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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X10,13,5,14 X8,15,9,16 X14,7,15,8 X16,9,11,10 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-10 - 2q-9 + 4q-8 - 4q-7 + 6q-6 - 4q-5 + 4q-4 - 2q-3 + q-2

A2 (sl(3)) Invariant:

q-32 + 2q-30 + q-28 + 3q-26 + 3q-24 + 3q-22 + 5q-20 + 3q-18 + 4q-16 + q-14 + q-10 - q-8 + q-6

HOMFLY-PT Polynomial:

2a4z2 + a4z4 + a6z-2 + 5a6 + 6a6z2 + 2a6z4 - 2a8z-2 - 6a8 - 3a8z2 + a10z-2 + a10

Kauffman Polynomial:

- 2a4z2 + a4z4 - 3a5z3 + 2a5z5 + a6z-2 - 5a6 + 10a6z2 - 8a6z4 + 3a6z6 - 2a7z-1 + 6a7z - 6a7z3 + 2a7z5 + a7z7 + 2a8z-2 - 8a8 + 15a8z2 - 12a8z4 + 5a8z6 - 2a9z-1 + 6a9z - 6a9z3 + 2a9z5 + a9z7 + a10z-2 - 3a10 + a10z2 - 2a10z4 + 2a10z6 - 3a11z3 + 2a11z5 + a12 - 2a12z2 + a12z4

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -3                 1 j = -5               2 1 j = -7             2     j = -9           2 2     j = -11         4 2       j = -13       1 3         j = -15     3 3           j = -17   1 3             j = -19   1               j = -21 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[8, Alternating, 17]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[8, Alternating, 17]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[10, 13, 5, 14], X[8, 15, 9, 16], > X[14, 7, 15, 8], X[16, 9, 11, 10], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -7, 2, -8}, {7, -1, 5, -4, 6, -3}, {8, -2, 3, -5, 4, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-10 2 4 4 6 4 4 2 -2 q - -- + -- - -- + -- - -- + -- - -- + q 9 8 7 6 5 4 3 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 2 -28 3 3 3 5 3 4 -14 -10 -8 -6 q + --- + q + --- + --- + --- + --- + --- + --- + q + q - q + q 30 26 24 22 20 18 16 q q q q q q q
In[8]:=	HOMFLYPT[Link[8, Alternating, 17]][a, z]
Out[8]=	6 8 10 6 8 10 a 2 a a 4 2 6 2 8 2 4 4 5 a - 6 a + a + -- - ---- + --- + 2 a z + 6 a z - 3 a z + a z + 2 2 2 z z z 6 4 > 2 a z
In[9]:=	Kauffman[Link[8, Alternating, 17]][a, z]
Out[9]=	6 8 10 7 9 6 8 10 12 a 2 a a 2 a 2 a 7 9 -5 a - 8 a - 3 a + a + -- + ---- + --- - ---- - ---- + 6 a z + 6 a z - 2 2 2 z z z z z 4 2 6 2 8 2 10 2 12 2 5 3 7 3 > 2 a z + 10 a z + 15 a z + a z - 2 a z - 3 a z - 6 a z - 9 3 11 3 4 4 6 4 8 4 10 4 12 4 > 6 a z - 3 a z + a z - 8 a z - 12 a z - 2 a z + a z + 5 5 7 5 9 5 11 5 6 6 8 6 10 6 > 2 a z + 2 a z + 2 a z + 2 a z + 3 a z + 5 a z + 2 a z + 7 7 9 7 > a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-5 -3 1 1 1 3 3 3 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q t q t q t q t q t q t q t 3 4 2 2 2 2 2 > ------ + ------ + ------ + ----- + ----- + ----- + ---- 13 4 11 4 11 3 9 3 9 2 7 2 5 q t q t q t q t q t q t q t

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