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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X20,16,21,15 X14,8,15,7 X10,22,5,21 X11,19,12,18 X9,17,10,16 X17,11,18,22 X19,9,20,8 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

q - q2 + 3q3 - q4 + 3q5 - q6 + q7 - q8

A2 (sl(3)) Invariant:

q4 + q6 + 2q8 + 4q10 + 5q12 + 6q14 + 6q16 + 6q18 + 2q20 + q22 - 2q24 - 2q26 - 2q28 - q30 + q32 - q34

HOMFLY-PT Polynomial:

- a-10z-2 - a-10 - a-10z2 + 4a-8z-2 + 8a-8 + 10a-8z2 + 6a-8z4 + a-8z6 - 5a-6z-2 - 15a-6 - 21a-6z2 - 17a-6z4 - 7a-6z6 - a-6z8 + 2a-4z-2 + 8a-4 + 11a-4z2 + 6a-4z4 + a-4z6

Kauffman Polynomial:

a-11z-1 - 2a-11z - a-10z-2 + 2a-10 - 3a-10z2 + 5a-9z-1 - 13a-9z + 19a-9z3 - 12a-9z5 + 2a-9z7 - 4a-8z-2 + 13a-8 - 29a-8z2 + 36a-8z4 - 19a-8z6 + 3a-8z8 + 9a-7z-1 - 27a-7z + 31a-7z3 - 9a-7z5 - 3a-7z7 + a-7z9 - 5a-6z-2 + 20a-6 - 45a-6z2 + 53a-6z4 - 26a-6z6 + 4a-6z8 + 5a-5z-1 - 16a-5z + 12a-5z3 + 3a-5z5 - 5a-5z7 + a-5z9 - 2a-4z-2 + 10a-4 - 19a-4z2 + 17a-4z4 - 7a-4z6 + a-4z8

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 17               2 1 j = 15             1 2 1 j = 13           2 2     j = 11         1 1 2     j = 9       1 3         j = 7     2 1 1         j = 5   1 3             j = 3                   j = 1 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 294]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 294]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[20, 16, 21, 15], X[14, 8, 15, 7], > X[10, 22, 5, 21], X[11, 19, 12, 18], X[9, 17, 10, 16], X[17, 11, 18, 22], > X[19, 9, 20, 8], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, 9, -7, -5}, > {-6, -2, 11, -4, 3, 7, -8, 6, -9, -3, 5, 8}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 4 5 6 7 8 q - q + 3 q - q + 3 q - q + q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	4 6 8 10 12 14 16 18 20 22 24 q + q + 2 q + 4 q + 5 q + 6 q + 6 q + 6 q + 2 q + q - 2 q - 26 28 30 32 34 > 2 q - 2 q - q + q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 294]][a, z]
Out[8]=	2 2 2 -10 8 15 8 1 4 5 2 z 10 z 21 z -a + -- - -- + -- - ------ + ----- - ----- + ----- - --- + ----- - ----- + 8 6 4 10 2 8 2 6 2 4 2 10 8 6 a a a a z a z a z a z a a a 2 4 4 4 6 6 6 8 11 z 6 z 17 z 6 z z 7 z z z > ----- + ---- - ----- + ---- + -- - ---- + -- - -- 4 8 6 4 8 6 4 6 a a a a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 294]][a, z]
Out[9]=	2 13 20 10 1 4 5 2 1 5 9 --- + -- + -- + -- - ------ - ----- - ----- - ----- + ----- + ---- + ---- + 10 8 6 4 10 2 8 2 6 2 4 2 11 9 7 a a a a a z a z a z a z a z a z a z 2 2 2 2 3 5 2 z 13 z 27 z 16 z 3 z 29 z 45 z 19 z 19 z > ---- - --- - ---- - ---- - ---- - ---- - ----- - ----- - ----- + ----- + 5 11 9 7 5 10 8 6 4 9 a z a a a a a a a a a 3 3 4 4 4 5 5 5 6 31 z 12 z 36 z 53 z 17 z 12 z 9 z 3 z 19 z > ----- + ----- + ----- + ----- + ----- - ----- - ---- + ---- - ----- - 7 5 8 6 4 9 7 5 8 a a a a a a a a a 6 6 7 7 7 8 8 8 9 9 26 z 7 z 2 z 3 z 5 z 3 z 4 z z z z > ----- - ---- + ---- - ---- - ---- + ---- + ---- + -- + -- + -- 6 4 9 7 5 8 6 4 7 5 a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	5 5 7 q q 7 9 7 2 9 2 11 2 11 3 3 q + 2 q + -- + -- + q t + q t + q t + 3 q t + q t + q t + 2 t t 13 3 11 4 13 4 15 4 15 5 17 5 15 6 > 2 q t + 2 q t + 2 q t + q t + 2 q t + 2 q t + q t + 17 6 > q t

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