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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 2q-3/2 - 6q-1/2 + 8q1/2 - 12q3/2 + 14q5/2 - 15q7/2 + 13q9/2 - 10q11/2 + 7q13/2 - 3q15/2 + q17/2

A2 (sl(3)) Invariant:

q-8 + q-6 + 3q-2 + 1 + 4q2 + 4q4 + 4q8 - 3q10 - 2q14 - 3q16 + q18 - 2q20 + q22 - q26

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-10z2 - a-10z4 + 2a-9z3 - 3a-9z5 + 2a-8 - 6a-8z2 + 8a-8z4 - 6a-8z6 + a-7z - 5a-7z3 + 9a-7z5 - 7a-7z7 + a-6 - 4a-6z2 + 2a-6z4 + 6a-6z6 - 6a-6z8 + 2a-5z-1 - 5a-5z + 2a-5z3 + 3a-5z5 + 3a-5z7 - 4a-5z9 - 6a-4 + 27a-4z2 - 44a-4z4 + 35a-4z6 - 9a-4z8 - a-4z10 + 4a-3z-1 - 17a-3z + 27a-3z3 - 27a-3z5 + 22a-3z7 - 7a-3z9 - 5a-2 + 23a-2z2 - 42a-2z4 + 30a-2z6 - 5a-2z8 - a-2z10 + a-1z-1 - 6a-1z + 10a-1z3 - 13a-1z5 + 11a-1z7 - 3a-1z9 + 1 - z2 - 5z4 + 7z6 - 2z8 - az-1 + 5az - 8az3 + 5az5 - 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                 5 2     j = 10               8 5       j = 8             7 5         j = 6           7 8           j = 4         5 7             j = 2       4 8               j = 0     2 4                 j = -2     4                   j = -4 1 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, 31]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 31]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 8, 17, 7], X[22, 18, 5, 17], > X[18, 12, 19, 11], X[20, 14, 21, 13], X[12, 20, 13, 19], X[14, 22, 15, 21], > X[8, 16, 9, 15], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 5, -7, 6, -8, 9, -3, 4, -5, > 7, -6, 8, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 2 6 3/2 5/2 7/2 9/2 -q + ---- - ------- + 8 Sqrt[q] - 12 q + 14 q - 15 q + 13 q - 3/2 Sqrt[q] q 11/2 13/2 15/2 17/2 > 10 q + 7 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 3 2 4 8 10 14 16 18 20 1 + q + q + -- + 4 q + 4 q + 4 q - 3 q - 2 q - 3 q + q - 2 q + 2 q 22 26 > q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 31]][a, z]
Out[8]=	3 3 3 2 4 1 a 2 z 5 z 2 z z 5 z 6 z 3 ---- - ---- + --- + - + --- - --- - --- + 3 a z + -- - ---- - ---- + a z - 5 3 a z z 7 3 a 7 5 a a z a z a a a a 5 5 5 7 2 z 3 z 2 z z > ---- + ---- - ---- + -- 5 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 31]][a, z]
Out[9]=	2 -6 6 5 2 4 1 a z 5 z 17 z 6 z 1 + -- + a - -- - -- + ---- + ---- + --- - - + -- - --- - ---- - --- + 8 4 2 5 3 a z z 7 5 3 a a a a a z a z a a a 2 2 2 2 2 3 3 3 2 z 6 z 4 z 27 z 23 z 2 z 5 z 2 z > 5 a z - 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 27 z 10 z 3 4 z 8 z 2 z 44 z 42 z 3 z > ----- + ----- - 8 a z - 5 z - --- + ---- + ---- - ----- - ----- - ---- + 3 a 10 8 6 4 2 9 a a a a a a a 5 5 5 5 6 6 6 6 9 z 3 z 27 z 13 z 5 6 6 z 6 z 35 z 30 z > ---- + ---- - ----- - ----- + 5 a z + 7 z - ---- + ---- + ----- + ----- - 7 5 3 a 8 6 4 2 a a a a a a a 7 7 7 7 8 8 8 9 7 z 3 z 22 z 11 z 7 8 6 z 9 z 5 z 4 z > ---- + ---- + ----- + ----- - a z - 2 z - ---- - ---- - ---- - ---- - 7 5 3 a 6 4 2 5 a a a a a a a 9 9 10 10 7 z 3 z z z > ---- - ---- - --- - --- 3 a 4 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 2 2 4 4 4 q 4 6 8 q + 5 q + ----- + ----- + ----- + -- + ----- + - + ---- + 7 q t + 7 q t + 6 4 4 4 4 3 2 2 2 t t q t q t q t t q t 6 2 8 2 8 3 10 3 10 4 12 4 12 5 > 8 q t + 7 q t + 5 q t + 8 q t + 5 q t + 5 q t + 2 q t + 14 5 14 6 16 6 18 7 > 5 q t + q t + 2 q t + q t

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