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L10n109 L10n111

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- 3q-3/2 + 4q-1/2 - 8q1/2 + 7q3/2 - 10q5/2 + 6q7/2 - 6q9/2 + 3q11/2 - q13/2

A2 (sl(3)) Invariant:

3q-6 + 3q-4 + 3q-2 + 9 + 8q2 + 12q4 + 12q6 + 10q8 + 10q10 + 4q12 + 5q14 + 2q16 - q18 + q20

HOMFLY-PT Polynomial:

- a-5z-3 - 2a-5z-1 - a-5z - a-5z3 + 3a-3z-3 + 7a-3z-1 + 7a-3z + 3a-3z3 + a-3z5 - 3a-1z-3 - 8a-1z-1 - 9a-1z - 4a-1z3 + az-3 + 3az-1 + 3az

Kauffman Polynomial:

- a-7z + 2a-7z3 - a-7z5 - a-6z2 + 6a-6z4 - 3a-6z6 + a-5z-3 - 5a-5z-1 + 8a-5z - 7a-5z3 + 9a-5z5 - 4a-5z7 - 3a-4z-2 + 10a-4 - 13a-4z2 + 11a-4z4 - a-4z6 - 2a-4z8 + 3a-3z-3 - 12a-3z-1 + 25a-3z - 36a-3z3 + 27a-3z5 - 9a-3z7 - 6a-2z-2 + 19a-2 - 23a-2z2 + 11a-2z4 - a-2z6 - 2a-2z8 + 3a-1z-3 - 12a-1z-1 + 27a-1z - 33a-1z3 + 17a-1z5 - 5a-1z7 - 3z-2 + 10 - 11z2 + 6z4 - 3z6 + az-3 - 5az-1 + 11az - 6az3

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14                 1 j = 12               2   j = 10             4 1   j = 8           4 4     j = 6         6 2       j = 4       3 6         j = 2     5 4           j = 0   1 5             j = -2 2 3               j = -4 3

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]=	4
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 110]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 110]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 13, 4, 12], X[13, 17, 14, 20], X[19, 11, 20, 16], > X[7, 19, 8, 18], X[15, 8, 16, 9], X[9, 14, 10, 15], X[17, 5, 18, 10], > X[2, 5, 3, 6], X[11, 1, 12, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, -2, 10}, {-8, 5, -4, 3}, {9, -1, -5, 6, -7, 8}, > {-10, 2, -3, 7, -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-3 4 3/2 5/2 7/2 9/2 11/2 ---- + ------- - 8 Sqrt[q] + 7 q - 10 q + 6 q - 6 q + 3 q - 3/2 Sqrt[q] q 13/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	3 3 3 2 4 6 8 10 12 14 9 + -- + -- + -- + 8 q + 12 q + 12 q + 10 q + 10 q + 4 q + 5 q + 6 4 2 q q q 16 18 20 > 2 q - q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 110]][a, z]
Out[8]=	1 3 3 a 2 7 8 3 a z 7 z 9 z -(-----) + ----- - ---- + -- - ---- + ---- - --- + --- - -- + --- - --- + 5 3 3 3 3 3 5 3 a z z 5 3 a a z a z a z z a z a z a a 3 3 3 5 z 3 z 4 z z > 3 a z - -- + ---- - ---- + -- 5 3 a 3 a a a
In[9]:=	Kauffman[Link[10, NonAlternating, 110]][a, z]
Out[9]=	10 19 1 3 3 a 3 3 6 5 12 10 + -- + -- + ----- + ----- + ---- + -- - -- - ----- - ----- - ---- - ---- - 4 2 5 3 3 3 3 3 2 4 2 2 2 5 3 a a a z a z a z z z a z a z a z a z 2 2 2 12 5 a z 8 z 25 z 27 z 2 z 13 z 23 z > --- - --- - -- + --- + ---- + ---- + 11 a z - 11 z - -- - ----- - ----- + a z z 7 5 3 a 6 4 2 a a a a a a 3 3 3 3 4 4 4 5 2 z 7 z 36 z 33 z 3 4 6 z 11 z 11 z z > ---- - ---- - ----- - ----- - 6 a z + 6 z + ---- + ----- + ----- - -- + 7 5 3 a 6 4 2 7 a a a a a a a 5 5 5 6 6 6 7 7 7 8 9 z 27 z 17 z 6 3 z z z 4 z 9 z 5 z 2 z > ---- + ----- + ----- - 3 z - ---- - -- - -- - ---- - ---- - ---- - ---- - 5 3 a 6 4 2 5 3 a 4 a a a a a a a a 8 2 z > ---- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 2 1 3 2 4 4 2 6 2 5 + 5 q + ----- + ----- + - + ---- + 4 q t + 3 q t + 6 q t + 6 q t + 4 2 2 2 t 2 q t q t q t 6 3 8 3 8 4 10 4 10 5 12 5 14 6 > 2 q t + 4 q t + 4 q t + 4 q t + q t + 2 q t + q t

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