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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 6q-11/2 - 9q-9/2 + 12q-7/2 - 14q-5/2 + 12q-3/2 - 11q-1/2 + 7q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant:

- q-24 + 2q-20 - 2q-18 - 4q-12 + 2q-10 + q-8 + 4q-6 + 4q-4 + 4 - 2q2 + 2q6 - q8

HOMFLY-PT Polynomial:

a-1z3 - 2az-1 - 2az - az3 - az5 + 3a3z-1 + 2a3z - a3z5 - a5z-1 + a5z + 2a5z3 - a7z

Kauffman Polynomial:

- a-2z4 + 3a-1z3 - 4a-1z5 + 7z4 - 7z6 + 2az-1 - 3az - 2az3 + 10az5 - 8az7 - 3a2 + 5a2z2 - 4a2z4 + 7a2z6 - 6a2z8 + 3a3z-1 - 3a3z - 12a3z3 + 21a3z5 - 7a3z7 - 2a3z9 - 3a4 + 11a4z2 - 24a4z4 + 26a4z6 - 10a4z8 + a5z-1 + 2a5z - 14a5z3 + 16a5z5 - 2a5z7 - 2a5z9 - a6 + 4a6z2 - 9a6z4 + 11a6z6 - 4a6z8 + 2a7z - 7a7z3 + 9a7z5 - 3a7z7 - 2a8z2 + 3a8z4 - a8z6

Khovanov Homology:

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

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