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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,8,13,7 X16,10,17,9 X17,5,18,20 X13,19,14,18 X19,15,20,14 X8,16,9,15 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

- q1/2 + q3/2 - 3q5/2 + 2q7/2 - 3q9/2 + 2q11/2 - 2q13/2 + 2q15/2

A2 (sl(3)) Invariant:

q2 + q4 + 2q6 + 3q8 + 3q10 + 3q12 + q14 + 2q16 - q18 - q20 - 2q22 - 2q24 - q26 - q28 + q30

HOMFLY-PT Polynomial:

- a-9z + 2a-7z-1 + 6a-7z + 5a-7z3 + a-7z5 - 5a-5z-1 - 12a-5z - 12a-5z3 - 6a-5z5 - a-5z7 + 3a-3z-1 + 7a-3z + 5a-3z3 + a-3z5

Kauffman Polynomial:

- a-10 + 2a-9z - a-9z3 - 6a-8z2 + 8a-8z4 - 2a-8z6 - 2a-7z-1 + 11a-7z - 21a-7z3 + 15a-7z5 - 3a-7z7 + 5a-6 - 13a-6z2 + 7a-6z4 + 2a-6z6 - a-6z8 - 5a-5z-1 + 19a-5z - 32a-5z3 + 21a-5z5 - 4a-5z7 + 5a-4 - 7a-4z2 - a-4z4 + 4a-4z6 - a-4z8 - 3a-3z-1 + 10a-3z - 12a-3z3 + 6a-3z5 - a-3z7

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 16               2 j = 14             1 1 j = 12           2 2   j = 10         1 1 1   j = 8       1 2       j = 6     2 1         j = 4   1 3           j = 2                 j = 0 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, NonAlternating, 23]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 23]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[16, 10, 17, 9], > X[17, 5, 18, 20], X[13, 19, 14, 18], X[19, 15, 20, 14], X[8, 16, 9, 15], > X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -8, 4, -2, 10, -3, -6, 7, 8, -4, -5, 6, > -7, 5}]
In[6]:=	Jones[L][q]
Out[6]=	3/2 5/2 7/2 9/2 11/2 13/2 15/2 -Sqrt[q] + q - 3 q + 2 q - 3 q + 2 q - 2 q + 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 4 6 8 10 12 14 16 18 20 22 q + q + 2 q + 3 q + 3 q + 3 q + q + 2 q - q - q - 2 q - 24 26 28 30 > 2 q - q - q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 23]][a, z]
Out[8]=	3 3 3 5 5 2 5 3 z 6 z 12 z 7 z 5 z 12 z 5 z z 6 z ---- - ---- + ---- - -- + --- - ---- + --- + ---- - ----- + ---- + -- - ---- + 7 5 3 9 7 5 3 7 5 3 7 5 a z a z a z a a a a a a a a a 5 7 z z > -- - -- 3 5 a a
In[9]:=	Kauffman[Link[10, NonAlternating, 23]][a, z]
Out[9]=	2 -10 5 5 2 5 3 2 z 11 z 19 z 10 z 6 z -a + -- + -- - ---- - ---- - ---- + --- + ---- + ---- + ---- - ---- - 6 4 7 5 3 9 7 5 3 8 a a a z a z a z a a a a a 2 2 3 3 3 3 4 4 4 5 13 z 7 z z 21 z 32 z 12 z 8 z 7 z z 15 z > ----- - ---- - -- - ----- - ----- - ----- + ---- + ---- - -- + ----- + 6 4 9 7 5 3 8 6 4 7 a a a a a a a a a a 5 5 6 6 6 7 7 7 8 8 21 z 6 z 2 z 2 z 4 z 3 z 4 z z z z > ----- + ---- - ---- + ---- + ---- - ---- - ---- - -- - -- - -- 5 3 8 6 4 7 5 3 6 4 a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	4 4 6 -2 q 6 8 8 2 10 2 10 3 12 3 3 q + 2 q + t + -- + q t + q t + 2 q t + q t + q t + 2 q t + t 10 4 12 4 14 4 14 5 16 5 > q t + 2 q t + q t + q t + 2 q t

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