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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-25/2 + q-21/2 - 2q-19/2 + q-17/2 - 2q-15/2 + q-13/2 - 2q-11/2 + q-9/2 - q-7/2

A2 (sl(3)) Invariant:

- q-42 - q-40 - 2q-38 - 3q-36 - q-34 + 2q-30 + 2q-28 + 3q-26 + 3q-24 + 2q-22 + 2q-20 + q-18 + q-16 + q-12

HOMFLY-PT Polynomial:

- a7z-1 - 5a7z - 10a7z3 - 6a7z5 - a7z7 - a9z-1 - 5a9z - 10a9z3 - 6a9z5 - a9z7 + 4a11z-1 + 9a11z + 6a11z3 + a11z5 - 2a13z-1 - a13z

Kauffman Polynomial:

- a7z-1 + 5a7z - 10a7z3 + 6a7z5 - a7z7 + a8 - a8z2 - 5a8z4 + 5a8z6 - a8z8 + a9z-1 - 4a9z + 9a9z3 - 11a9z5 + 6a9z7 - a9z9 - 5a10 + 17a10z2 - 22a10z4 + 12a10z6 - 2a10z8 + 4a11z-1 - 15a11z + 24a11z3 - 18a11z5 + 7a11z7 - a11z9 - 6a12 + 18a12z2 - 17a12z4 + 7a12z6 - a12z8 + 2a13z-1 - 7a13z + 5a13z3 - a13z5 + a14 - a14z2 - a15z + 2a16 - 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               1     j = -12           1 1 1     j = -14         1 3 1       j = -16       1 1 1         j = -18       3 2           j = -20   1 1 1             j = -22   1 2               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, 19]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 19]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[5, 12, 6, 13], > X[3, 8, 4, 9], X[13, 22, 14, 5], X[21, 14, 22, 15], X[9, 18, 10, 19], > X[11, 20, 12, 21], X[19, 10, 20, 11], X[15, 2, 16, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -3}, {-4, -1, 2, 5, -8, 10, -9, 4, -6, 7, -11, -2, 3, 8, > -10, 9, -7, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) -(21/2) 2 -(17/2) 2 -(13/2) 2 -(9/2) q + q - ----- + q - ----- + q - ----- + q - 19/2 15/2 11/2 q q q -(7/2) > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -40 2 3 -34 2 2 3 3 2 2 -18 -q - q - --- - --- - q + --- + --- + --- + --- + --- + --- + q + 38 36 30 28 26 24 22 20 q q q q q q q q -16 -12 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 19]][a, z]
Out[8]=	7 9 11 13 a a 4 a 2 a 7 9 11 13 7 3 -(--) - -- + ----- - ----- - 5 a z - 5 a z + 9 a z - a z - 10 a z - z z z z 9 3 11 3 7 5 9 5 11 5 7 7 9 7 > 10 a z + 6 a z - 6 a z - 6 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 19]][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 - -- + -- + ----- + ----- + 5 a z - 4 a z - z z z z 11 13 15 8 2 10 2 12 2 14 2 > 15 a z - 7 a z - a z - a z + 17 a z + 18 a z - a z - 16 2 7 3 9 3 11 3 13 3 8 4 10 4 > a z - 10 a z + 9 a z + 24 a z + 5 a z - 5 a z - 22 a z - 12 4 7 5 9 5 11 5 13 5 8 6 10 6 > 17 a z + 6 a z - 11 a z - 18 a z - a z + 5 a z + 12 a z + 12 6 7 7 9 7 11 7 8 8 10 8 12 8 9 9 > 7 a z - a z + 6 a z + 7 a z - a z - 2 a z - a z - a z - 11 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 1 1 1 2 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 26 9 24 9 22 8 20 8 22 7 20 7 20 6 q t q t q t q t q t q t q t 3 1 2 1 1 1 3 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 6 16 6 18 5 16 5 14 5 16 4 14 4 12 4 q t q t q t q t q t q t q t q t 1 1 1 1 1 > ------ + ------ + ------ + ------ + ---- 14 3 12 3 12 2 10 2 8 q t q t q t q t q t

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