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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X5,10,6,11 X3849 X13,22,14,5 X11,20,12,21 X21,12,22,13 X19,14,20,15 X9,18,10,19 X15,2,16,3

Gauss Code:

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

Jones Polynomial:

q-25/2 - 2q-23/2 + 4q-21/2 - 4q-19/2 + 4q-17/2 - 5q-15/2 + 3q-13/2 - 3q-11/2 + q-9/2 - q-7/2

A2 (sl(3)) Invariant:

- q-42 - q-38 - q-36 - q-34 - 2q-32 - q-28 + 3q-26 + 3q-24 + 3q-22 + 3q-20 + q-18 + 2q-16 + q-12

HOMFLY-PT Polynomial:

- 2a7z-1 - 7a7z - 11a7z3 - 6a7z5 - a7z7 + 2a9z-1 - 6a9z3 - 5a9z5 - a9z7 + a11z-1 + 6a11z + 5a11z3 + a11z5 - a13z-1 - a13z

Kauffman Polynomial:

- 2a7z-1 + 7a7z - 11a7z3 + 6a7z5 - a7z7 + 3a8 - 4a8z2 - 2a8z4 + 4a8z6 - a8z8 - 2a9z-1 + 5a9z - 2a9z3 - 3a9z5 + 4a9z7 - a9z9 + 6a10z2 - 15a10z4 + 13a10z6 - 3a10z8 + a11z-1 - 2a11z + 3a11z3 - 2a11z5 + 3a11z7 - a11z9 - 3a12 + 8a12z2 - 11a12z4 + 8a12z6 - 2a12z8 + a13z-1 + a13z - 8a13z3 + 7a13z5 - 2a13z7 - 3a14z2 + 2a14z4 - a14z6 + a15z - 2a15z3 + a16 - a16z2

Khovanov Homology:

trqj 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 1 j = -10               2     j = -12           1 2 1     j = -14           4 2       j = -16       1 2 2         j = -18       3 3           j = -20     2 2             j = -22   1 3               j = -24   1                 j = -26 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, 10]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 10]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[5, 10, 6, 11], > X[3, 8, 4, 9], X[13, 22, 14, 5], X[11, 20, 12, 21], X[21, 12, 22, 13], > X[19, 14, 20, 15], X[9, 18, 10, 19], X[15, 2, 16, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -3}, {-4, -1, 2, 5, -10, 4, -7, 8, -6, 9, -11, -2, 3, 10, > -9, 7, -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) 2 4 4 4 5 3 3 -(9/2) q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + q - 23/2 21/2 19/2 17/2 15/2 13/2 11/2 q q q q q q q -(7/2) > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -38 -36 -34 2 -28 3 3 3 3 -18 2 -q - q - q - q - --- - q + --- + --- + --- + --- + q + --- + 32 26 24 22 20 16 q q q q q q -12 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 10]][a, z]
Out[8]=	7 9 11 13 -2 a 2 a a a 7 11 13 7 3 9 3 ----- + ---- + --- - --- - 7 a z + 6 a z - a z - 11 a z - 6 a z + z z z z 11 3 7 5 9 5 11 5 7 7 9 7 > 5 a z - 6 a z - 5 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 10]][a, z]
Out[9]=	7 9 11 13 8 12 16 2 a 2 a a a 7 9 11 3 a - 3 a + a - ---- - ---- + --- + --- + 7 a z + 5 a z - 2 a z + z z z z 13 15 8 2 10 2 12 2 14 2 16 2 > a z + a z - 4 a z + 6 a z + 8 a z - 3 a z - a z - 7 3 9 3 11 3 13 3 15 3 8 4 10 4 > 11 a z - 2 a z + 3 a z - 8 a z - 2 a z - 2 a z - 15 a z - 12 4 14 4 7 5 9 5 11 5 13 5 8 6 > 11 a z + 2 a z + 6 a z - 3 a z - 2 a z + 7 a z + 4 a z + 10 6 12 6 14 6 7 7 9 7 11 7 13 7 > 13 a z + 8 a z - a z - a z + 4 a z + 3 a z - 2 a z - 8 8 10 8 12 8 9 9 11 9 > a z - 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 1 1 3 2 2 3 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 26 9 24 8 22 8 22 7 20 7 20 6 18 6 q t q t q t q t q t q t q t 1 3 2 2 4 1 2 2 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 16 6 18 5 16 5 16 4 14 4 12 4 14 3 12 3 q t q t q t q t q t q t q t q t 1 2 1 > ------ + ------ + ---- 12 2 10 2 8 q t q t q t

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