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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X14,8,15,7 X16,13,17,14 X6,18,1,17 X18,12,19,11 X12,6,13,5 X20,16,7,15 X4,19,5,20

Gauss Code:

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

Jones Polynomial:

q-11/2 - 4q-9/2 + 7q-7/2 - 11q-5/2 + 13q-3/2 - 14q-1/2 + 12q1/2 - 10q3/2 + 6q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

- q-16 + 2q-14 - q-12 + 2q-10 + 3q-8 - q-6 + 4q-4 - 2q-2 + 2 - q4 + 3q6 - q8 + q10 - q14

HOMFLY-PT Polynomial:

2a-3z + a-3z3 - 4a-1z - 6a-1z3 - 2a-1z5 - az-1 + az + 5az3 + 4az5 + az7 + a3z-1 - 2a3z3 - a3z5

Kauffman Polynomial:

- 2a-4z2 + 3a-4z4 - a-4z6 + 3a-3z - 8a-3z3 + 9a-3z5 - 3a-3z7 + a-2z2 - 6a-2z4 + 10a-2z6 - 4a-2z8 + 5a-1z - 18a-1z3 + 19a-1z5 - 3a-1z7 - 2a-1z9 + 3z2 - 15z4 + 23z6 - 10z8 + az-1 - 14az3 + 23az5 - 8az7 - 2az9 - a2 - a2z2 + 2a2z4 + 5a2z6 - 6a2z8 + a3z-1 - 2a3z - a3z3 + 9a3z5 - 8a3z7 - a4z2 + 7a4z4 - 7a4z6 + 3a5z3 - 4a5z5 - a6z4

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                     1 j = 8                   2   j = 6                 4 1   j = 4               6 2     j = 2             6 4       j = 0           8 6         j = -2         6 7           j = -4       5 7             j = -6     3 7               j = -8   1 4                 j = -10   3                   j = -12 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, 55]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 55]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[14, 8, 15, 7], > X[16, 13, 17, 14], X[6, 18, 1, 17], X[18, 12, 19, 11], X[12, 6, 13, 5], > X[20, 16, 7, 15], X[4, 19, 5, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -10, 8, -6}, > {4, -1, 2, -3, 7, -8, 5, -4, 9, -5, 6, -7, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 4 7 11 13 14 3/2 q - ---- + ---- - ---- + ---- - ------- + 12 Sqrt[q] - 10 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 > 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 -12 2 3 -6 4 2 4 6 8 10 14 2 - q + --- - q + --- + -- - q + -- - -- - q + 3 q - q + q - q 14 10 8 4 2 q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 55]][a, z]
Out[8]=	3 3 3 5 a a 2 z 4 z z 6 z 3 3 3 2 z 5 -(-) + -- + --- - --- + a z + -- - ---- + 5 a z - 2 a z - ---- + 4 a z - z z 3 a 3 a a a a 3 5 7 > a z + a z
In[9]:=	Kauffman[Link[10, Alternating, 55]][a, z]
Out[9]=	3 2 2 3 2 a a 3 z 5 z 3 2 2 z z 2 2 4 2 8 z -a + - + -- + --- + --- - 2 a z + 3 z - ---- + -- - a z - a z - ---- - z z 3 a 4 2 3 a a a a 3 4 4 18 z 3 3 3 5 3 4 3 z 6 z 2 4 > ----- - 14 a z - a z + 3 a z - 15 z + ---- - ---- + 2 a z + a 4 2 a a 5 5 6 4 4 6 4 9 z 19 z 5 3 5 5 5 6 z > 7 a z - a z + ---- + ----- + 23 a z + 9 a z - 4 a z + 23 z - -- + 3 a 4 a a 6 7 7 8 10 z 2 6 4 6 3 z 3 z 7 3 7 8 4 z > ----- + 5 a z - 7 a z - ---- - ---- - 8 a z - 8 a z - 10 z - ---- - 2 3 a 2 a a a 9 2 8 2 z 9 > 6 a z - ---- - 2 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	7 1 3 1 4 3 7 5 7 8 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 6 2 2 2 4 2 4 3 6 3 6 4 > ---- + 6 t + 6 q t + 4 q t + 6 q t + 2 q t + 4 q t + q t + 2 q t 8 4 10 5 > 2 q t + q t

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