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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X11,19,12,18 X5,12,6,13 X17,4,18,5 X14,7,15,8 X16,14,17,13 X20,15,7,16 X6,19,1,20

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-30 + q-28 + 2q-24 + q-18 - q-16 + q-14 - q-12 + q-10 + 2q-8 + 2q-4

HOMFLY-PT Polynomial:

- a3z-1 - 4a3z - 2a3z3 + 2a5z-1 + 4a5z + 3a5z3 + a5z5 - 2a7z-1 - 4a7z - 2a7z3 + a9z-1 + a9z

Kauffman Polynomial:

- a3z-1 + 5a3z - 3a3z3 - a4z6 - 2a5z-1 + 9a5z - 13a5z3 + 6a5z5 - 2a5z7 - a6 + 2a6z2 - 3a6z4 + a6z6 - a6z8 - 2a7z-1 + 9a7z - 15a7z3 + 11a7z5 - 4a7z7 + 2a8z4 - a8z8 - a9z-1 + 3a9z - 2a9z3 + 4a9z5 - 2a9z7 - 2a10z2 + 5a10z4 - 2a10z6 - 2a11z + 3a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                 2 j = -4               2 1 j = -6             3 1   j = -8           3 2     j = -10         3 3       j = -12       2 3         j = -14     2 3           j = -16   1 3             j = -18   1               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, 51]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 51]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[11, 19, 12, 18], > X[5, 12, 6, 13], X[17, 4, 18, 5], X[14, 7, 15, 8], X[16, 14, 17, 13], > X[20, 15, 7, 16], X[6, 19, 1, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, 6, -5, -10}, > {7, -1, 2, -3, -4, 5, 8, -7, 9, -8, -6, 4, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 2 4 5 6 6 5 3 2 -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 -28 2 -18 -16 -14 -12 -10 2 2 q + q + --- + q - q + q - q + q + -- + -- 24 8 4 q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 51]][a, z]
Out[8]=	3 5 7 9 a 2 a 2 a a 3 5 7 9 3 3 -(--) + ---- - ---- + -- - 4 a z + 4 a z - 4 a z + a z - 2 a z + z z z z 5 3 7 3 5 5 > 3 a z - 2 a z + a z
In[9]:=	Kauffman[Link[10, NonAlternating, 51]][a, z]
Out[9]=	3 5 7 9 6 a 2 a 2 a a 3 5 7 9 11 -a - -- - ---- - ---- - -- + 5 a z + 9 a z + 9 a z + 3 a z - 2 a z + z z z z 6 2 10 2 3 3 5 3 7 3 9 3 11 3 > 2 a z - 2 a z - 3 a z - 13 a z - 15 a z - 2 a z + 3 a z - 6 4 8 4 10 4 5 5 7 5 9 5 11 5 > 3 a z + 2 a z + 5 a z + 6 a z + 11 a z + 4 a z - a z - 4 6 6 6 10 6 5 7 7 7 9 7 6 8 8 8 > a z + a z - 2 a z - 2 a z - 4 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 1 1 1 3 2 3 2 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 20 8 18 7 16 7 16 6 14 6 14 5 12 5 q q t q t q t q t q t q t q t 3 3 3 3 2 3 1 2 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- 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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