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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 3q-3/2 - 7q-1/2 + 11q1/2 - 16q3/2 + 17q5/2 - 18q7/2 + 16q9/2 - 12q11/2 + 7q13/2 - 3q15/2 + q17/2

A2 (sl(3)) Invariant:

q-8 + 3q-2 - 2 + 2q2 + 2q4 - q6 + 5q8 - 2q10 + 3q12 - q16 + 3q18 - 3q20 - q26

HOMFLY-PT Polynomial:

a-7z-1 + 2a-7z + a-7z3 - 3a-5z-1 - 7a-5z - 6a-5z3 - 2a-5z5 + 3a-3z-1 + 9a-3z + 8a-3z3 + 4a-3z5 + a-3z7 - 2a-1z-1 - 6a-1z - 6a-1z3 - 2a-1z5 + az-1 + 2az + az3

Kauffman Polynomial:

a-10z2 - a-10z4 + 2a-9z3 - 3a-9z5 + a-8 - 4a-8z2 + 6a-8z4 - 6a-8z6 - a-7z-1 + 5a-7z - 13a-7z3 + 14a-7z5 - 9a-7z7 + 2a-6 - 4a-6z2 - 4a-6z4 + 13a-6z6 - 9a-6z8 - 3a-5z-1 + 16a-5z - 36a-5z3 + 33a-5z5 - 4a-5z7 - 5a-5z9 + 11a-4z2 - 40a-4z4 + 49a-4z6 - 16a-4z8 - a-4z10 - 3a-3z-1 + 20a-3z - 40a-3z3 + 25a-3z5 + 10a-3z7 - 8a-3z9 - 2a-2 + 15a-2z2 - 42a-2z4 + 41a-2z6 - 10a-2z8 - a-2z10 - 2a-1z-1 + 13a-1z - 25a-1z3 + 13a-1z5 + 4a-1z7 - 3a-1z9 + 5z2 - 13z4 + 11z6 - 3z8 - az-1 + 4az - 6az3 + 4az5 - az7

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 = 18                       1 j = 16                     2   j = 14                   5 1   j = 12                 7 2     j = 10               9 5       j = 8             9 7         j = 6           8 9           j = 4         8 9             j = 2       5 10               j = 0     2 6                 j = -2   1 5                   j = -4   2                     j = -6 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, 19]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 19]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[16, 10, 17, 9], > X[20, 14, 21, 13], X[22, 17, 5, 18], X[18, 21, 19, 22], X[14, 20, 15, 19], > X[8, 16, 9, 15], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 4, -2, 11, -3, 5, -8, 9, -4, 6, -7, > 8, -5, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 3 7 3/2 5/2 7/2 -q + ---- - ------- + 11 Sqrt[q] - 16 q + 17 q - 18 q + 3/2 Sqrt[q] q 9/2 11/2 13/2 15/2 17/2 > 16 q - 12 q + 7 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 3 2 4 6 8 10 12 16 18 20 -2 + q + -- + 2 q + 2 q - q + 5 q - 2 q + 3 q - q + 3 q - 3 q - 2 q 26 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 19]][a, z]
Out[8]=	3 3 1 3 3 2 a 2 z 7 z 9 z 6 z z 6 z ---- - ---- + ---- - --- + - + --- - --- + --- - --- + 2 a z + -- - ---- + 7 5 3 a z z 7 5 3 a 7 5 a z a z a z a a a a a 3 3 5 5 5 7 8 z 6 z 3 2 z 4 z 2 z z > ---- - ---- + a z - ---- + ---- - ---- + -- 3 a 5 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 19]][a, z]
Out[9]=	-8 2 2 1 3 3 2 a 5 z 16 z 20 z 13 z a + -- - -- - ---- - ---- - ---- - --- - - + --- + ---- + ---- + ---- + 6 2 7 5 3 a z z 7 5 3 a a a a z a z a z a a a 2 2 2 2 2 3 3 3 2 z 4 z 4 z 11 z 15 z 2 z 13 z 36 z > 4 a z + 5 z + --- - ---- - ---- + ----- + ----- + ---- - ----- - ----- - 10 8 6 4 2 9 7 5 a a a a a a a a 3 3 4 4 4 4 4 5 40 z 25 z 3 4 z 6 z 4 z 40 z 42 z 3 z > ----- - ----- - 6 a z - 13 z - --- + ---- - ---- - ----- - ----- - ---- + 3 a 10 8 6 4 2 9 a a a a a a a 5 5 5 5 6 6 6 14 z 33 z 25 z 13 z 5 6 6 z 13 z 49 z > ----- + ----- + ----- + ----- + 4 a z + 11 z - ---- + ----- + ----- + 7 5 3 a 8 6 4 a a a a a a 6 7 7 7 7 8 8 8 41 z 9 z 4 z 10 z 4 z 7 8 9 z 16 z 10 z > ----- - ---- - ---- + ----- + ---- - a z - 3 z - ---- - ----- - ----- - 2 7 5 3 a 6 4 2 a a a a a a a 9 9 9 10 10 5 z 8 z 3 z z z > ---- - ---- - ---- - --- - --- 5 3 a 4 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 2 1 2 5 6 5 q 4 10 q + 8 q + ----- + ----- + ----- + -- + ----- + - + ---- + 9 q t + 6 4 4 3 2 3 2 2 2 t t q t q t q t t q t 6 6 2 8 2 8 3 10 3 10 4 12 4 > 8 q t + 9 q t + 9 q t + 7 q t + 9 q t + 5 q t + 7 q t + 12 5 14 5 14 6 16 6 18 7 > 2 q t + 5 q t + q t + 2 q t + q t

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