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DrawMorseLink

PD Presentation:

X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X9,18,10,19 X17,1,18,4 X19,12,20,5 X11,20,12,13 X13,10,14,11

Gauss Code:

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

Jones Polynomial:

q-9 - q-8 + 3q-7 - 4q-6 + 5q-5 - 4q-4 + 5q-3 - 3q-2 + 2q-1

A2 (sl(3)) Invariant:

q-28 + q-26 + 2q-24 + 4q-22 + 2q-20 + 3q-18 + 3q-16 + 2q-14 + 3q-12 + q-10 + 2q-8 + q-6 + 2q-2

HOMFLY-PT Polynomial:

2a2 + 2a2z2 + a4z-2 + a4 - a4z2 - a4z4 - 2a6z-2 - 5a6 - 3a6z2 - a6z4 + a8z-2 + 2a8 + a8z2

Kauffman Polynomial:

- 2a2 + 3a2z2 + a3z + a3z3 + a3z5 - a4z-2 + 3a4 - 2a4z4 + 2a4z6 + 2a5z-1 - 8a5z + 9a5z3 - 5a5z5 + 2a5z7 - 2a6z-2 + 9a6 - 14a6z2 + 6a6z4 - 2a6z6 + a6z8 + 2a7z-1 - 8a7z + 9a7z3 - 9a7z5 + 3a7z7 - a8z-2 + 3a8 - 4a8z2 + 3a8z4 - 3a8z6 + a8z8 + a9z + a9z3 - 3a9z5 + a9z7 - 2a10 + 7a10z2 - 5a10z4 + a10z6

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -1                 2 j = -3               3 2 j = -5             2     j = -7           2 3     j = -9         3 2       j = -11       2 3         j = -13     1 2           j = -15     2             j = -17 1 1               j = -19 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]=	3
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 75]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 75]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 14, 6, 15], X[3, 8, 4, 9], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[9, 18, 10, 19], X[17, 1, 18, 4], X[19, 12, 20, 5], > X[11, 20, 12, 13], X[13, 10, 14, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 7}, {-2, -1, 5, 3, -6, 10, -9, 8}, > {-10, 2, 4, -5, -7, 6, -8, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-9 -8 3 4 5 4 5 3 2 q - q + -- - -- + -- - -- + -- - -- + - 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 2 4 2 3 3 2 3 -10 2 -6 2 q + q + --- + --- + --- + --- + --- + --- + --- + q + -- + q + -- 24 22 20 18 16 14 12 8 2 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 75]][a, z]
Out[8]=	4 6 8 2 4 6 8 a 2 a a 2 2 4 2 6 2 8 2 2 a + a - 5 a + 2 a + -- - ---- + -- + 2 a z - a z - 3 a z + a z - 2 2 2 z z z 4 4 6 4 > a z - a z
In[9]:=	Kauffman[Link[10, NonAlternating, 75]][a, z]
Out[9]=	4 6 8 5 7 2 4 6 8 10 a 2 a a 2 a 2 a 3 -2 a + 3 a + 9 a + 3 a - 2 a - -- - ---- - -- + ---- + ---- + a z - 2 2 2 z z z z z 5 7 9 2 2 6 2 8 2 10 2 3 3 > 8 a z - 8 a z + a z + 3 a z - 14 a z - 4 a z + 7 a z + a z + 5 3 7 3 9 3 4 4 6 4 8 4 10 4 > 9 a z + 9 a z + a z - 2 a z + 6 a z + 3 a z - 5 a z + 3 5 5 5 7 5 9 5 4 6 6 6 8 6 > a z - 5 a z - 9 a z - 3 a z + 2 a z - 2 a z - 3 a z + 10 6 5 7 7 7 9 7 6 8 8 8 > a z + 2 a z + 3 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 2 1 2 2 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 8 17 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 3 3 2 2 3 2 3 > ------ + ----- + ----- + ----- + ----- + ----- + ---- 11 4 9 4 9 3 7 3 7 2 5 2 3 q t q t q t q t q t q t q t

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