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DrawMorseLink

PD Presentation:

X6172 X3,13,4,12 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 X11,1,12,4

Gauss Code:

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

Jones Polynomial:

3q-5 - 5q-4 + 8q-3 - 7q-2 + 9q-1 - 7 + 5q - 3q2 + q3

A2 (sl(3)) Invariant:

q-20 + q-18 + 4q-16 + 2q-14 + 3q-12 + 6q-10 + 3q-8 + 6q-6 + q-4 + q-2 - 2q2 + q4 - q6 + q10

HOMFLY-PT Polynomial:

a-2 + a-2z2 - 3 - 5z2 - 2z4 + a2z-2 + 6a2 + 7a2z2 + 4a2z4 + a2z6 - 2a4z-2 - 5a4 - 3a4z2 - a4z4 + a6z-2 + a6

Kauffman Polynomial:

- a-2 + 3a-2z2 - 3a-2z4 + a-2z6 - a-1z + 9a-1z3 - 10a-1z5 + 3a-1z7 - 3 + 11z2 - 11z4 - z6 + 2z8 - 3az + 16az3 - 25az5 + 9az7 + a2z-2 - 7a2 + 18a2z2 - 23a2z4 + 5a2z6 + 2a2z8 - 2a3z-1 + a3z + 7a3z3 - 12a3z5 + 6a3z7 + 2a4z-2 - 8a4 + 16a4z2 - 15a4z4 + 7a4z6 - 2a5z-1 + 3a5z + 3a5z5 + a6z-2 - 4a6 + 6a6z2

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                 1 j = 5               2   j = 3             3 1   j = 1           4 2     j = -1         5 3       j = -3       4 6         j = -5     4 3           j = -7   1 4             j = -9 2 4               j = -11 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]=	3
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 90]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 90]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 13, 4, 12], 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[11, 1, 12, 4]]
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]=	3 5 8 7 9 2 3 -7 + -- - -- + -- - -- + - + 5 q - 3 q + q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 4 2 3 6 3 6 -4 -2 2 4 6 10 q + q + --- + --- + --- + --- + -- + -- + q + q - 2 q + q - q + q 16 14 12 10 8 6 q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 90]][a, z]
Out[8]=	2 4 6 2 -2 2 4 6 a 2 a a 2 z 2 2 4 2 -3 + a + 6 a - 5 a + a + -- - ---- + -- - 5 z + -- + 7 a z - 3 a z - 2 2 2 2 z z z a 4 2 4 4 4 2 6 > 2 z + 4 a z - a z + a z
In[9]:=	Kauffman[Link[10, NonAlternating, 90]][a, z]
Out[9]=	2 4 6 3 5 -2 2 4 6 a 2 a a 2 a 2 a z -3 - a - 7 a - 8 a - 4 a + -- + ---- + -- - ---- - ---- - - - 3 a z + 2 2 2 z z a z z z 2 3 3 5 2 3 z 2 2 4 2 6 2 9 z > a z + 3 a z + 11 z + ---- + 18 a z + 16 a z + 6 a z + ---- + 2 a a 4 5 3 3 3 4 3 z 2 4 4 4 10 z 5 > 16 a z + 7 a z - 11 z - ---- - 23 a z - 15 a z - ----- - 25 a z - 2 a a 6 7 3 5 5 5 6 z 2 6 4 6 3 z 7 > 12 a z + 3 a z - z + -- + 5 a z + 7 a z + ---- + 9 a z + 2 a a 3 7 8 2 8 > 6 a z + 2 z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 5 3 2 4 1 4 4 3 4 3 t -- + - + ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 3 q 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q q t q t q t q t q t q t q t q t 2 3 2 3 3 5 3 7 4 > 4 q t + 2 q t + 3 q t + q t + 2 q t + q t

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