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DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X4,19,1,20 X11,14,12,15 X3,10,4,11 X5,13,6,12 X13,5,14,20 X16,9,17,10 X15,2,16,3 X8,17,9,18

Gauss Code:

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

Jones Polynomial:

- 2q-15/2 + 3q-13/2 - 5q-11/2 + 7q-9/2 - 6q-7/2 + 5q-5/2 - 5q-3/2 + 2q-1/2 - q1/2

A2 (sl(3)) Invariant:

q-28 + q-26 + 2q-24 + q-22 - q-20 + q-18 - 2q-16 - q-14 - q-12 - q-10 + 3q-8 + q-6 + 3q-4 + q-2 + q2

HOMFLY-PT Polynomial:

- az-1 - 2az - az3 + a3z-1 + 2a3z3 + a3z5 + a5z-1 + 3a5z + 3a5z3 + a5z5 - 2a7z-1 - 3a7z - a7z3 + a9z-1

Kauffman Polynomial:

az-1 - 3az + 3az3 - az5 - a2 + 4a2z4 - 2a2z6 + a3z-1 - 2a3z + 4a3z3 + 2a3z5 - 2a3z7 - 2a4 + 4a4z4 - a4z6 - a4z8 - a5z-1 + 8a5z - 11a5z3 + 9a5z5 - 4a5z7 - 3a6 - a6z8 - 2a7z-1 + 12a7z - 15a7z3 + 6a7z5 - 2a7z7 - a8 - a8z6 - a9z-1 + 5a9z - 3a9z3

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                 1 j = 0               1   j = -2             4 1   j = -4           2 2     j = -6         4 3       j = -8       3 2         j = -10     2 4           j = -12   2 4             j = -14   1               j = -16 2

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]=	2
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 31]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 31]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[4, 19, 1, 20], X[11, 14, 12, 15], > X[3, 10, 4, 11], X[5, 13, 6, 12], X[13, 5, 14, 20], X[16, 9, 17, 10], > X[15, 2, 16, 3], X[8, 17, 9, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 9, -5, -3}, {-6, -1, 2, -10, 8, 5, -4, 6, -7, 4, -9, -8, 10, -2, > 3, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-2 3 5 7 6 5 5 2 ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - Sqrt[q] 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 2 -22 -20 -18 2 -14 -12 -10 3 -6 q + q + --- + q - q + q - --- - q - q - q + -- + q + 24 16 8 q q q 3 -2 2 > -- + q + q 4 q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 31]][a, z]
Out[8]=	3 5 7 9 a a a 2 a a 5 7 3 3 3 -(-) + -- + -- - ---- + -- - 2 a z + 3 a z - 3 a z - a z + 2 a z + z z z z z 5 3 7 3 3 5 5 5 > 3 a z - a z + a z + a z
In[9]:=	Kauffman[Link[10, NonAlternating, 31]][a, z]
Out[9]=	3 5 7 9 2 4 6 8 a a a 2 a a 3 5 -a - 2 a - 3 a - a + - + -- - -- - ---- - -- - 3 a z - 2 a z + 8 a z + z z z z z 7 9 3 3 3 5 3 7 3 9 3 > 12 a z + 5 a z + 3 a z + 4 a z - 11 a z - 15 a z - 3 a z + 2 4 4 4 5 3 5 5 5 7 5 2 6 4 6 > 4 a z + 4 a z - a z + 2 a z + 9 a z + 6 a z - 2 a z - a z - 8 6 3 7 5 7 7 7 4 8 6 8 > a z - 2 a z - 4 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 2 1 2 4 2 4 3 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 4 2 16 6 14 5 12 5 12 4 10 4 10 3 8 3 8 2 q q q t q t q t q t q t q t q t q t 4 3 2 t 2 2 > ----- + ---- + ---- + t + -- + q t 6 2 6 4 2 q t q t q t q

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