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DrawMorseLink

PD Presentation:

X6172 X20,7,21,8 X4,21,1,22 X14,10,15,9 X8493 X5,13,6,12 X13,5,14,22 X18,16,19,15 X16,11,17,12 X10,17,11,18 X2,20,3,19

Gauss Code:

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

Jones Polynomial:

q-9/2 - 4q-7/2 + 7q-5/2 - 12q-3/2 + 14q-1/2 - 15q1/2 + 14q3/2 - 11q5/2 + 7q7/2 - 3q9/2

A2 (sl(3)) Invariant:

- q-14 + q-12 + 2q-10 + 5q-6 + q-2 + 1 - 4q2 + 2q4 - 4q6 + 2q8 + q10 - q12 + 3q14 + q18

HOMFLY-PT Polynomial:

- a-5z-1 - a-5z + 2a-3z-1 + 4a-3z + 3a-3z3 + a-3z5 - a-1z-1 - 4a-1z - 6a-1z3 - 4a-1z5 - a-1z7 - az-1 + 2az + 5az3 + 2az5 + a3z-1 - a3z - a3z3

Kauffman Polynomial:

- a-5z-1 + 5a-5z - 6a-5z3 - a-4 + 3a-4z2 - 3a-4z4 - 3a-4z6 - 2a-3z-1 + 15a-3z - 24a-3z3 + 16a-3z5 - 8a-3z7 - 3a-2 + 6a-2z2 - 8a-2z4 + 10a-2z6 - 7a-2z8 - a-1z-1 + 14a-1z - 36a-1z3 + 40a-1z5 - 13a-1z7 - 2a-1z9 - 2 + 3z2 - 9z4 + 24z6 - 12z8 + az-1 + 5az - 27az3 + 35az5 - 9az7 - 2az9 - a2 - a2z2 - 2a2z4 + 10a2z6 - 5a2z8 + a3z-1 + a3z - 9a3z3 + 11a3z5 - 4a3z7 - a4z2 + 2a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 10                   3 j = 8                 4   j = 6               7 3   j = 4             7 4     j = 2           8 7       j = 0         8 9         j = -2       4 6           j = -4     3 8             j = -6   1 4               j = -8   3                 j = -10 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, NonAlternating, 34]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 34]]
Out[4]=	PD[X[6, 1, 7, 2], X[20, 7, 21, 8], X[4, 21, 1, 22], X[14, 10, 15, 9], > X[8, 4, 9, 3], X[5, 13, 6, 12], X[13, 5, 14, 22], X[18, 16, 19, 15], > X[16, 11, 17, 12], X[10, 17, 11, 18], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-6, -1, 2, -5, 4, -10, 9, 6, -7, -4, 8, -9, 10, -8, > 11, -2, 3, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 4 7 12 14 3/2 5/2 q - ---- + ---- - ---- + ------- - 15 Sqrt[q] + 14 q - 11 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 > 7 q - 3 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 2 5 -2 2 4 6 8 10 12 1 - q + q + --- + -- + q - 4 q + 2 q - 4 q + 2 q + q - q + 10 6 q q 14 18 > 3 q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 34]][a, z]
Out[8]=	3 3 3 1 2 1 a a z 4 z 4 z 3 3 z 6 z -(----) + ---- - --- - - + -- - -- + --- - --- + 2 a z - a z + ---- - ---- + 5 3 a z z z 5 3 a 3 a a z a z a a a 5 5 7 3 3 3 z 4 z 5 z > 5 a z - a z + -- - ---- + 2 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 34]][a, z]
Out[9]=	3 -4 3 2 1 2 1 a a 5 z 15 z 14 z -2 - a - -- - a - ---- - ---- - --- + - + -- + --- + ---- + ---- + 5 a z + 2 5 3 a z z z 5 3 a a a z a z a a 2 2 3 3 3 3 2 3 z 6 z 2 2 4 2 6 z 24 z 36 z > a z + 3 z + ---- + ---- - a z - a z - ---- - ----- - ----- - 4 2 5 3 a a a a a 4 4 5 3 3 3 4 3 z 8 z 2 4 4 4 16 z > 27 a z - 9 a z - 9 z - ---- - ---- - 2 a z + 2 a z + ----- + 4 2 3 a a a 5 6 6 40 z 5 3 5 6 3 z 10 z 2 6 4 6 > ----- + 35 a z + 11 a z + 24 z - ---- + ----- + 10 a z - a z - a 4 2 a a 7 7 8 9 8 z 13 z 7 3 7 8 7 z 2 8 2 z 9 > ---- - ----- - 9 a z - 4 a z - 12 z - ---- - 5 a z - ---- - 2 a z 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 4 3 8 4 8 6 9 + 8 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t q t 2 4 4 2 6 2 6 3 8 3 10 4 > 7 q t + 7 q t + 4 q t + 7 q t + 3 q t + 4 q t + 3 q t

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