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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 2q-5/2 - 5q-3/2 + 8q-1/2 - 12q1/2 + 14q3/2 - 14q5/2 + 12q7/2 - 10q9/2 + 6q11/2 - 3q13/2 + q15/2

A2 (sl(3)) Invariant:

q-12 + q-10 + 3q-6 - 2q-2 + 3 - 2q2 + q4 + q6 + 3q10 - 2q12 + 2q14 + q16 - 2q18 + 2q20 - q24

HOMFLY-PT Polynomial:

a-7z - a-5z - 2a-5z3 - a-3z-1 - a-3z + a-3z5 + 2a-1z-1 + 3a-1z + a-1z3 + a-1z5 - 2az-1 - 3az - 2az3 + a3z-1 + a3z

Kauffman Polynomial:

- 2a-8z2 + 3a-8z4 - a-8z6 + 2a-7z - 7a-7z3 + 9a-7z5 - 3a-7z7 + a-6z2 - 5a-6z4 + 10a-6z6 - 4a-6z8 + 2a-5z - 5a-5z3 + 4a-5z5 + 4a-5z7 - 3a-5z9 + 2a-4z2 - 10a-4z4 + 13a-4z6 - 4a-4z8 - a-4z10 + a-3z-1 - 3a-3z + 6a-3z3 - 10a-3z5 + 10a-3z7 - 5a-3z9 - 3a-2z2 + a-2z4 + 2a-2z6 - 2a-2z8 - a-2z10 + 2a-1z-1 - 8a-1z + 10a-1z3 - 5a-1z5 + a-1z7 - 2a-1z9 + 1 - 4z2 + 7z4 - 2z6 - 2z8 + 2az-1 - 8az + 9az3 - az5 - 2az7 - 2a2z2 + 4a2z4 - 2a2z6 + a3z-1 - 3a3z + 3a3z3 - a3z5

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 16                       1 j = 14                     2   j = 12                   4 1   j = 10                 6 2     j = 8               6 4       j = 6             8 6         j = 4           6 6           j = 2         6 8             j = 0       4 8               j = -2     1 4                 j = -4   1 4                   j = -6   1                     j = -8 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, 94]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 94]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[20, 8, 21, 7], X[22, 17, 5, 18], > X[18, 21, 19, 22], X[16, 10, 17, 9], X[14, 12, 15, 11], X[10, 16, 11, 15], > X[8, 20, 9, 19], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 6, -8, 7, -2, 11, -7, 8, -6, 4, -5, > 9, -3, 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 2 5 8 3/2 5/2 7/2 -q + ---- - ---- + ------- - 12 Sqrt[q] + 14 q - 14 q + 12 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 10 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 3 2 2 4 6 10 12 14 16 3 + q + q + -- - -- - 2 q + q + q + 3 q - 2 q + 2 q + q - 6 2 q q 18 20 24 > 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 94]][a, z]
Out[8]=	3 3 3 1 2 2 a a z z z 3 z 3 2 z z -(----) + --- - --- + -- + -- - -- - -- + --- - 3 a z + a z - ---- + -- - 3 a z z z 7 5 3 a 5 a a z a a a a 5 5 3 z z > 2 a z + -- + -- 3 a a
In[9]:=	Kauffman[Link[11, Alternating, 94]][a, z]
Out[9]=	3 1 2 2 a a 2 z 2 z 3 z 8 z 3 2 1 + ---- + --- + --- + -- + --- + --- - --- - --- - 8 a z - 3 a z - 4 z - 3 a z z z 7 5 3 a a z a a a 2 2 2 2 3 3 3 3 2 z z 2 z 3 z 2 2 7 z 5 z 6 z 10 z 3 > ---- + -- + ---- - ---- - 2 a z - ---- - ---- + ---- + ----- + 9 a z + 8 6 4 2 7 5 3 a a a a a a a a 4 4 4 4 5 5 5 3 3 4 3 z 5 z 10 z z 2 4 9 z 4 z 10 z > 3 a z + 7 z + ---- - ---- - ----- + -- + 4 a z + ---- + ---- - ----- - 8 6 4 2 7 5 3 a a a a a a a 5 6 6 6 6 7 5 z 5 3 5 6 z 10 z 13 z 2 z 2 6 3 z > ---- - a z - a z - 2 z - -- + ----- + ----- + ---- - 2 a z - ---- + a 8 6 4 2 7 a a a a a 7 7 7 8 8 8 9 9 4 z 10 z z 7 8 4 z 4 z 2 z 3 z 5 z > ---- + ----- + -- - 2 a z - 2 z - ---- - ---- - ---- - ---- - ---- - 5 3 a 6 4 2 5 3 a a a a a a a 9 10 10 2 z z z > ---- - --- - --- a 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 4 1 4 4 2 4 8 + 6 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 8 q t + 6 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 2 6 2 6 3 8 3 8 4 10 4 10 5 > 6 q t + 8 q t + 6 q t + 6 q t + 4 q t + 6 q t + 2 q t + 12 5 12 6 14 6 16 7 > 4 q t + q t + 2 q t + q t

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