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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X14,10,15,9 X22,20,5,19 X20,11,21,12 X10,21,11,22 X18,15,19,16 X16,7,17,8 X8,17,9,18 X2536 X4,14,1,13

Gauss Code:

{{1, -10, 2, -11}, {10, -1, 8, -9, 3, -6, 5, -2, 11, -3, 7, -8, 9, -7, 4, -5, 6, -4}}

Jones Polynomial:

q-15/2 - 3q-13/2 + 8q-11/2 - 14q-9/2 + 18q-7/2 - 22q-5/2 + 22q-3/2 - 20q-1/2 + 14q1/2 - 9q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-24 - q-22 + q-20 - 3q-18 + 4q-14 - 2q-12 + 5q-10 + 3q-4 - 3q-2 + 6 - 2q2 + 3q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- a-1z-1 - 2a-1z - 2a-1z3 - a-1z5 + 2az-1 + 6az + 8az3 + 4az5 + az7 - 3a3z-1 - 8a3z - 8a3z3 - 3a3z5 + 3a5z-1 + 5a5z + 3a5z3 - a7z-1 - a7z

Kauffman Polynomial:

a-3z3 - a-3z5 - a-2z2 + 5a-2z4 - 4a-2z6 - a-1z-1 + 4a-1z - 9a-1z3 + 13a-1z5 - 8a-1z7 + 2z2 - 6z4 + 12z6 - 9z8 - 2az-1 + 13az - 35az3 + 36az5 - 9az7 - 5az9 - 2a2 + 11a2z2 - 37a2z4 + 47a2z6 - 20a2z8 - a2z10 - 3a3z-1 + 18a3z - 41a3z3 + 36a3z5 - 9a3z9 + 9a4z2 - 32a4z4 + 41a4z6 - 16a4z8 - a4z10 - 3a5z-1 + 12a5z - 22a5z3 + 21a5z5 - 2a5z7 - 4a5z9 + 2a6 - 2a6z2 - 3a6z4 + 9a6z6 - 5a6z8 - a7z-1 + 3a7z - 6a7z3 + 7a7z5 - 3a7z7 + a8 - 3a8z2 + 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 r = 4 j = 8                       1 j = 6                     3   j = 4                   6 1   j = 2                 8 3     j = 0               12 6       j = -2             12 10         j = -4           10 10           j = -6         8 12             j = -8       6 10               j = -10     2 8                 j = -12   1 6                   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[11, Alternating, 69]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 69]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 10, 15, 9], X[22, 20, 5, 19], > X[20, 11, 21, 12], X[10, 21, 11, 22], X[18, 15, 19, 16], X[16, 7, 17, 8], > X[8, 17, 9, 18], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 8, -9, 3, -6, 5, -2, 11, -3, 7, -8, 9, -7, > 4, -5, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 8 14 18 22 22 20 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 14 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 7/2 > 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -20 3 4 2 5 3 3 2 6 8 6 - q - q + q - --- + --- - --- + --- + -- - -- - 2 q + 3 q - 2 q + 18 14 12 10 4 2 q q q q q q 10 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 69]][a, z]
Out[8]=	3 5 7 3 1 2 a 3 a 3 a a 2 z 3 5 7 2 z -(---) + --- - ---- + ---- - -- - --- + 6 a z - 8 a z + 5 a z - a z - ---- + a z z z z z a a 5 3 3 3 5 3 z 5 3 5 7 > 8 a z - 8 a z + 3 a z - -- + 4 a z - 3 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 69]][a, z]
Out[9]=	3 5 7 2 6 8 1 2 a 3 a 3 a a 4 z 3 -2 a + 2 a + a - --- - --- - ---- - ---- - -- + --- + 13 a z + 18 a z + a z z z z z a 2 5 7 2 z 2 2 4 2 6 2 8 2 > 12 a z + 3 a z + 2 z - -- + 11 a z + 9 a z - 2 a z - 3 a z + 2 a 3 3 4 z 9 z 3 3 3 5 3 7 3 4 5 z > -- - ---- - 35 a z - 41 a z - 22 a z - 6 a z - 6 z + ---- - 3 a 2 a a 5 5 2 4 4 4 6 4 8 4 z 13 z 5 3 5 > 37 a z - 32 a z - 3 a z + 3 a z - -- + ----- + 36 a z + 36 a z + 3 a a 6 5 5 7 5 6 4 z 2 6 4 6 6 6 8 6 > 21 a z + 7 a z + 12 z - ---- + 47 a z + 41 a z + 9 a z - a z - 2 a 7 8 z 7 5 7 7 7 8 2 8 4 8 6 8 > ---- - 9 a z - 2 a z - 3 a z - 9 z - 20 a z - 16 a z - 5 a z - a 9 3 9 5 9 2 10 4 10 > 5 a z - 9 a z - 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 2 1 6 2 8 6 10 12 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 8 12 10 10 12 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 6 t + 8 q t + 3 q t + 6 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 3 q t + q t

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