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DrawMorseLink

PD Presentation:

X6172 X2536 X11,19,12,18 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X15,17,16,20 X19,13,20,16 X17,9,18,12

Gauss Code:

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

Jones Polynomial:

4q - 5q2 + 10q3 - 5q4 + 11q5 - 5q6 + 6q7 - q8 + q9

A2 (sl(3)) Invariant:

4q2 + 2q4 + 7q6 + 14q8 + 18q10 + 28q12 + 30q14 + 34q16 + 32q18 + 26q20 + 23q22 + 13q24 + 7q26 + 4q28 + q30

HOMFLY-PT Polynomial:

a-10z-4 + a-10z-2 - 4a-8z-4 - 7a-8z-2 - 4a-8 + 6a-6z-4 + 15a-6z-2 + 14a-6 + 6a-6z2 - 4a-4z-4 - 13a-4z-2 - 16a-4 - 10a-4z2 - 3a-4z4 + a-2z-4 + 4a-2z-2 + 6a-2 + 4a-2z2

Kauffman Polynomial:

- a-10z-4 + 5a-10z-2 - 10a-10 + 10a-10z2 - 5a-10z4 + a-10z6 + 4a-9z-3 - 15a-9z-1 + 20a-9z - 10a-9z3 + a-9z7 - 4a-8z-4 + 14a-8z-2 - 25a-8 + 30a-8z2 - 19a-8z4 + 3a-8z6 + a-8z8 + 12a-7z-3 - 41a-7z-1 + 55a-7z - 30a-7z3 - 6a-7z5 + 6a-7z7 - 6a-6z-4 + 18a-6z-2 - 31a-6 + 40a-6z2 - 39a-6z4 + 12a-6z6 + a-6z8 + 12a-5z-3 - 41a-5z-1 + 55a-5z - 30a-5z3 + 5a-5z7 - 4a-4z-4 + 14a-4z-2 - 25a-4 + 30a-4z2 - 25a-4z4 + 10a-4z6 + 4a-3z-3 - 15a-3z-1 + 20a-3z - 10a-3z3 + 6a-3z5 - a-2z-4 + 5a-2z-2 - 10a-2 + 10a-2z2

Khovanov Homology:

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                 1 j = 17               1 1 j = 15             5     j = 13             1     j = 11         11 5       j = 9       4 10         j = 7     6 1           j = 5 1   4             j = 3 5 6               j = 1 4

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]=	5
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 112]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 112]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[11, 19, 12, 18], X[3, 11, 4, 10], > X[9, 1, 10, 4], X[7, 15, 8, 14], X[13, 5, 14, 8], X[15, 17, 16, 20], > X[19, 13, 20, 16], X[17, 9, 18, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 10}, {-7, 6, -8, 9}, > {-10, 3, -9, 8}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 4 5 6 7 8 9 4 q - 5 q + 10 q - 5 q + 11 q - 5 q + 6 q - q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 4 6 8 10 12 14 16 18 4 q + 2 q + 7 q + 14 q + 18 q + 28 q + 30 q + 34 q + 32 q + 20 22 24 26 28 30 > 26 q + 23 q + 13 q + 7 q + 4 q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 112]][a, z]
Out[8]=	-4 14 16 6 1 4 6 4 1 1 7 -- + -- - -- + -- + ------ - ----- + ----- - ----- + ----- + ------ - ----- + 8 6 4 2 10 4 8 4 6 4 4 4 2 4 10 2 8 2 a a a a a z a z a z a z a z a z a z 2 2 2 4 15 13 4 6 z 10 z 4 z 3 z > ----- - ----- + ----- + ---- - ----- + ---- - ---- 6 2 4 2 2 2 6 4 2 4 a z a z a z a a a a
In[9]:=	Kauffman[Link[10, NonAlternating, 112]][a, z]
Out[9]=	-10 25 31 25 10 1 4 6 4 1 4 --- - -- - -- - -- - -- - ------ - ----- - ----- - ----- - ----- + ----- + 10 8 6 4 2 10 4 8 4 6 4 4 4 2 4 9 3 a a a a a a z a z a z a z a z a z 12 12 4 5 14 18 14 5 15 > ----- + ----- + ----- + ------ + ----- + ----- + ----- + ----- - ---- - 7 3 5 3 3 3 10 2 8 2 6 2 4 2 2 2 9 a z a z a z a z a z a z a z a z a z 2 2 2 41 41 15 20 z 55 z 55 z 20 z 10 z 30 z 40 z > ---- - ---- - ---- + ---- + ---- + ---- + ---- + ----- + ----- + ----- + 7 5 3 9 7 5 3 10 8 6 a z a z a z a a a a a a a 2 2 3 3 3 3 4 4 4 30 z 10 z 10 z 30 z 30 z 10 z 5 z 19 z 39 z > ----- + ----- - ----- - ----- - ----- - ----- - ---- - ----- - ----- - 4 2 9 7 5 3 10 8 6 a a a a a a a a a 4 5 5 6 6 6 6 7 7 7 8 25 z 6 z 6 z z 3 z 12 z 10 z z 6 z 5 z z > ----- - ---- + ---- + --- + ---- + ----- + ----- + -- + ---- + ---- + -- + 4 7 3 10 8 6 4 9 7 5 8 a a a a a a a a a a a 8 z > -- 6 a
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 3 5 2 7 2 7 3 9 3 9 4 4 q + 5 q + q + 6 q t + 4 q t + 6 q t + q t + 4 q t + 10 q t + 11 4 11 5 13 6 15 6 17 7 17 8 19 8 > 11 q t + 5 q t + q t + 5 q t + q t + q t + q t

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