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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X11,18,12,19 X7,14,8,15 X13,8,14,9 X15,13,16,20 X19,17,20,16 X17,12,18,5 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-9 - q-8 + 4q-7 - 4q-6 + 6q-5 - 5q-4 + 5q-3 - 4q-2 + 2q-1

A2 (sl(3)) Invariant:

q-30 + 3q-28 + 3q-26 + 4q-24 + 6q-22 + 3q-20 + 3q-18 + 2q-16 + q-12 - q-10 + q-8 - q-4 + 2q-2

HOMFLY-PT Polynomial:

a2 + 2a2z2 - a4 - a4z2 - a4z4 + a6z-2 + 3a6 + 3a6z2 - 2a8z-2 - 3a8 + a10z-2

Kauffman Polynomial:

- a2 + 3a2z2 - 2a3z + 3a3z3 + a3z5 + 4a4z2 - 5a4z4 + 3a4z6 - 6a5z + 11a5z3 - 8a5z5 + 3a5z7 + a6z-2 - 2a6 + 3a6z2 - 6a6z4 + a6z6 + a6z8 - 2a7z-1 + 2a7z + 5a7z3 - 11a7z5 + 4a7z7 + 2a8z-2 - 7a8 + 10a8z2 - 6a8z4 - a8z6 + a8z8 - 2a9z-1 + 6a9z - 3a9z3 - 2a9z5 + a9z7 + a10z-2 - 5a10 + 8a10z2 - 5a10z4 + a10z6

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -1                 2 j = -3               3 1 j = -5             2 1   j = -7           3 3     j = -9         3 2       j = -11       3 5         j = -13     1 1           j = -15     3             j = -17 1 1               j = -19 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, 80]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 80]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 18, 12, 19], X[7, 14, 8, 15], > X[13, 8, 14, 9], X[15, 13, 16, 20], X[19, 17, 20, 16], X[17, 12, 18, 5], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, -4, 5, 10, -2, -3, 8}, > {-5, 4, -6, 7, -8, 3, -7, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-9 -8 4 4 6 5 5 4 2 q - q + -- - -- + -- - -- + -- - -- + - 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 3 3 4 6 3 3 2 -12 -10 -8 -4 2 q + --- + --- + --- + --- + --- + --- + --- + q - q + q - q + -- 28 26 24 22 20 18 16 2 q q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 80]][a, z]
Out[8]=	6 8 10 2 4 6 8 a 2 a a 2 2 4 2 6 2 4 4 a - a + 3 a - 3 a + -- - ---- + --- + 2 a z - a z + 3 a z - a z 2 2 2 z z z
In[9]:=	Kauffman[Link[10, NonAlternating, 80]][a, z]
Out[9]=	6 8 10 7 9 2 6 8 10 a 2 a a 2 a 2 a 3 5 -a - 2 a - 7 a - 5 a + -- + ---- + --- - ---- - ---- - 2 a z - 6 a z + 2 2 2 z z z z z 7 9 2 2 4 2 6 2 8 2 10 2 > 2 a z + 6 a z + 3 a z + 4 a z + 3 a z + 10 a z + 8 a z + 3 3 5 3 7 3 9 3 4 4 6 4 8 4 > 3 a z + 11 a z + 5 a z - 3 a z - 5 a z - 6 a z - 6 a z - 10 4 3 5 5 5 7 5 9 5 4 6 6 6 8 6 > 5 a z + a z - 8 a z - 11 a z - 2 a z + 3 a z + a z - a z + 10 6 5 7 7 7 9 7 6 8 8 8 > a z + 3 a z + 4 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-3 2 1 1 1 3 1 1 3 q + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + q 19 8 17 8 17 7 15 6 13 6 13 5 11 5 q t q t q t q t q t q t q t 5 3 2 3 3 2 1 3 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q t q t q t q t q t q t q t q t

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