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DrawMorseLink

PD Presentation:

X8192 X10,3,11,4 X14,17,15,18 X16,5,17,6 X4,15,5,16 X18,11,19,12 X20,13,7,14 X12,19,13,20 X2738 X6,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-25/2 + 3q-23/2 - 5q-21/2 + 8q-19/2 - 10q-17/2 + 10q-15/2 - 10q-13/2 + 7q-11/2 - 5q-9/2 + 2q-7/2 - q-5/2

A2 (sl(3)) Invariant:

q-38 - q-36 - q-34 - 3q-30 + q-28 + q-26 + 2q-24 + 4q-22 + 3q-18 + 2q-12 - q-10 + q-8

HOMFLY-PT Polynomial:

- 2a5z - 3a5z3 - a5z5 - 2a7z-1 - 5a7z - 6a7z3 - 2a7z5 + 3a9z-1 + 3a9z - a9z3 - a9z5 - a11z-1 + a11z + a11z3

Kauffman Polynomial:

- 2a5z + 3a5z3 - a5z5 - a6z2 + 4a6z4 - 2a6z6 - 2a7z-1 + 7a7z - 8a7z3 + 7a7z5 - 3a7z7 + 3a8 - 7a8z2 + 6a8z4 - 2a8z8 - 3a9z-1 + 12a9z - 16a9z3 + 10a9z5 - 3a9z7 - a9z9 + 3a10 - 5a10z2 - 5a10z4 + 9a10z6 - 5a10z8 - a11z-1 + 5a11z - 13a11z3 + 12a11z5 - 4a11z7 - a11z9 + a12 - 2a12z2 + 4a12z6 - 3a12z8 + 2a13z - 6a13z3 + 9a13z5 - 4a13z7 - 3a14z2 + 7a14z4 - 3a14z6 + 2a15z3 - a15z5

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 = -4                     1 j = -6                   2 1 j = -8                 3     j = -10               4 2     j = -12             6 3       j = -14           4 4         j = -16         6 6           j = -18       3 5             j = -20     2 5               j = -22   1 3                 j = -24   2                   j = -26 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]=	2
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 72]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 72]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[14, 17, 15, 18], X[16, 5, 17, 6], > X[4, 15, 5, 16], X[18, 11, 19, 12], X[20, 13, 7, 14], X[12, 19, 13, 20], > X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -5, 4, -10}, > {9, -1, 10, -2, 6, -8, 7, -3, 5, -4, 3, -6, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) 3 5 8 10 10 10 7 5 -q + ----- - ----- + ----- - ----- + ----- - ----- + ----- - ---- + 23/2 21/2 19/2 17/2 15/2 13/2 11/2 9/2 q q q q q q q q 2 -(5/2) > ---- - q 7/2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-38 -36 -34 3 -28 -26 2 4 3 2 -10 -8 q - q - q - --- + q + q + --- + --- + --- + --- - q + q 30 24 22 18 12 q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 72]][a, z]
Out[8]=	7 9 11 -2 a 3 a a 5 7 9 11 5 3 7 3 ----- + ---- - --- - 2 a z - 5 a z + 3 a z + a z - 3 a z - 6 a z - z z z 9 3 11 3 5 5 7 5 9 5 > a z + a z - a z - 2 a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 72]][a, z]
Out[9]=	7 9 11 8 10 12 2 a 3 a a 5 7 9 11 3 a + 3 a + a - ---- - ---- - --- - 2 a z + 7 a z + 12 a z + 5 a z + z z z 13 6 2 8 2 10 2 12 2 14 2 5 3 > 2 a z - a z - 7 a z - 5 a z - 2 a z - 3 a z + 3 a z - 7 3 9 3 11 3 13 3 15 3 6 4 8 4 > 8 a z - 16 a z - 13 a z - 6 a z + 2 a z + 4 a z + 6 a z - 10 4 14 4 5 5 7 5 9 5 11 5 13 5 > 5 a z + 7 a z - a z + 7 a z + 10 a z + 12 a z + 9 a z - 15 5 6 6 10 6 12 6 14 6 7 7 9 7 > a z - 2 a z + 9 a z + 4 a z - 3 a z - 3 a z - 3 a z - 11 7 13 7 8 8 10 8 12 8 9 9 11 9 > 4 a z - 4 a z - 2 a z - 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 2 1 3 2 5 3 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 26 10 24 9 22 9 22 8 20 8 20 7 18 7 q t q t q t q t q t q t q t 5 6 6 4 4 6 3 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 6 16 6 16 5 14 5 14 4 12 4 12 3 10 3 q t q t q t q t q t q t q t q t 2 3 2 > ------ + ----- + ---- 10 2 8 2 6 q t q t q t

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