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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X13,20,14,17 X7,18,8,19 X17,10,18,11 X9,15,10,14 X15,9,16,8 X19,16,20,5 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-30 + 3q-28 + 2q-26 + 4q-24 + 5q-22 + 5q-20 + 5q-18 + 2q-16 + q-14 - 2q-12 - q-10 + q-2 + 1

HOMFLY-PT Polynomial:

2a2 + 4a2z2 + a2z4 - 4a4 - 6a4z2 - 5a4z4 - a4z6 + a6z-2 + 6a6 + 6a6z2 + a6z4 - 2a8z-2 - 4a8 + a10z-2

Kauffman Polynomial:

- 2a2 + 6a2z2 - 5a2z4 + a2z6 - 4a3z + 6a3z3 - 5a3z5 + a3z7 - 2a4 + 12a4z2 - 11a4z4 + 2a4z6 - 12a5z + 22a5z3 - 13a5z5 + 2a5z7 + a6z-2 - 2a6 + 6a6z2 + 2a6z4 - 5a6z6 + a6z8 - 2a7z-1 - 4a7z + 16a7z3 - 12a7z5 + 2a7z7 + 2a8z-2 - 5a8 + 6a8z2 + 3a8z4 - 5a8z6 + a8z8 - 2a9z-1 + 4a9z - 4a9z5 + a9z7 + a10z-2 - 4a10 + 6a10z2 - 5a10z4 + a10z6

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                     1 j = -1                       j = -3               1 2 1   j = -5               1 1     j = -7           2 2 1       j = -9         1 1 1         j = -11       1 4 1           j = -13     1 1 3             j = -15     1                 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, 91]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 91]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[13, 20, 14, 17], X[7, 18, 8, 19], > X[17, 10, 18, 11], X[9, 15, 10, 14], X[15, 9, 16, 8], X[19, 16, 20, 5], > X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {-5, 4, -8, 3}, > {9, -1, -4, 7, -6, 5, 10, -2, -3, 6, -7, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-9 -8 2 -6 -5 -3 -2 1 1 + q - q + -- + q + q - q + q - - 7 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 3 2 4 5 5 5 2 -14 2 -10 -2 1 + q + --- + --- + --- + --- + --- + --- + --- + q - --- - q + q 28 26 24 22 20 18 16 12 q q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 91]][a, z]
Out[8]=	6 8 10 2 4 6 8 a 2 a a 2 2 4 2 6 2 2 a - 4 a + 6 a - 4 a + -- - ---- + --- + 4 a z - 6 a z + 6 a z + 2 2 2 z z z 2 4 4 4 6 4 4 6 > a z - 5 a z + a z - a z
In[9]:=	Kauffman[Link[10, NonAlternating, 91]][a, z]
Out[9]=	6 8 10 7 9 2 4 6 8 10 a 2 a a 2 a 2 a 3 -2 a - 2 a - 2 a - 5 a - 4 a + -- + ---- + --- - ---- - ---- - 4 a z - 2 2 2 z z z z z 5 7 9 2 2 4 2 6 2 8 2 > 12 a z - 4 a z + 4 a z + 6 a z + 12 a z + 6 a z + 6 a z + 10 2 3 3 5 3 7 3 2 4 4 4 6 4 > 6 a z + 6 a z + 22 a z + 16 a z - 5 a z - 11 a z + 2 a z + 8 4 10 4 3 5 5 5 7 5 9 5 2 6 > 3 a z - 5 a z - 5 a z - 13 a z - 12 a z - 4 a z + a z + 4 6 6 6 8 6 10 6 3 7 5 7 7 7 9 7 > 2 a z - 5 a z - 5 a z + a z + a z + 2 a z + 2 a z + a z + 6 8 8 8 > a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-5 2 1 1 1 1 1 1 1 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 19 8 17 8 17 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 3 4 1 1 1 2 1 2 1 > ------ + ------ + ----- + ------ + ----- + ----- + ----- + ----- + ---- + 13 4 11 4 9 4 11 3 9 3 7 3 9 2 7 2 7 q t q t q t q t q t q t q t q t q t 1 1 t 2 > ---- + ---- + -- + q t 5 3 3 q t q t q

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