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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X7,16,8,17 X17,22,18,5 X11,18,12,19 X13,20,14,21 X19,12,20,13 X21,14,22,15 X15,8,16,9 X2536 X4,9,1,10

Gauss Code:

{{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -5, 7, -6, 8, -9, 3, -4, 5, -7, 6, -8, 4}}

Jones Polynomial:

q-27/2 - q-25/2 + 2q-23/2 - 2q-21/2 + 2q-19/2 - 3q-17/2 + q-15/2 - q-13/2 - q-7/2

A2 (sl(3)) Invariant:

- q-42 - 2q-40 - 2q-38 - 3q-36 + q-32 + 2q-30 + 3q-28 + 2q-26 + 3q-24 + q-22 + q-20 + q-18 + q-16 + q-14 + q-12

HOMFLY-PT Polynomial:

- a7z-1 - 8a7z - 14a7z3 - 7a7z5 - a7z7 - a9z-1 + a9z + a9z3 + 4a11z-1 + 6a11z + 2a11z3 - 2a13z-1 - a13z

Kauffman Polynomial:

- a7z-1 + 8a7z - 14a7z3 + 7a7z5 - a7z7 + a8 - 2a8z2 + a8z4 + a9z-1 - a9z - 2a9z3 + a9z5 - 5a10 + 18a10z2 - 20a10z4 + 8a10z6 - a10z8 + 4a11z-1 - 18a11z + 30a11z3 - 23a11z5 + 8a11z7 - a11z9 - 6a12 + 24a12z2 - 28a12z4 + 13a12z6 - 2a12z8 + 2a13z-1 - 10a13z + 16a13z3 - 13a13z5 + 6a13z7 - a13z9 + a14 - 2a14z2 - 2a14z4 + 4a14z6 - a14z8 - a15z - 2a15z3 + 4a15z5 - a15z7 + 2a16 - 6a16z2 + 5a16z4 - a16z6

Khovanov Homology:

trqj r = -11 r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -6                       1 j = -8                       1 j = -10                 1 1     j = -12               1         j = -14             2 3 1       j = -16           2             j = -18         1 3 1           j = -20       2 2               j = -22       1 1               j = -24   1 2                   j = -26                         j = -28 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[11, NonAlternating, 68]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 68]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[7, 16, 8, 17], X[17, 22, 18, 5], > X[11, 18, 12, 19], X[13, 20, 14, 21], X[19, 12, 20, 13], X[21, 14, 22, 15], > X[15, 8, 16, 9], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -5, 7, -6, 8, -9, 3, -4, 5, > -7, 6, -8, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) -(25/2) 2 2 2 3 -(15/2) -(13/2) q - q + ----- - ----- + ----- - ----- + q - q - 23/2 21/2 19/2 17/2 q q q q -(7/2) > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 2 2 3 -32 2 3 2 3 -22 -20 -18 -q - --- - --- - --- + q + --- + --- + --- + --- + q + q + q + 40 38 36 30 28 26 24 q q q q q q q -16 -14 -12 > q + q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 68]][a, z]
Out[8]=	7 9 11 13 a a 4 a 2 a 7 9 11 13 7 3 -(--) - -- + ----- - ----- - 8 a z + a z + 6 a z - a z - 14 a z + z z z z 9 3 11 3 7 5 7 7 > a z + 2 a z - 7 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 68]][a, z]
Out[9]=	7 9 11 13 8 10 12 14 16 a a 4 a 2 a 7 9 a - 5 a - 6 a + a + 2 a - -- + -- + ----- + ----- + 8 a z - a z - z z z z 11 13 15 8 2 10 2 12 2 14 2 > 18 a z - 10 a z - a z - 2 a z + 18 a z + 24 a z - 2 a z - 16 2 7 3 9 3 11 3 13 3 15 3 8 4 > 6 a z - 14 a z - 2 a z + 30 a z + 16 a z - 2 a z + a z - 10 4 12 4 14 4 16 4 7 5 9 5 11 5 > 20 a z - 28 a z - 2 a z + 5 a z + 7 a z + a z - 23 a z - 13 5 15 5 10 6 12 6 14 6 16 6 7 7 > 13 a z + 4 a z + 8 a z + 13 a z + 4 a z - a z - a z + 11 7 13 7 15 7 10 8 12 8 14 8 11 9 13 9 > 8 a z + 6 a z - a z - a z - 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 1 2 1 2 1 2 q + q + ------- + ------- + ------ + ------ + ------ + ------ + ------ + 28 11 24 10 24 9 22 8 20 8 22 7 20 7 q t q t q t q t q t q t q t 1 3 2 1 2 3 1 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 7 18 6 16 6 18 5 14 5 14 4 12 4 14 3 q t q t q t q t q t q t q t q t 1 1 > ------ + ------ 10 3 10 2 q t q t

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