a2z4 - 10a2z6 + 2a2z8 + 6a3z - 18a3z3 + 15a3z5 - 6a3z7 + a3z9 - 2a4z2 + 4a4z4 - 4a4z6 + a4z8 + 3a5z3 - 4a5z5 + a5z7 - a6 + 6a6z2 - 5a6z4 + a6z6

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

{-6, 3}

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=0 is the signature of 103. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                     1 j = 7                       j = 5                 2 1   j = 3               1       j = 1             3 2       j = -1           2 2         j = -3         1 2           j = -5       2 2             j = -7       1               j = -9   1 2                 j = -11                       j = -13 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-18 - q-17 + 2q-15 - 3q-14 - q-13 + 5q-12 - 4q-11 - 3q-10 + 8q-9 - 4q-8 - 5q-7 + 9q-6 - 2q-5 - 7q-4 + 9q-3 - q-2 - 8q-1 + 9 - q - 6q2 + 6q3 - q4 - 3q5 + 3q6 - q7 - 2q8 + 2q9 - q11 + q12
3	q-36 - q-35 + q-32 - 2q-31 + q-29 + 2q-28 - 4q-27 - q-26 + 3q-25 + 5q-24 - 5q-23 - 5q-22 + 3q-21 + 8q-20 - 3q-19 - 8q-18 + 2q-17 + 8q-16 - q-15 - 7q-14 + q-13 + 4q-12 + q-11 - 4q-10 + q-9 + q-7 - q-6 - q-4 + q-3 + q-2 - q-1 - q + q2 + 2q3 - 3q5 - q6 + 4q7 + 3q8 - 5q9 - 2q10 + 2q11 + 5q12 - 3q13 - 2q14 - q15 + 4q16 - q17 - q18 - 2q19 + 2q20 - q23 + q24

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[10, 3]]
Out[2]=	PD[X[1, 6, 2, 7], X[11, 16, 12, 17], X[5, 13, 6, 12], X[3, 15, 4, 14], > X[13, 5, 14, 4], X[15, 3, 16, 2], X[7, 20, 8, 1], X[9, 18, 10, 19], > X[17, 10, 18, 11], X[19, 8, 20, 9]]
In[3]:=	GaussCode[Knot[10, 3]]
Out[3]=	GaussCode[-1, 6, -4, 5, -3, 1, -7, 10, -8, 9, -2, 3, -5, 4, -6, 2, -9, 8, -10, > 7]
In[4]:=	DTCode[Knot[10, 3]]
Out[4]=	DTCode[6, 14, 12, 20, 18, 16, 4, 2, 10, 8]
In[5]:=	br = BR[Knot[10, 3]]
Out[5]=	BR[6, {-1, -1, -2, 1, -2, -3, 2, 4, -3, 4, 5, -4, 5}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{6, 13}
In[7]:=	BraidIndex[Knot[10, 3]]
Out[7]=	6
In[8]:=	Show[DrawMorseLink[Knot[10, 3]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 3]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 1, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 3]][t]
Out[10]=	6 13 - - - 6 t t
In[11]:=	Conway[Knot[10, 3]][z]
Out[11]=	2 1 - 6 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 3]}
In[13]:=	{KnotDet[Knot[10, 3]], KnotSignature[Knot[10, 3]]}
Out[13]=	{25, 0}
In[14]:=	Jones[Knot[10, 3]][q]
Out[14]=	-6 -5 2 3 3 4 2 3 4 4 + q - q + -- - -- + -- - - - 3 q + 2 q - q + q 4 3 2 q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 3]}
In[16]:=	A2Invariant[Knot[10, 3]][q]
Out[16]=	-20 -18 -14 -10 -6 -4 2 4 8 12 14 q + q + q - q - q - q + q - q + q + q + q
In[17]:=	HOMFLYPT[Knot[10, 3]][a, z]
Out[17]=	2 -4 2 6 2 z 2 2 4 2 a - a + a - 2 z - -- - 2 a z - a z 2 a
In[18]:=	Kauffman[Knot[10, 3]][a, z]
Out[18]=	2 2 -4 2 6 3 3 z z 2 2 4 2 6 2 a + a - a + 6 a z + 6 a z - ---- + -- - 12 a z - 2 a z + 6 a z - 4 2 a a 3 3 4 4 2 z 4 z 3 3 3 5 3 4 z 2 z 2 4 > ---- + ---- - 15 a z - 18 a z + 3 a z + 6 z + -- - ---- + 18 a z + 3 a 4 2 a a a 5 5 6 4 4 6 4 z 3 z 5 3 5 5 5 6 z > 4 a z - 5 a z + -- - ---- + 15 a z + 15 a z - 4 a z - 4 z + -- - 3 a 2 a a 7 2 6 4 6 6 6 z 7 3 7 5 7 8 2 8 > 10 a z - 4 a z + a z + -- - 6 a z - 6 a z + a z + z + 2 a z + a 4 8 9 3 9 > a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 3]], Vassiliev[3][Knot[10, 3]]}
Out[19]=	{-6, 3}
In[20]:=	Kh[Knot[10, 3]][q, t]
Out[20]=	2 1 1 2 1 2 2 1 2 2 - + 3 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 13 6 9 5 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 5 2 5 3 9 4 > 2 q t + q t + 2 q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 3], 2][q]
Out[21]=	-18 -17 2 3 -13 5 4 3 8 4 5 9 2 9 + q - q + --- - --- - q + --- - --- - --- + -- - -- - -- + -- - -- - 15 14 12 11 10 9 8 7 6 5 q q q q q q q q q q 7 9 -2 8 2 3 4 5 6 7 8 9 > -- + -- - q - - - q - 6 q + 6 q - q - 3 q + 3 q - q - 2 q + 2 q - 4 3 q q q 11 12 > q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 103

102 104