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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X7,14,8,15 X15,17,16,20 X11,19,12,18 X17,13,18,12 X19,5,20,16 X13,8,14,9 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-6 - 2q-5 + 3q-4 - 3q-3 + 3q-2 - 2q-1 + 3 + q3

A2 (sl(3)) Invariant:

q-20 + q-18 - q-16 - q-12 + 2q-8 + 2q-6 + 4q-4 + 3q-2 + 4 + 4q2 + 2q4 + 3q6 + 2q8 + q10

HOMFLY-PT Polynomial:

a-2z-2 + 3a-2 + a-2z2 - 2z-2 - 6 - 6z2 - z4 + a2z-2 + 4a2 + 3a2z2 + a2z4 - 2a4 - 2a4z2 + a6

Kauffman Polynomial:

a-2z-2 - 5a-2 + 9a-2z2 - 6a-2z4 + a-2z6 - 2a-1z-1 + 6a-1z - 3a-1z3 + 2z-2 - 7 + 10z2 - 3z4 - 2az-1 + 2az + 5az3 - 5az5 + az7 + a2z-2 - 2a2 + a2z2 + 3a2z4 - 4a2z6 + a2z8 - 6a3z + 15a3z3 - 13a3z5 + 3a3z7 + 4a4z2 - 4a4z4 - 2a4z6 + a4z8 - 2a5z + 7a5z3 - 8a5z5 + 2a5z7 - a6 + 4a6z2 - 4a6z4 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                     1 j = 5                     1 j = 3               1 1     j = 1             3         j = -1           2 4 1       j = -3         2 1           j = -5       1 2 1           j = -7     2 2               j = -9   1 2                 j = -11   1                   j = -13 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, 88]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 88]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[7, 14, 8, 15], X[15, 17, 16, 20], > X[11, 19, 12, 18], X[17, 13, 18, 12], X[19, 5, 20, 16], X[13, 8, 14, 9], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {-6, 5, -7, 4}, > {9, -1, -3, 8, 10, -2, -5, 6, -8, 3, -4, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-6 2 3 3 3 2 3 3 + q - -- + -- - -- + -- - - + q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 -16 -12 2 2 4 3 2 4 6 8 4 + q + q - q - q + -- + -- + -- + -- + 4 q + 2 q + 3 q + 2 q + 8 6 4 2 q q q q 10 > q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 88]][a, z]
Out[8]=	2 2 3 2 4 6 2 1 a 2 z 2 2 4 2 -6 + -- + 4 a - 2 a + a - -- + ----- + -- - 6 z + -- + 3 a z - 2 a z - 2 2 2 2 2 2 a z a z z a 4 2 4 > z + a z
In[9]:=	Kauffman[Link[10, NonAlternating, 88]][a, z]
Out[9]=	2 5 2 6 2 1 a 2 2 a 6 z 3 -7 - -- - 2 a - a + -- + ----- + -- - --- - --- + --- + 2 a z - 6 a z - 2 2 2 2 2 a z z a a z a z z 2 3 5 2 9 z 2 2 4 2 6 2 3 z 3 > 2 a z + 10 z + ---- + a z + 4 a z + 4 a z - ---- + 5 a z + 2 a a 4 3 3 5 3 4 6 z 2 4 4 4 6 4 5 > 15 a z + 7 a z - 3 z - ---- + 3 a z - 4 a z - 4 a z - 5 a z - 2 a 6 3 5 5 5 z 2 6 4 6 6 6 7 3 7 > 13 a z - 8 a z + -- - 4 a z - 2 a z + a z + a z + 3 a z + 2 a 5 7 2 8 4 8 > 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 1 1 2 2 2 1 2 - + 3 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 2 1 1 2 t 3 3 2 5 4 7 4 > ----- + ---- + ---- + --- + - + q t + q t + q t + q t 3 2 5 3 q t q q t q t q t

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