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DrawMorseLink

PD Presentation:

X6172 X3,10,4,11 X9,20,10,21 X13,18,14,19 X7,14,8,15 X17,8,18,9 X19,12,20,13 X15,22,16,5 X21,16,22,17 X2536 X11,4,12,1

Gauss Code:

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

Jones Polynomial:

3q-23/2 - 6q-21/2 + 10q-19/2 - 14q-17/2 + 14q-15/2 - 14q-13/2 + 11q-11/2 - 8q-9/2 + 3q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-40 - q-38 - 3q-36 - 2q-34 + 2q-32 - 2q-30 + 3q-28 + 2q-26 + q-24 + 5q-22 - q-20 + 4q-18 + 3q-12 - 2q-10 + q-8

HOMFLY-PT Polynomial:

- a5z - 2a5z3 - a5z5 - 2a7z-1 - 7a7z - 9a7z3 - 3a7z5 + 2a9z-1 + 4a9z - a9z5 + a11z-1 + 2a11z + a11z3 - a13z-1

Kauffman Polynomial:

- a5z + 2a5z3 - a5z5 - a6z2 + 4a6z4 - 3a6z6 - 2a7z-1 + 9a7z - 12a7z3 + 12a7z5 - 6a7z7 + 2a8 - 7a8z2 + 8a8z4 + 3a8z6 - 5a8z8 - 2a9z-1 + 12a9z - 24a9z3 + 24a9z5 - 8a9z7 - 2a9z9 - 4a10 + 11a10z2 - 17a10z4 + 19a10z6 - 10a10z8 + a11z-1 + a11z - 11a11z3 + 14a11z5 - 5a11z7 - 2a11z9 - 9a12 + 28a12z2 - 27a12z4 + 13a12z6 - 5a12z8 + a13z-1 - a13z - a13z3 + 3a13z5 - 3a13z7 - 4a14 + 11a14z2 - 6a14z4

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                   1 j = -6                 3 1 j = -8               5     j = -10             6 3     j = -12           8 5       j = -14         7 7         j = -16       7 7           j = -18     3 7             j = -20   3 7               j = -22   3                 j = -24 3

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, 67]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 67]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 10, 4, 11], X[9, 20, 10, 21], X[13, 18, 14, 19], > X[7, 14, 8, 15], X[17, 8, 18, 9], X[19, 12, 20, 13], X[15, 22, 16, 5], > X[21, 16, 22, 17], X[2, 5, 3, 6], X[11, 4, 12, 1]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -5, 6, -3, 2, -11, 7, -4, 5, -8, 9, -6, 4, > -7, 3, -9, 8}]
In[6]:=	Jones[L][q]
Out[6]=	3 6 10 14 14 14 11 8 3 -(5/2) ----- - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - q 23/2 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 -38 3 2 2 2 3 2 -24 5 -20 4 -q - q - --- - --- + --- - --- + --- + --- + q + --- - q + --- + 36 34 32 30 28 26 22 18 q q q q q q q q 3 2 -8 > --- - --- + q 12 10 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 67]][a, z]
Out[8]=	7 9 11 13 -2 a 2 a a a 5 7 9 11 5 3 ----- + ---- + --- - --- - a z - 7 a z + 4 a z + 2 a z - 2 a z - z z z z 7 3 11 3 5 5 7 5 9 5 > 9 a z + a z - a z - 3 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 67]][a, z]
Out[9]=	7 9 11 13 8 10 12 14 2 a 2 a a a 5 7 2 a - 4 a - 9 a - 4 a - ---- - ---- + --- + --- - a z + 9 a z + z z z z 9 11 13 6 2 8 2 10 2 12 2 > 12 a z + a z - a z - a z - 7 a z + 11 a z + 28 a z + 14 2 5 3 7 3 9 3 11 3 13 3 6 4 > 11 a z + 2 a z - 12 a z - 24 a z - 11 a z - a z + 4 a z + 8 4 10 4 12 4 14 4 5 5 7 5 9 5 > 8 a z - 17 a z - 27 a z - 6 a z - a z + 12 a z + 24 a z + 11 5 13 5 6 6 8 6 10 6 12 6 > 14 a z + 3 a z - 3 a z + 3 a z + 19 a z + 13 a z - 7 7 9 7 11 7 13 7 8 8 10 8 12 8 > 6 a z - 8 a z - 5 a z - 3 a z - 5 a z - 10 a z - 5 a z - 9 9 11 9 > 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 3 3 3 7 3 7 7 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q t q t q t q t q t q t q t 7 7 7 8 5 6 3 5 3 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ---- 16 5 14 5 14 4 12 4 12 3 10 3 10 2 8 2 6 q t q t q t q t q t q t q t q t q t

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