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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,3,15,4 X20,8,5,7 X18,10,19,9 X16,12,17,11 X12,16,13,15 X10,18,11,17 X8,20,9,19 X2536 X4,13,1,14

Gauss Code:

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

Jones Polynomial:

- q-9/2 + q-7/2 - 3q-5/2 + 3q-3/2 - 4q-1/2 + 4q1/2 - 4q3/2 + 3q5/2 - 2q7/2 + 2q9/2 - q11/2

A2 (sl(3)) Invariant:

q-16 + 2q-14 + q-12 + 2q-10 + 2q-8 + q-4 + q6 - q10 - q14 + q18

HOMFLY-PT Polynomial:

- a-5z + a-3z + a-3z3 + a-1z3 + az3 - a3z-1 - 2a3z + a5z-1

Kauffman Polynomial:

a-5z - 6a-5z3 + 5a-5z5 - a-5z7 + 7a-4z2 - 17a-4z4 + 11a-4z6 - 2a-4z8 - 3a-3z5 + 4a-3z7 - a-3z9 + 7a-2z2 - 18a-2z4 + 14a-2z6 - 3a-2z8 - a-1z + 6a-1z3 - 6a-1z5 + 4a-1z7 - a-1z9 + 2z6 - z8 + az5 - az7 - a2z6 - a3z-1 + 2a3z - a3z3 - a3z5 + a4 - a4z4 - a5z-1 + 2a5z - a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 12                     1 j = 10                   1   j = 8                 1 1   j = 6               2 1     j = 4             2 1       j = 2           2 2         j = 0         2 2           j = -2       2 3             j = -4     1 1               j = -6     2                 j = -8 1 1                   j = -10 1

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]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 48]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 48]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[20, 8, 5, 7], X[18, 10, 19, 9], > X[16, 12, 17, 11], X[12, 16, 13, 15], X[10, 18, 11, 17], X[8, 20, 9, 19], > X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -8, 4, -7, 5, -6, 10, -2, 6, -5, 7, -4, > 8, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) -(7/2) 3 3 4 3/2 5/2 -q + q - ---- + ---- - ------- + 4 Sqrt[q] - 4 q + 3 q - 5/2 3/2 Sqrt[q] q q 7/2 9/2 11/2 > 2 q + 2 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 -12 2 2 -4 6 10 14 18 q + --- + q + --- + -- + q + q - q - q + q 14 10 8 q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 48]][a, z]
Out[8]=	3 5 3 3 a a z z 3 z z 3 -(--) + -- - -- + -- - 2 a z + -- + -- + a z z z 5 3 3 a a a a
In[9]:=	Kauffman[Link[10, Alternating, 48]][a, z]
Out[9]=	3 5 2 2 3 3 4 a a z z 3 5 7 z 7 z 6 z 6 z 3 3 a - -- - -- + -- - - + 2 a z + 2 a z + ---- + ---- - ---- + ---- - a z - z z 5 a 4 2 5 a a a a a 4 4 5 5 5 5 3 17 z 18 z 4 4 5 z 3 z 6 z 5 3 5 6 > a z - ----- - ----- - a z + ---- - ---- - ---- + a z - a z + 2 z + 4 2 5 3 a a a a a 6 6 7 7 7 8 8 9 9 11 z 14 z 2 6 z 4 z 4 z 7 8 2 z 3 z z z > ----- + ----- - a z - -- + ---- + ---- - a z - z - ---- - ---- - -- - -- 4 2 5 3 a 4 2 3 a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 2 1 1 2 2 2 + -- + ------ + ----- + ----- + ----- + ----- + ---- + ---- + 2 t + 2 q t + 2 10 4 8 4 8 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t 2 2 4 2 4 3 6 3 6 4 8 4 8 5 10 5 > 2 q t + 2 q t + q t + 2 q t + q t + q t + q t + q t + 12 6 > q t

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