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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-19/2 + q-17/2 - 5q-15/2 + 5q-13/2 - 11q-11/2 + 10q-9/2 - 11q-7/2 + 8q-5/2 - 7q-3/2 + 4q-1/2 - q1/2

A2 (sl(3)) Invariant:

q-32 + 4q-30 + 6q-28 + 8q-26 + 13q-24 + 12q-22 + 10q-20 + 12q-18 + 6q-16 + 5q-14 + 2q-12 + 3q-8 - 2q-6 + 2q-4 - 2 + q2

HOMFLY-PT Polynomial:

- az3 + a3z3 + a3z5 - a5z-3 - 4a5z-1 - 6a5z - 4a5z3 + 3a7z-3 + 8a7z-1 + 6a7z - 3a9z-3 - 4a9z-1 + a11z-3

Kauffman Polynomial:

az3 - az5 + 7a2z4 - 4a2z6 - 5a3z3 + 13a3z5 - 6a3z7 - 2a4z4 + 6a4z6 - 4a4z8 - a5z-3 + 4a5z-1 - 6a5z - a5z3 + 10a5z5 - 5a5z7 - a5z9 + 3a6z-2 - 8a6 + 6a6z2 - 9a6z4 + 9a6z6 - 5a6z8 - 3a7z-3 + 9a7z-1 - 14a7z + 13a7z3 - 6a7z5 - a7z9 + 6a8z-2 - 15a8 + 12a8z2 - 2a8z6 - a8z8 - 3a9z-3 + 9a9z-1 - 14a9z + 12a9z3 - 3a9z5 - a9z7 + 3a10z-2 - 8a10 + 6a10z2 - a10z6 - a11z-3 + 4a11z-1 - 6a11z + 4a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                     1 j = 0                   3   j = -2                 4 1   j = -4               5 4     j = -6             6 3       j = -8           4 5         j = -10         7 6           j = -12       4 10             j = -14     1 1               j = -16     4                 j = -18 1 1                   j = -20 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]=	4
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 165]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 165]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[18, 11, 19, 12], X[10, 3, 11, 4], > X[4, 9, 1, 10], X[14, 7, 15, 8], X[8, 13, 5, 14], X[20, 16, 13, 15], > X[16, 20, 17, 19], X[12, 17, 9, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -10}, > {7, -6, 8, -9, 10, -3, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) -(17/2) 5 5 11 10 11 8 7 -q + q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q 4 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 4 6 8 13 12 10 12 6 5 2 3 -2 + q + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + -- - 30 28 26 24 22 20 18 16 14 12 8 q q q q q q q q q q q 2 2 2 > -- + -- + q 6 4 q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 165]][a, z]
Out[8]=	5 7 9 11 5 7 9 a 3 a 3 a a 4 a 8 a 4 a 5 7 3 -(--) + ---- - ---- + --- - ---- + ---- - ---- - 6 a z + 6 a z - a z + 3 3 3 3 z z z z z z z 3 3 5 3 3 5 > a z - 4 a z + a z
In[9]:=	Kauffman[Link[10, Alternating, 165]][a, z]
Out[9]=	5 7 9 11 6 8 10 5 6 8 10 a 3 a 3 a a 3 a 6 a 3 a 4 a -8 a - 15 a - 8 a - -- - ---- - ---- - --- + ---- + ---- + ----- + ---- + 3 3 3 3 2 2 2 z z z z z z z z 7 9 11 9 a 9 a 4 a 5 7 9 11 6 2 > ---- + ---- + ----- - 6 a z - 14 a z - 14 a z - 6 a z + 6 a z + z z z 8 2 10 2 3 3 3 5 3 7 3 9 3 > 12 a z + 6 a z + a z - 5 a z - a z + 13 a z + 12 a z + 11 3 2 4 4 4 6 4 5 3 5 5 5 > 4 a z + 7 a z - 2 a z - 9 a z - a z + 13 a z + 10 a z - 7 5 9 5 11 5 2 6 4 6 6 6 8 6 > 6 a z - 3 a z - a z - 4 a z + 6 a z + 9 a z - 2 a z - 10 6 3 7 5 7 9 7 4 8 6 8 8 8 5 9 > a z - 6 a z - 5 a z - a z - 4 a z - 5 a z - a z - a z - 7 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 4 1 1 1 4 1 1 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 8 18 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 10 7 6 4 5 6 3 5 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 3 t + -- + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q t q t q 2 2 > q t

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