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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,4,13,3 X20,12,9,11 X2,9,3,10 X18,14,19,13 X14,7,15,8 X16,5,17,6 X6,15,7,16 X4,17,5,18 X8,20,1,19

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 2q-9/2 - 4q-7/2 + 5q-5/2 - 8q-3/2 + 8q-1/2 - 8q1/2 + 7q3/2 - 5q5/2 + 3q7/2 - q9/2

A2 (sl(3)) Invariant:

q-18 + q-16 + 2q-12 + q-10 + q-8 + 3q-6 + q-2 - q2 + q4 - 2q6 + q8 - q12 + q14

HOMFLY-PT Polynomial:

- a-3z - a-3z3 + a-1z + 2a-1z3 + a-1z5 + az + 2az3 + az5 - a3z-1 - 4a3z - 2a3z3 + a5z-1 + a5z

Kauffman Polynomial:

- a-5z3 + a-4z2 - 3a-4z4 - a-3z + 4a-3z3 - 5a-3z5 - 3a-2z2 + 8a-2z4 - 6a-2z6 + a-1z - a-1z3 + 7a-1z5 - 5a-1z7 - 4z2 + 8z4 + 2z6 - 3z8 + 6az - 19az3 + 20az5 - 4az7 - az9 + 2a2z2 - 14a2z4 + 17a2z6 - 5a2z8 - a3z-1 + 9a3z - 21a3z3 + 13a3z5 - a3z9 + a4 + 2a4z2 - 11a4z4 + 9a4z6 - 2a4z8 - a5z-1 + 5a5z - 8a5z3 + 5a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 10                     1 j = 8                   2   j = 6                 3 1   j = 4               4 2     j = 2             4 3       j = 0           5 5         j = -2         3 3           j = -4       2 5             j = -6     2 3               j = -8   1 3                 j = -10   1                   j = -12 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, 99]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 99]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[20, 12, 9, 11], X[2, 9, 3, 10], > X[18, 14, 19, 13], X[14, 7, 15, 8], X[16, 5, 17, 6], X[6, 15, 7, 16], > X[4, 17, 5, 18], X[8, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -9, 7, -8, 6, -10}, > {4, -1, 3, -2, 5, -6, 8, -7, 9, -5, 10, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 4 5 8 8 3/2 5/2 -q + ---- - ---- + ---- - ---- + ------- - 8 Sqrt[q] + 7 q - 5 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 7/2 9/2 > 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 2 -10 -8 3 -2 2 4 6 8 12 14 q + q + --- + q + q + -- + q - q + q - 2 q + q - q + q 12 6 q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 99]][a, z]
Out[8]=	3 5 3 3 5 a a z z 3 5 z 2 z 3 3 3 z -(--) + -- - -- + - + a z - 4 a z + a z - -- + ---- + 2 a z - 2 a z + -- + z z 3 a 3 a a a a 5 > a z
In[9]:=	Kauffman[Link[10, Alternating, 99]][a, z]
Out[9]=	3 5 2 2 4 a a z z 3 5 2 z 3 z 2 2 a - -- - -- - -- + - + 6 a z + 9 a z + 5 a z - 4 z + -- - ---- + 2 a z + z z 3 a 4 2 a a a 3 3 3 4 4 2 z 4 z z 3 3 3 5 3 4 3 z > 2 a z - -- + ---- - -- - 19 a z - 21 a z - 8 a z + 8 z - ---- + 5 3 a 4 a a a 4 5 5 8 z 2 4 4 4 5 z 7 z 5 3 5 5 5 > ---- - 14 a z - 11 a z - ---- + ---- + 20 a z + 13 a z + 5 a z + 2 3 a a a 6 7 6 6 z 2 6 4 6 5 z 7 5 7 8 2 8 > 2 z - ---- + 17 a z + 9 a z - ---- - 4 a z - a z - 3 z - 5 a z - 2 a a 4 8 9 3 9 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 3 2 3 2 5 5 + 4 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 3 5 3 2 4 4 2 6 2 6 3 8 3 > ----- + - + ---- + 3 q t + 4 q t + 2 q t + 3 q t + q t + 2 q t + 2 2 t 2 q t q t 10 4 > q t

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