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DrawMorseLink

PD Presentation:

X12,1,13,2 X20,5,11,6 X16,3,17,4 X10,15,1,16 X8,19,9,20 X18,9,19,10 X4,17,5,18 X2,11,3,12 X6,13,7,14 X14,7,15,8

Gauss Code:

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

Jones Polynomial:

- q-25/2 + 3q-23/2 - 8q-21/2 + 11q-19/2 - 14q-17/2 + 16q-15/2 - 14q-13/2 + 11q-11/2 - 8q-9/2 + 3q-7/2 - q-5/2

A2 (sl(3)) Invariant:

q-38 - q-36 + q-34 + 5q-32 + 3q-28 - q-26 - 3q-24 + q-22 - 3q-20 + 4q-18 + 3q-12 - 2q-10 + q-8

HOMFLY-PT Polynomial:

- a5z - 2a5z3 - a5z5 - 7a7z - 9a7z3 - 3a7z5 - a9z-1 + 2a9z - a9z5 + a11z-1 + a11z + a11z3

Kauffman Polynomial:

- a5z + 2a5z3 - a5z5 - a6z2 + 4a6z4 - 3a6z6 + 7a7z - 12a7z3 + 12a7z5 - 6a7z7 - 5a8z2 + 6a8z4 + 3a8z6 - 5a8z8 + a9z-1 + a9z - 9a9z3 + 18a9z5 - 8a9z7 - 2a9z9 - a10 - 8a10z2 + 4a10z4 + 12a10z6 - 10a10z8 + a11z-1 + a11z - 9a11z3 + 18a11z5 - 8a11z7 - 2a11z9 - 5a12z2 + 6a12z4 + 3a12z6 - 5a12z8 + 7a13z - 12a13z3 + 12a13z5 - 6a13z7 - a14z2 + 4a14z4 - 3a14z6 - a15z + 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                   3 1 j = -8                 5     j = -10               6 3     j = -12             8 5       j = -14           8 6         j = -16         6 8           j = -18       5 8             j = -20     3 6               j = -22     5                 j = -24 1 3                   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, 121]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 121]]
Out[4]=	PD[X[12, 1, 13, 2], X[20, 5, 11, 6], X[16, 3, 17, 4], X[10, 15, 1, 16], > X[8, 19, 9, 20], X[18, 9, 19, 10], X[4, 17, 5, 18], X[2, 11, 3, 12], > X[6, 13, 7, 14], X[14, 7, 15, 8]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 3, -7, 2, -9, 10, -5, 6, -4}, > {8, -1, 9, -10, 4, -3, 7, -6, 5, -2}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) 3 8 11 14 16 14 11 8 -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 3 -(5/2) > ---- - q 7/2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-38 -36 -34 5 3 -26 3 -22 3 4 3 2 -8 q - q + q + --- + --- - q - --- + q - --- + --- + --- - --- + q 32 28 24 20 18 12 10 q q q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 121]][a, z]
Out[8]=	9 11 a a 5 7 9 11 5 3 7 3 11 3 -(--) + --- - a z - 7 a z + 2 a z + a z - 2 a z - 9 a z + a z - z z 5 5 7 5 9 5 > a z - 3 a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 121]][a, z]
Out[9]=	9 11 10 a a 5 7 9 11 13 15 6 2 -a + -- + --- - a z + 7 a z + a z + a z + 7 a z - a z - a z - z z 8 2 10 2 12 2 14 2 5 3 7 3 9 3 > 5 a z - 8 a z - 5 a z - a z + 2 a z - 12 a z - 9 a z - 11 3 13 3 15 3 6 4 8 4 10 4 12 4 > 9 a z - 12 a z + 2 a z + 4 a z + 6 a z + 4 a z + 6 a z + 14 4 5 5 7 5 9 5 11 5 13 5 15 5 > 4 a z - a z + 12 a z + 18 a z + 18 a z + 12 a z - a z - 6 6 8 6 10 6 12 6 14 6 7 7 9 7 > 3 a z + 3 a z + 12 a z + 3 a z - 3 a z - 6 a z - 8 a z - 11 7 13 7 8 8 10 8 12 8 9 9 11 9 > 8 a z - 6 a z - 5 a z - 10 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 3 5 3 6 5 q + q + ------- + ------- + ------ + ------ + ------ + ------ + ------ + 26 10 24 10 24 9 22 8 20 8 20 7 18 7 q t q t q t q t q t q t q t 8 6 8 8 6 8 5 6 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 3 5 3 > ------ + ----- + ---- 10 2 8 2 6 q t q t q t

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