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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X18,14,19,13 X14,18,15,17 X4,19,1,20 X10,5,11,6 X12,3,13,4 X20,11,5,12 X2,9,3,10 X8,15,9,16

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 3q-17/2 - 6q-15/2 + 10q-13/2 - 13q-11/2 + 14q-9/2 - 14q-7/2 + 10q-5/2 - 8q-3/2 + 4q-1/2 - q1/2

A2 (sl(3)) Invariant:

q-30 - 2q-26 + 2q-24 - q-22 - 2q-20 + 3q-18 - q-16 + 3q-14 + q-12 + q-10 + 4q-8 - 2q-6 + 3q-4 - 2 + q2

HOMFLY-PT Polynomial:

- az3 - a3z-1 - 2a3z + a3z5 + a5z-1 + a5z5 - a7z - 2a7z3 + a9z

Kauffman Polynomial:

az3 - az5 + 6a2z4 - 4a2z6 - a3z-1 + 2a3z - 7a3z3 + 14a3z5 - 7a3z7 + a4 + 2a4z2 - 6a4z4 + 10a4z6 - 6a4z8 - a5z-1 + 2a5z - 14a5z3 + 20a5z5 - 7a5z7 - 2a5z9 + 6a6z2 - 19a6z4 + 21a6z6 - 10a6z8 + a7z - 8a7z3 + 11a7z5 - 4a7z7 - 2a7z9 + a8z2 - a8z4 + 4a8z6 - 4a8z8 + 5a9z5 - 4a9z7 - 3a10z2 + 6a10z4 - 3a10z6 - a11z + 2a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                     1 j = 0                   3   j = -2                 5 1   j = -4               6 4     j = -6             8 4       j = -8           6 6         j = -10         7 8           j = -12       4 7             j = -14     2 6               j = -16   1 4                 j = -18   2                   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, Alternating, 42]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 42]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[18, 14, 19, 13], X[14, 18, 15, 17], > X[4, 19, 1, 20], X[10, 5, 11, 6], X[12, 3, 13, 4], X[20, 11, 5, 12], > X[2, 9, 3, 10], X[8, 15, 9, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 7, -5}, {6, -1, 2, -10, 9, -6, 8, -7, 3, -4, 10, -2, 4, -3, > 5, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 6 10 13 14 14 10 8 -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 4 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 2 2 -22 2 3 -16 3 -12 -10 4 2 -2 + q - --- + --- - q - --- + --- - q + --- + q + q + -- - -- + 26 24 20 18 14 8 6 q q q q q q q 3 2 > -- + q 4 q
In[8]:=	HOMFLYPT[Link[10, Alternating, 42]][a, z]
Out[8]=	3 5 a a 3 7 9 3 7 3 3 5 5 5 -(--) + -- - 2 a z - a z + a z - a z - 2 a z + a z + a z z z
In[9]:=	Kauffman[Link[10, Alternating, 42]][a, z]
Out[9]=	3 5 4 a a 3 5 7 11 4 2 6 2 8 2 a - -- - -- + 2 a z + 2 a z + a z - a z + 2 a z + 6 a z + a z - z z 10 2 3 3 3 5 3 7 3 11 3 2 4 > 3 a z + a z - 7 a z - 14 a z - 8 a z + 2 a z + 6 a z - 4 4 6 4 8 4 10 4 5 3 5 5 5 > 6 a z - 19 a z - a z + 6 a z - a z + 14 a z + 20 a z + 7 5 9 5 11 5 2 6 4 6 6 6 8 6 > 11 a z + 5 a z - a z - 4 a z + 10 a z + 21 a z + 4 a z - 10 6 3 7 5 7 7 7 9 7 4 8 6 8 > 3 a z - 7 a z - 7 a z - 4 a z - 4 a z - 6 a z - 10 a z - 8 8 5 9 7 9 > 4 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 5 1 2 1 4 2 6 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 7 7 8 6 6 8 4 6 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 3 t + -- + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q t q t q 2 2 > q t

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