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DrawMorseLink

PD Presentation:

X8192 X14,9,15,10 X4758 X16,6,17,5 X18,16,19,15 X6,18,1,17 X20,11,7,12 X10,19,11,20 X2,14,3,13 X12,4,13,3

Gauss Code:

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

Jones Polynomial:

q-11/2 - 3q-9/2 + 7q-7/2 - 11q-5/2 + 13q-3/2 - 15q-1/2 + 13q1/2 - 11q3/2 + 7q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

- q-16 + q-14 - 3q-12 + 2q-8 - 2q-6 + 4q-4 - q-2 + 3 + 2q2 + 4q6 - q8 + q10 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 - 2a-1z-1 - 6a-1z - 6a-1z3 - 2a-1z5 + 3az-1 + 10az + 11az3 + 5az5 + az7 - a3z-1 - 4a3z - 3a3z3 - a3z5

Kauffman Polynomial:

- a-4z2 + 2a-4z4 - a-4z6 + a-3z - 8a-3z3 + 11a-3z5 - 4a-3z7 - 2a-2z2 - 3a-2z4 + 11a-2z6 - 5a-2z8 - 2a-1z-1 + 10a-1z - 26a-1z3 + 30a-1z5 - 7a-1z7 - 2a-1z9 + 3 - 5z2 - 6z4 + 22z6 - 11z8 - 3az-1 + 15az - 33az3 + 35az5 - 11az7 - 2az9 + 3a2 - 10a2z2 + 7a2z4 + 4a2z6 - 6a2z8 - a3z-1 + 6a3z - 13a3z3 + 13a3z5 - 8a3z7 + a4 - 5a4z2 + 7a4z4 - 6a4z6 + 2a5z3 - 3a5z5 + a6z2 - 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                   3   j = 6                 4 1   j = 4               7 3     j = 2             7 5       j = 0           8 6         j = -2         6 8           j = -4       5 7             j = -6     2 6               j = -8   1 5                 j = -10   2                   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, 84]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 84]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 9, 15, 10], X[4, 7, 5, 8], X[16, 6, 17, 5], > X[18, 16, 19, 15], X[6, 18, 1, 17], X[20, 11, 7, 12], X[10, 19, 11, 20], > X[2, 14, 3, 13], X[12, 4, 13, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 10, -3, 4, -6}, > {3, -1, 2, -8, 7, -10, 9, -2, 5, -4, 6, -5, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 7 11 13 15 3/2 q - ---- + ---- - ---- + ---- - ------- + 13 Sqrt[q] - 11 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 > 7 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 3 2 2 4 -2 2 6 8 10 12 14 3 - q + q - --- + -- - -- + -- - q + 2 q + 4 q - q + q + q - q 12 8 6 4 q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 84]][a, z]
Out[8]=	3 3 3 -2 3 a a z 6 z 3 z 6 z 3 3 3 --- + --- - -- + -- - --- + 10 a z - 4 a z + -- - ---- + 11 a z - 3 a z - a z z z 3 a 3 a a a 5 2 z 5 3 5 7 > ---- + 5 a z - a z + a z a
In[9]:=	Kauffman[Link[10, Alternating, 84]][a, z]
Out[9]=	3 2 2 4 2 3 a a z 10 z 3 2 z 3 + 3 a + a - --- - --- - -- + -- + ---- + 15 a z + 6 a z - 5 z - -- - a z z z 3 a 4 a a 2 3 3 2 z 2 2 4 2 6 2 8 z 26 z 3 3 3 > ---- - 10 a z - 5 a z + a z - ---- - ----- - 33 a z - 13 a z + 2 3 a a a 4 4 5 5 5 3 4 2 z 3 z 2 4 4 4 6 4 11 z 30 z > 2 a z - 6 z + ---- - ---- + 7 a z + 7 a z - a z + ----- + ----- + 4 2 3 a a a a 6 6 5 3 5 5 5 6 z 11 z 2 6 4 6 > 35 a z + 13 a z - 3 a z + 22 z - -- + ----- + 4 a z - 6 a z - 4 2 a a 7 7 8 9 4 z 7 z 7 3 7 8 5 z 2 8 2 z 9 > ---- - ---- - 11 a z - 8 a z - 11 z - ---- - 6 a z - ---- - 2 a z 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	8 1 2 1 5 2 6 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 + 7 q t + 5 q t + 7 q t + 3 q t + 4 q t + q t + 2 q t 8 4 10 5 > 3 q t + q t

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