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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 2q-5/2 - 6q-3/2 + 8q-1/2 - 12q1/2 + 13q3/2 - 12q5/2 + 10q7/2 - 7q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

q-12 + q-10 + 4q-6 + 2q-4 + q-2 + 5 - q2 + q4 - 2q6 - 2q8 + q10 - 3q12 + 2q14 - 2q18 + q20

HOMFLY-PT Polynomial:

- a-5z3 + a-3z-1 + a-3z + a-3z3 + a-3z5 - 2a-1z-1 - 2a-1z + a-1z5 - 2az - 2az3 + a3z-1 + a3z

Kauffman Polynomial:

a-7z3 - a-7z5 - 2a-6z2 + 7a-6z4 - 4a-6z6 - 4a-5z3 + 11a-5z5 - 6a-5z7 + 2a-4 - 7a-4z2 + 9a-4z4 + a-4z6 - 4a-4z8 - a-3z-1 + a-3z - 8a-3z3 + 18a-3z5 - 9a-3z7 - a-3z9 + 5a-2 - 16a-2z2 + 12a-2z4 + 3a-2z6 - 6a-2z8 - 2a-1z-1 + 5a-1z - 8a-1z3 + 11a-1z5 - 6a-1z7 - a-1z9 + 3 - 11z2 + 13z4 - 4z6 - 2z8 + az - 2az3 + 4az5 - 3az7 - a2 + 3a2z4 - 2a2z6 + a3z-1 - 3a3z + 3a3z3 - a3z5

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 = 14                     1 j = 12                   3   j = 10                 4 1   j = 8               6 3     j = 6             6 4       j = 4           7 6         j = 2         5 6           j = 0       4 8             j = -2     2 4               j = -4     4                 j = -6 1 2                   j = -8 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, 30]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 30]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[14, 8, 15, 7], X[18, 10, 19, 9], > X[20, 16, 5, 15], X[16, 20, 17, 19], X[8, 18, 9, 17], X[10, 14, 11, 13], > X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -7, 4, -8, 10, -2, 8, -3, 5, -6, 7, -4, > 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 2 6 8 3/2 5/2 7/2 -q + ---- - ---- + ------- - 12 Sqrt[q] + 13 q - 12 q + 10 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 > 7 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 4 2 -2 2 4 6 8 10 12 14 5 + q + q + -- + -- + q - q + q - 2 q - 2 q + q - 3 q + 2 q - 6 4 q q 18 20 > 2 q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 30]][a, z]
Out[8]=	3 3 3 5 5 1 2 a z 2 z 3 z z 3 z z ---- - --- + -- + -- - --- - 2 a z + a z - -- + -- - 2 a z + -- + -- 3 a z z 3 a 5 3 3 a a z a a a a
In[9]:=	Kauffman[Link[10, Alternating, 30]][a, z]
Out[9]=	3 2 2 5 2 1 2 a z 5 z 3 2 2 z 3 + -- + -- - a - ---- - --- + -- + -- + --- + a z - 3 a z - 11 z - ---- - 4 2 3 a z z 3 a 6 a a a z a a 2 2 3 3 3 3 4 7 z 16 z z 4 z 8 z 8 z 3 3 3 4 7 z > ---- - ----- + -- - ---- - ---- - ---- - 2 a z + 3 a z + 13 z + ---- + 4 2 7 5 3 a 6 a a a a a a 4 4 5 5 5 5 9 z 12 z 2 4 z 11 z 18 z 11 z 5 3 5 > ---- + ----- + 3 a z - -- + ----- + ----- + ----- + 4 a z - a z - 4 2 7 5 3 a a a a a a 6 6 6 7 7 7 6 4 z z 3 z 2 6 6 z 9 z 6 z 7 8 > 4 z - ---- + -- + ---- - 2 a z - ---- - ---- - ---- - 3 a z - 2 z - 6 4 2 5 3 a a a a a a 8 8 9 9 4 z 6 z z z > ---- - ---- - -- - -- 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 2 4 2 4 4 2 4 8 + 5 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 6 q t + 7 q t + 8 4 6 4 6 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 2 6 2 6 3 8 3 8 4 10 4 10 5 > 6 q t + 6 q t + 4 q t + 6 q t + 3 q t + 4 q t + q t + 12 5 14 6 > 3 q t + q t

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