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DrawMorseLink

PD Presentation:

X8192 X14,4,15,3 X20,12,13,11 X18,10,19,9 X10,14,11,13 X12,20,7,19 X16,6,17,5 X2738 X4,16,5,15 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

1 - 2q + 4q2 - 5q3 + 8q4 - 7q5 + 8q6 - 6q7 + 4q8 - 2q9 + q10

A2 (sl(3)) Invariant:

1 + q6 + 4q10 + 2q12 + 4q14 + 4q16 + 2q18 + 4q20 + q22 + 2q24 + q26 + q30

HOMFLY-PT Polynomial:

a-8z-2 + 2a-8 + 3a-8z2 + a-8z4 - 2a-6z-2 - 6a-6 - 6a-6z2 - 4a-6z4 - a-6z6 + a-4z-2 + 3a-4 - 3a-4z4 - a-4z6 + a-2 + 3a-2z2 + a-2z4

Kauffman Polynomial:

- 2a-12z2 + a-12z4 - 3a-11z3 + 2a-11z5 - a-10 + 5a-10z2 - 6a-10z4 + 3a-10z6 + 6a-9z3 - 6a-9z5 + 3a-9z7 - a-8z-2 + 5a-8 - 14a-8z2 + 19a-8z4 - 10a-8z6 + 3a-8z8 + 2a-7z-1 - 9a-7z + 16a-7z3 - 9a-7z5 + a-7z7 + a-7z9 - 2a-6z-2 + 11a-6 - 31a-6z2 + 35a-6z4 - 20a-6z6 + 5a-6z8 + 2a-5z-1 - 9a-5z + 12a-5z3 - 8a-5z5 + a-5z9 - a-4z-2 + 5a-4 - 6a-4z2 + 5a-4z4 - 6a-4z6 + 2a-4z8 + 5a-3z3 - 7a-3z5 + 2a-3z7 - a-2 + 4a-2z2 - 4a-2z4 + a-2z6

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 21                     1 j = 19                   2 1 j = 17                 2     j = 15               4 2     j = 13             4 2       j = 11           3 4         j = 9         5 4           j = 7       2 5             j = 5     2 3               j = 3   1 3                 j = 1   1                   j = -1 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]=	3
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 160]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 160]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 4, 15, 3], X[20, 12, 13, 11], X[18, 10, 19, 9], > X[10, 14, 11, 13], X[12, 20, 7, 19], X[16, 6, 17, 5], X[2, 7, 3, 8], > X[4, 16, 5, 15], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9, 7, -10}, {8, -1, 4, -5, 3, -6}, > {5, -2, 9, -7, 10, -4, 6, -3}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 4 5 6 7 8 9 10 1 - 2 q + 4 q - 5 q + 8 q - 7 q + 8 q - 6 q + 4 q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	6 10 12 14 16 18 20 22 24 26 30 1 + q + 4 q + 2 q + 4 q + 4 q + 2 q + 4 q + q + 2 q + q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 160]][a, z]
Out[8]=	2 2 2 4 4 2 6 3 -2 1 2 1 3 z 6 z 3 z z 4 z -- - -- + -- + a + ----- - ----- + ----- + ---- - ---- + ---- + -- - ---- - 8 6 4 8 2 6 2 4 2 8 6 2 8 6 a a a a z a z a z a a a a a 4 4 6 6 3 z z z z > ---- + -- - -- - -- 4 2 6 4 a a a a
In[9]:=	Kauffman[Link[10, Alternating, 160]][a, z]
Out[9]=	-10 5 11 5 -2 1 2 1 2 2 9 z 9 z -a + -- + -- + -- - a - ----- - ----- - ----- + ---- + ---- - --- - --- - 8 6 4 8 2 6 2 4 2 7 5 7 5 a a a a z a z a z a z a z a a 2 2 2 2 2 2 3 3 3 3 2 z 5 z 14 z 31 z 6 z 4 z 3 z 6 z 16 z 12 z > ---- + ---- - ----- - ----- - ---- + ---- - ---- + ---- + ----- + ----- + 12 10 8 6 4 2 11 9 7 5 a a a a a a a a a a 3 4 4 4 4 4 4 5 5 5 5 z z 6 z 19 z 35 z 5 z 4 z 2 z 6 z 9 z > ---- + --- - ---- + ----- + ----- + ---- - ---- + ---- - ---- - ---- - 3 12 10 8 6 4 2 11 9 7 a a a a a a a a a a 5 5 6 6 6 6 6 7 7 7 8 8 z 7 z 3 z 10 z 20 z 6 z z 3 z z 2 z 3 z > ---- - ---- + ---- - ----- - ----- - ---- + -- + ---- + -- + ---- + ---- + 5 3 10 8 6 4 2 9 7 3 8 a a a a a a a a a a a 8 8 9 9 5 z 2 z z z > ---- + ---- + -- + -- 6 4 7 5 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 1 q q 5 7 7 2 9 2 9 3 3 q + 2 q + ---- + - + -- + 3 q t + 2 q t + 5 q t + 5 q t + 4 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 3 q t + 4 q t + 4 q t + 2 q t + 4 q t + 2 q t + 17 6 19 7 19 8 21 8 > 2 q t + 2 q t + q t + q t

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