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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-1/2 + 3q1/2 - 8q3/2 + 14q5/2 - 21q7/2 + 22q9/2 - 24q11/2 + 21q13/2 - 15q15/2 + 10q17/2 - 4q19/2 + q21/2

A2 (sl(3)) Invariant:

q-2 - 1 + 4q4 - 4q6 + 4q8 + 4q10 + 7q14 - q16 + 4q18 - q20 - 4q22 + 2q24 - 6q26 - q28 + 2q30 - q32

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-12z2 + 2a-12z4 - a-12z6 - 4a-11z3 + 8a-11z5 - 4a-11z7 - 4a-10 + 9a-10z2 - 16a-10z4 + 19a-10z6 - 8a-10z8 + a-9z-1 + 2a-9z - 3a-9z3 + 3a-9z5 + 9a-9z7 - 7a-9z9 - 9a-8 + 25a-8z2 - 44a-8z4 + 48a-8z6 - 16a-8z8 - 2a-8z10 + a-7z-1 + 9a-7z - 19a-7z3 + 8a-7z5 + 17a-7z7 - 13a-7z9 - 4a-6 + 14a-6z2 - 37a-6z4 + 42a-6z6 - 16a-6z8 - 2a-6z10 - 2a-5z-1 + 15a-5z - 31a-5z3 + 23a-5z5 - 2a-5z7 - 6a-5z9 + 2a-4 - 2a-4z2 - 7a-4z4 + 11a-4z6 - 8a-4z8 - 2a-3z-1 + 7a-3z - 9a-3z3 + 9a-3z5 - 6a-3z7 - a-2z2 + 4a-2z4 - 3a-2z6 - a-1z + 2a-1z3 - a-1z5

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 22                       1 j = 20                     3   j = 18                   7 1   j = 16                 8 3     j = 14               13 7       j = 12             11 8         j = 10           11 13           j = 8         10 11             j = 6       4 11               j = 4     4 10                 j = 2   1 6                   j = 0   2                     j = -2 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, 29]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 29]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[20, 10, 21, 9], X[18, 14, 19, 13], > X[14, 8, 15, 7], X[8, 18, 9, 17], X[12, 20, 13, 19], X[22, 16, 5, 15], > X[16, 22, 17, 21], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 5, -6, 3, -2, 11, -7, 4, -5, 8, -9, 6, -4, > 7, -3, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	1 3/2 5/2 7/2 9/2 11/2 -(-------) + 3 Sqrt[q] - 8 q + 14 q - 21 q + 22 q - 24 q + Sqrt[q] 13/2 15/2 17/2 19/2 21/2 > 21 q - 15 q + 10 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 4 6 8 10 14 16 18 20 22 -1 + q + 4 q - 4 q + 4 q + 4 q + 7 q - q + 4 q - q - 4 q + 24 26 28 30 32 > 2 q - 6 q - q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 29]][a, z]
Out[8]=	3 3 3 3 3 1 1 2 2 z 5 z 3 z z z z 6 z z z ---- - ---- - ---- + ---- + -- - --- + --- + - + -- + -- - ---- + -- + -- - 9 7 5 3 7 5 3 a 9 7 5 3 a a z a z a z a z a a a a a a a 5 5 5 z 3 z z > -- - ---- - -- 7 5 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 29]][a, z]
Out[9]=	-4 9 4 2 1 1 2 2 2 z 9 z 15 z 7 z z --- - -- - -- + -- + ---- + ---- - ---- - ---- + --- + --- + ---- + --- - - - 10 8 6 4 9 7 5 3 9 7 5 3 a a a a a a z a z a z a z a a a a 2 2 2 2 2 2 3 3 3 3 z 9 z 25 z 14 z 2 z z 4 z 3 z 19 z 31 z > --- + ---- + ----- + ----- - ---- - -- - ---- - ---- - ----- - ----- - 12 10 8 6 4 2 11 9 7 5 a a a a a a a a a a 3 3 4 4 4 4 4 4 5 5 9 z 2 z 2 z 16 z 44 z 37 z 7 z 4 z 8 z 3 z > ---- + ---- + ---- - ----- - ----- - ----- - ---- + ---- + ---- + ---- + 3 a 12 10 8 6 4 2 11 9 a a a a a a a a a 5 5 5 5 6 6 6 6 6 6 8 z 23 z 9 z z z 19 z 48 z 42 z 11 z 3 z > ---- + ----- + ---- - -- - --- + ----- + ----- + ----- + ----- - ---- - 7 5 3 a 12 10 8 6 4 2 a a a a a a a a a 7 7 7 7 7 8 8 8 8 9 4 z 9 z 17 z 2 z 6 z 8 z 16 z 16 z 8 z 7 z > ---- + ---- + ----- - ---- - ---- - ---- - ----- - ----- - ---- - ---- - 11 9 7 5 3 10 8 6 4 9 a a a a a a a a a a 9 9 10 10 13 z 6 z 2 z 2 z > ----- - ---- - ----- - ----- 7 5 8 6 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 2 q 4 6 6 2 8 2 6 q + 4 q + ----- + - + -- + 10 q t + 4 q t + 11 q t + 10 q t + 2 2 t t q t 8 3 10 3 10 4 12 4 12 5 14 5 > 11 q t + 11 q t + 13 q t + 11 q t + 8 q t + 13 q t + 14 6 16 6 16 7 18 7 18 8 20 8 22 9 > 7 q t + 8 q t + 3 q t + 7 q t + q t + 3 q t + q t

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