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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-17/2 - 3q-15/2 + 5q-13/2 - 8q-11/2 + 9q-9/2 - 10q-7/2 + 9q-5/2 - 7q-3/2 + 4q-1/2 - 3q1/2 + q3/2

A2 (sl(3)) Invariant:

- q-26 + q-22 + 3q-18 + q-14 + q-12 - q-10 + 3q-8 - q-6 + 2q-4 + q-2 + q2 - q4

HOMFLY-PT Polynomial:

az + 3az3 + az5 - a3z-1 - 7a3z - 9a3z3 - 5a3z5 - a3z7 + a5z-1 + 6a5z + 7a5z3 + 2a5z5 - 2a7z - a7z3

Kauffman Polynomial:

- z2 + 3z4 - z6 + 2az - 10az3 + 11az5 - 3az7 + a2z2 - 6a2z4 + 9a2z6 - 3a2z8 - a3z-1 + 10a3z - 26a3z3 + 22a3z5 - 4a3z7 - a3z9 + a4 + 4a4z2 - 16a4z4 + 17a4z6 - 6a4z8 - a5z-1 + 9a5z - 21a5z3 + 18a5z5 - 5a5z7 - a5z9 + a6z2 - 2a6z4 + 3a6z6 - 3a6z8 - a7z3 + 4a7z5 - 4a7z7 + 4a8z4 - 4a8z6 - a9z + 4a9z3 - 3a9z5 + a10z2 - a10z4

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                     1 j = 2                   2   j = 0                 2 1   j = -2               5 2     j = -4             5 3       j = -6           5 4         j = -8         4 5           j = -10       4 5             j = -12     2 5               j = -14   1 3                 j = -16   2                   j = -18 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, 93]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 93]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[20, 12, 9, 11], X[2, 9, 3, 10], > X[4, 20, 5, 19], X[14, 5, 15, 6], X[18, 13, 19, 14], X[16, 7, 17, 8], > X[6, 15, 7, 16], X[8, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -5, 6, -9, 8, -10}, > {4, -1, 3, -2, 7, -6, 9, -8, 10, -7, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 5 8 9 10 9 7 4 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 > 3 Sqrt[q] + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -22 3 -14 -12 -10 3 -6 2 -2 2 4 -q + q + --- + q + q - q + -- - q + -- + q + q - q 18 8 4 q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 93]][a, z]
Out[8]=	3 5 a a 3 5 7 3 3 3 5 3 -(--) + -- + a z - 7 a z + 6 a z - 2 a z + 3 a z - 9 a z + 7 a z - z z 7 3 5 3 5 5 5 3 7 > a z + a z - 5 a z + 2 a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 93]][a, z]
Out[9]=	3 5 4 a a 3 5 9 2 2 2 4 2 6 2 a - -- - -- + 2 a z + 10 a z + 9 a z - a z - z + a z + 4 a z + a z + z z 10 2 3 3 3 5 3 7 3 9 3 4 2 4 > a z - 10 a z - 26 a z - 21 a z - a z + 4 a z + 3 z - 6 a z - 4 4 6 4 8 4 10 4 5 3 5 5 5 > 16 a z - 2 a z + 4 a z - a z + 11 a z + 22 a z + 18 a z + 7 5 9 5 6 2 6 4 6 6 6 8 6 7 > 4 a z - 3 a z - z + 9 a z + 17 a z + 3 a z - 4 a z - 3 a z - 3 7 5 7 7 7 2 8 4 8 6 8 3 9 5 9 > 4 a z - 5 a z - 4 a z - 3 a z - 6 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 2 1 3 2 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 5 4 5 5 4 5 2 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 2 t + --- + t + 2 q t + 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q 4 3 > q t

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