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DrawMorseLink

PD Presentation:

X6172 X3,13,4,12 X7,16,8,17 X17,22,18,5 X9,15,10,14 X19,10,20,11 X21,9,22,8 X13,18,14,19 X15,21,16,20 X2536 X11,1,12,4

Gauss Code:

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

Jones Polynomial:

- 3q-9/2 + 8q-7/2 - 12q-5/2 + 15q-3/2 - 17q-1/2 + 15q1/2 - 13q3/2 + 8q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 + 3q-14 - 2q-12 + q-8 - 5q-6 + 3q-4 - 2q-2 + 4 + 3q2 + 4q6 - 2q8 + q10 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 - 2a-1z-1 - 4a-1z - 5a-1z3 - 2a-1z5 + 4az-1 + 7az + 7az3 + 4az5 + az7 - 3a3z-1 - 5a3z - 3a3z3 - a3z5 + a5z-1 + a5z

Kauffman Polynomial:

- a-4z2 + 2a-4z4 - a-4z6 + 3a-3z - 8a-3z3 + 10a-3z5 - 4a-3z7 - a-2 - 5a-2z4 + 13a-2z6 - 6a-2z8 - 2a-1z-1 + 13a-1z - 35a-1z3 + 37a-1z5 - 7a-1z7 - 3a-1z9 - 2 + 11z2 - 28z4 + 39z6 - 16z8 - 4az-1 + 22az - 49az3 + 48az5 - 13az7 - 3az9 - 3a2 + 17a2z2 - 27a2z4 + 22a2z6 - 10a2z8 - 3a3z-1 + 16a3z - 28a3z3 + 21a3z5 - 10a3z7 - a4 + 7a4z2 - 6a4z4 - 3a4z6 - a5z-1 + 4a5z - 6a5z3

Khovanov Homology:

trqj 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               5 1   j = 4             8 3     j = 2           7 5       j = 0         10 8         j = -2       7 9           j = -4     5 8             j = -6   3 7               j = -8   5                 j = -10 3

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, NonAlternating, 88]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 88]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 13, 4, 12], X[7, 16, 8, 17], X[17, 22, 18, 5], > X[9, 15, 10, 14], X[19, 10, 20, 11], X[21, 9, 22, 8], X[13, 18, 14, 19], > X[15, 21, 16, 20], X[2, 5, 3, 6], X[11, 1, 12, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -3, 7, -5, 6, -11, 2, -8, 5, -9, 3, -4, 8, > -6, 9, -7, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-3 8 12 15 17 3/2 5/2 7/2 ---- + ---- - ---- + ---- - ------- + 15 Sqrt[q] - 13 q + 8 q - 4 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 9/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 3 2 -8 5 3 2 2 6 8 10 12 14 4 + q + --- - --- + q - -- + -- - -- + 3 q + 4 q - 2 q + q + q - q 14 12 6 4 2 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 88]][a, z]
Out[8]=	3 5 3 3 -2 4 a 3 a a z 4 z 3 5 z 5 z 3 --- + --- - ---- + -- + -- - --- + 7 a z - 5 a z + a z + -- - ---- + 7 a z - a z z z z 3 a 3 a a a 5 3 3 2 z 5 3 5 7 > 3 a z - ---- + 4 a z - a z + a z a
In[9]:=	Kauffman[Link[11, NonAlternating, 88]][a, z]
Out[9]=	3 5 -2 2 4 2 4 a 3 a a 3 z 13 z 3 -2 - a - 3 a - a - --- - --- - ---- - -- + --- + ---- + 22 a z + 16 a z + a z z z z 3 a a 2 3 3 5 2 z 2 2 4 2 8 z 35 z 3 > 4 a z + 11 z - -- + 17 a z + 7 a z - ---- - ----- - 49 a z - 4 3 a a a 4 4 5 3 3 5 3 4 2 z 5 z 2 4 4 4 10 z > 28 a z - 6 a z - 28 z + ---- - ---- - 27 a z - 6 a z + ----- + 4 2 3 a a a 5 6 6 37 z 5 3 5 6 z 13 z 2 6 4 6 > ----- + 48 a z + 21 a z + 39 z - -- + ----- + 22 a z - 3 a z - a 4 2 a a 7 7 8 9 4 z 7 z 7 3 7 8 6 z 2 8 3 z 9 > ---- - ---- - 13 a z - 10 a z - 16 z - ---- - 10 a z - ---- - 3 a z 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	9 3 5 3 7 5 8 7 2 10 + -- + ------ + ----- + ----- + ----- + ----- + ---- + ---- + 8 t + 7 q t + 2 10 4 8 3 6 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t 2 2 4 2 4 3 6 3 6 4 8 4 10 5 > 5 q t + 8 q t + 3 q t + 5 q t + q t + 3 q t + q t

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