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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X7,17,8,16 X9,11,10,20 X11,18,12,19 X15,9,16,8 X19,5,20,10 X17,14,18,15 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

q-3 - 2q-2 + 2q-1 - 2 + 3q - q2 + 2q3 + q6

A2 (sl(3)) Invariant:

q-10 - q-2 + 2q4 + 4q6 + 4q8 + 5q10 + 3q12 + 3q14 + 3q16 + 2q18 + q20

HOMFLY-PT Polynomial:

a-6z-2 + a-6 - 2a-4z-2 - 4a-4 - a-4z2 + a-2z-2 + 4a-2 + 2a-2z2 - 2 - 3z2 - z4 + a2 + a2z2

Kauffman Polynomial:

a-6z-2 - 4a-6 + 9a-6z2 - 6a-6z4 + a-6z6 - 2a-5z-1 + 4a-5z - 2a-5z3 + 2a-4z-2 - 6a-4 + 6a-4z2 - 2a-4z4 - 2a-3z-1 + 8a-3z3 - 6a-3z5 + a-3z7 + a-2z-2 - 3a-2 - a-2z2 + 6a-2z4 - 5a-2z6 + a-2z8 - 6a-1z + 19a-1z3 - 15a-1z5 + 3a-1z7 - 1 + 5z2 - 2z4 - 3z6 + z8 - 2az + 9az3 - 9az5 + 2az7 - a2 + 3a2z2 - 4a2z4 + a2z6

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 j = 13                     1 j = 11                     1 j = 9               1 1     j = 7             2         j = 5           1 3 1       j = 3         3 1           j = 1       1 3 1           j = -1     1 2 1             j = -3   1 1                 j = -5   1                   j = -7 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[10, NonAlternating, 72]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 72]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[7, 17, 8, 16], X[9, 11, 10, 20], > X[11, 18, 12, 19], X[15, 9, 16, 8], X[19, 5, 20, 10], X[17, 14, 18, 15], > X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, -3, 6, -4, 7}, > {-5, -2, 10, 8, -6, 3, -8, 5, -7, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-3 2 2 2 3 6 -2 + q - -- + - + 3 q - q + 2 q + q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -2 4 6 8 10 12 14 16 18 20 q - q + 2 q + 4 q + 4 q + 5 q + 3 q + 3 q + 3 q + 2 q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 72]][a, z]
Out[8]=	2 2 -6 4 4 2 1 2 1 2 z 2 z 2 2 4 -2 + a - -- + -- + a + ----- - ----- + ----- - 3 z - -- + ---- + a z - z 4 2 6 2 4 2 2 2 4 2 a a a z a z a z a a
In[9]:=	Kauffman[Link[10, NonAlternating, 72]][a, z]
Out[9]=	4 6 3 2 1 2 1 2 2 4 z 6 z -1 - -- - -- - -- - a + ----- + ----- + ----- - ---- - ---- + --- - --- - 6 4 2 6 2 4 2 2 2 5 3 5 a a a a a z a z a z a z a z a 2 2 2 3 3 3 2 9 z 6 z z 2 2 2 z 8 z 19 z 3 > 2 a z + 5 z + ---- + ---- - -- + 3 a z - ---- + ---- + ----- + 9 a z - 6 4 2 5 3 a a a a a a 4 4 4 5 5 6 4 6 z 2 z 6 z 2 4 6 z 15 z 5 6 z > 2 z - ---- - ---- + ---- - 4 a z - ---- - ----- - 9 a z - 3 z + -- - 6 4 2 3 a 6 a a a a a 6 7 7 8 5 z 2 6 z 3 z 7 8 z > ---- + a z + -- + ---- + 2 a z + z + -- 2 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	1 3 1 1 1 1 1 2 q 3 - + 3 q + 3 q + ----- + ----- + ----- + ----- + ---- + --- + - + q t + q t + q 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 5 3 9 3 9 4 11 6 13 6 > q t + 3 q t + 2 q t + q t + q t + q t + q t + q t

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