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DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X4,19,1,20 X9,14,10,15 X8493 X12,5,13,6 X20,13,5,14 X11,16,12,17 X15,10,16,11 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 4q-17/2 - 6q-15/2 + 9q-13/2 - 10q-11/2 + 9q-9/2 - 9q-7/2 + 5q-5/2 - 3q-3/2

A2 (sl(3)) Invariant:

q-30 - 3q-26 - 3q-22 - 2q-20 + q-18 - q-16 + 4q-14 + q-12 + 4q-10 + 4q-8 + 3q-4

HOMFLY-PT Polynomial:

- 3a3z-1 - 6a3z - 3a3z3 + 5a5z-1 + 10a5z + 7a5z3 + 2a5z5 - 2a7z-1 - 5a7z - 3a7z3 + a9z

Kauffman Polynomial:

- 3a3z-1 + 8a3z - 6a3z3 + 5a4 - 7a4z2 + 3a4z4 - 3a4z6 - 5a5z-1 + 15a5z - 19a5z3 + 11a5z5 - 5a5z7 + 5a6 - 14a6z2 + 15a6z4 - 5a6z6 - 2a6z8 - 2a7z-1 + 9a7z - 17a7z3 + 21a7z5 - 10a7z7 - 10a8z2 + 20a8z4 - 6a8z6 - 2a8z8 + 2a9z - 3a9z3 + 9a9z5 - 5a9z7 - a10 - 3a10z2 + 8a10z4 - 4a10z6 + a11z3 - a11z5

Khovanov Homology:

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

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