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DrawMorseLink

PD Presentation:

X6172 X14,3,15,4 X12,15,5,16 X8,17,9,18 X16,7,17,8 X18,9,19,10 X20,11,13,12 X10,19,11,20 X2536 X4,13,1,14

Gauss Code:

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

Jones Polynomial:

q-13 - 2q-12 + 4q-11 - 5q-10 + 7q-9 - 7q-8 + 7q-7 - 4q-6 + 4q-5 - 2q-4 + q-3

A2 (sl(3)) Invariant:

q-40 + 2q-38 + q-36 + 2q-34 + q-32 + 2q-30 + 3q-28 + 2q-26 + 5q-24 + 2q-22 + 3q-20 + 2q-18 + q-14 - q-12 + q-10

HOMFLY-PT Polynomial:

3a6z2 + 4a6z4 + a6z6 + a8z-2 + 8a8 + 15a8z2 + 10a8z4 + 2a8z6 - 2a10z-2 - 11a10 - 12a10z2 - 3a10z4 + a12z-2 + 3a12 + a12z2

Kauffman Polynomial:

3a6z2 - 4a6z4 + a6z6 + 4a7z3 - 7a7z5 + 2a7z7 - a8z-2 + 8a8 - 20a8z2 + 23a8z4 - 14a8z6 + 3a8z8 + 2a9z-1 - 11a9z + 18a9z3 - 11a9z5 + a9z9 - 2a10z-2 + 13a10 - 31a10z2 + 38a10z4 - 22a10z6 + 5a10z8 + 2a11z-1 - 11a11z + 18a11z3 - 8a11z5 + a11z9 - a12z-2 + 5a12 - 8a12z2 + 9a12z4 - 5a12z6 + 2a12z8 + a13z3 - 2a13z5 + 2a13z7 - 2a14z2 - a14z4 + 2a14z6 - 3a15z3 + 2a15z5 + a16 - 2a16z2 + a16z4

Khovanov Homology:

trqj r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -5                     1 j = -7                   2 1 j = -9                 2     j = -11               2 2     j = -13             5 2       j = -15           3 3         j = -17         4 4           j = -19       1 3             j = -21     3 4               j = -23   1 3                 j = -25   1                   j = -27 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, Alternating, 142]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 142]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[12, 15, 5, 16], X[8, 17, 9, 18], > X[16, 7, 17, 8], X[18, 9, 19, 10], X[20, 11, 13, 12], X[10, 19, 11, 20], > X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 5, -4, 6, -8, 7, -3}, > {10, -2, 3, -5, 4, -6, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-13 2 4 5 7 7 7 4 4 2 -3 q - --- + --- - --- + -- - -- + -- - -- + -- - -- + q 12 11 10 9 8 7 6 5 4 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 2 -36 2 -32 2 3 2 5 2 3 2 q + --- + q + --- + q + --- + --- + --- + --- + --- + --- + --- + 38 34 30 28 26 24 22 20 18 q q q q q q q q q -14 -12 -10 > q - q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 142]][a, z]
Out[8]=	8 10 12 8 10 12 a 2 a a 6 2 8 2 10 2 8 a - 11 a + 3 a + -- - ----- + --- + 3 a z + 15 a z - 12 a z + 2 2 2 z z z 12 2 6 4 8 4 10 4 6 6 8 6 > a z + 4 a z + 10 a z - 3 a z + a z + 2 a z
In[9]:=	Kauffman[Link[10, Alternating, 142]][a, z]
Out[9]=	8 10 12 9 11 8 10 12 16 a 2 a a 2 a 2 a 9 8 a + 13 a + 5 a + a - -- - ----- - --- + ---- + ----- - 11 a z - 2 2 2 z z z z z 11 6 2 8 2 10 2 12 2 14 2 > 11 a z + 3 a z - 20 a z - 31 a z - 8 a z - 2 a z - 16 2 7 3 9 3 11 3 13 3 15 3 6 4 > 2 a z + 4 a z + 18 a z + 18 a z + a z - 3 a z - 4 a z + 8 4 10 4 12 4 14 4 16 4 7 5 9 5 > 23 a z + 38 a z + 9 a z - a z + a z - 7 a z - 11 a z - 11 5 13 5 15 5 6 6 8 6 10 6 12 6 > 8 a z - 2 a z + 2 a z + a z - 14 a z - 22 a z - 5 a z + 14 6 7 7 13 7 8 8 10 8 12 8 9 9 > 2 a z + 2 a z + 2 a z + 3 a z + 5 a z + 2 a z + a z + 11 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-7 -5 1 1 1 3 3 4 1 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 27 10 25 9 23 9 23 8 21 8 21 7 19 7 q t q t q t q t q t q t q t 3 4 4 3 3 5 2 2 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 19 6 17 6 17 5 15 5 15 4 13 4 13 3 11 3 q t q t q t q t q t q t q t q t 2 2 2 > ------ + ----- + ---- 11 2 9 2 7 q t q t q t

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