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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-36 - 3q-34 - 2q-32 - q-30 - 2q-28 + 4q-26 + 5q-24 + 6q-22 + 8q-20 + 4q-18 + 6q-16 + q-14 + q-12 + 2q-10 - 2q-8 + q-6

HOMFLY-PT Polynomial:

a4z2 + a4z4 + 2a6z-2 + 7a6 + 8a6z2 + 3a6z4 - 5a8z-2 - 11a8 - 6a8z2 + 4a10z-2 + 4a10 - a12z-2

Kauffman Polynomial:

- a4z2 + a4z4 - 2a5z3 + 3a5z5 + 2a6z-2 - 7a6 + 11a6z2 - 11a6z4 + 6a6z6 - 5a7z-1 + 11a7z - 10a7z3 + 4a7z7 + 5a8z-2 - 14a8 + 23a8z2 - 24a8z4 + 10a8z6 + a8z8 - 9a9z-1 + 21a9z - 16a9z3 - 3a9z5 + 6a9z7 + 4a10z-2 - 10a10 + 14a10z2 - 16a10z4 + 6a10z6 + a10z8 - 5a11z-1 + 13a11z - 11a11z3 + a11z5 + 2a11z7 + a12z-2 - 2a12 + 3a12z2 - 4a12z4 + 2a12z6 - a13z-1 + 3a13z - 3a13z3 + a13z5

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -3                   1 j = -5                 3 1 j = -7               4     j = -9             3 3     j = -11           7 4       j = -13         4 6         j = -15       4 4           j = -17     1 4             j = -19   1 4               j = -21   1                 j = -23 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[9, Alternating, 43]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 43]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 7, 15, 8], X[8, 13, 5, 14], > X[16, 11, 17, 12], X[18, 15, 9, 16], X[12, 17, 13, 18], X[2, 5, 3, 6], > X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {8, -1, 3, -4}, {9, -2, 5, -7, 4, -3, 6, -5, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-11 2 5 8 8 10 7 7 3 -2 -q + --- - -- + -- - -- + -- - -- + -- - -- + q 10 9 8 7 6 5 4 3 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-36 3 2 -30 2 4 5 6 8 4 6 -14 -q - --- - --- - q - --- + --- + --- + --- + --- + --- + --- + q + 34 32 28 26 24 22 20 18 16 q q q q q q q q q -12 2 2 -6 > q + --- - -- + q 10 8 q q
In[8]:=	HOMFLYPT[Link[9, Alternating, 43]][a, z]
Out[8]=	6 8 10 12 6 8 10 2 a 5 a 4 a a 4 2 6 2 8 2 7 a - 11 a + 4 a + ---- - ---- + ----- - --- + a z + 8 a z - 6 a z + 2 2 2 2 z z z z 4 4 6 4 > a z + 3 a z
In[9]:=	Kauffman[Link[9, Alternating, 43]][a, z]
Out[9]=	6 8 10 12 7 9 6 8 10 12 2 a 5 a 4 a a 5 a 9 a -7 a - 14 a - 10 a - 2 a + ---- + ---- + ----- + --- - ---- - ---- - 2 2 2 2 z z z z z z 11 13 5 a a 7 9 11 13 4 2 6 2 > ----- - --- + 11 a z + 21 a z + 13 a z + 3 a z - a z + 11 a z + z z 8 2 10 2 12 2 5 3 7 3 9 3 > 23 a z + 14 a z + 3 a z - 2 a z - 10 a z - 16 a z - 11 3 13 3 4 4 6 4 8 4 10 4 12 4 > 11 a z - 3 a z + a z - 11 a z - 24 a z - 16 a z - 4 a z + 5 5 9 5 11 5 13 5 6 6 8 6 10 6 > 3 a z - 3 a z + a z + a z + 6 a z + 10 a z + 6 a z + 12 6 7 7 9 7 11 7 8 8 10 8 > 2 a z + 4 a z + 6 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-5 -3 1 1 1 4 1 4 4 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q t q t q t q t q t q t q t 4 4 6 7 4 3 3 4 3 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- 15 5 13 5 13 4 11 4 11 3 9 3 9 2 7 2 5 q t q t q t q t q t q t q t q t q t

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