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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9/2 - 2q-7/2 + 3q-5/2 - 6q-3/2 + 6q-1/2 - 7q1/2 + 6q3/2 - 5q5/2 + 3q7/2 - q9/2

A2 (sl(3)) Invariant:

- q-14 + 3q-6 + q-4 + 2q-2 + 2 + 2q4 - q6 + q8 - q12 + q14

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-5z3 + a-4z2 - 3a-4z4 - 2a-3z + 5a-3z3 - 5a-3z5 - 2a-2z2 + 6a-2z4 - 5a-2z6 - a-1z-1 + 4a-1z3 + a-1z5 - 3a-1z7 + 1 - 5z2 + 12z4 - 4z6 - z8 - az-1 + 6az - 12az3 + 14az5 - 5az7 - 6a2z2 + 7a2z4 - a2z8 + 4a3z - 10a3z3 + 8a3z5 - 2a3z7 - 4a4z2 + 4a4z4 - a4z6

Khovanov Homology:

trqj 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             3 2     j = 2           4 3       j = 0         4 5         j = -2       2 2           j = -4     1 4             j = -6   1 2               j = -8   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[9, Alternating, 35]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 35]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[16, 8, 17, 7], X[14, 6, 15, 5], > X[18, 13, 9, 14], X[6, 16, 7, 15], X[4, 18, 5, 17], X[2, 9, 3, 10], > X[8, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -7, 4, -6, 3, -9}, {8, -1, 9, -2, 5, -4, 6, -3, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 2 3 6 6 3/2 5/2 7/2 q - ---- + ---- - ---- + ------- - 7 Sqrt[q] + 6 q - 5 q + 3 q - 7/2 5/2 3/2 Sqrt[q] q q q 9/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 3 -4 2 4 6 8 12 14 2 - q + -- + q + -- + 2 q - q + q - q + q 6 2 q q
In[8]:=	HOMFLYPT[Link[9, Alternating, 35]][a, z]
Out[8]=	3 3 5 1 a z 3 z 2 z 3 3 3 z 5 -(---) + - - -- + 3 a z - 2 a z - -- + ---- + 3 a z - a z + -- + a z a z z 3 3 a a a a
In[9]:=	Kauffman[Link[9, Alternating, 35]][a, z]
Out[9]=	2 2 1 a 2 z 3 2 z 2 z 2 2 4 2 1 - --- - - - --- + 6 a z + 4 a z - 5 z + -- - ---- - 6 a z - 4 a z - a z z 3 4 2 a a a 3 3 3 4 4 z 5 z 4 z 3 3 3 4 3 z 6 z 2 4 > -- + ---- + ---- - 12 a z - 10 a z + 12 z - ---- + ---- + 7 a z + 5 3 a 4 2 a a a a 5 5 6 7 4 4 5 z z 5 3 5 6 5 z 4 6 3 z > 4 a z - ---- + -- + 14 a z + 8 a z - 4 z - ---- - a z - ---- - 3 a 2 a a a 7 3 7 8 2 8 > 5 a z - 2 a z - z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 2 1 4 2 4 2 5 + 4 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t q t 2 4 4 2 6 2 6 3 8 3 10 4 > 3 q t + 3 q t + 2 q t + 3 q t + q t + 2 q t + q t

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