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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 2q-9/2 - 5q-7/2 + 6q-5/2 - 9q-3/2 + 10q-1/2 - 9q1/2 + 8q3/2 - 6q5/2 + 3q7/2 - q9/2

A2 (sl(3)) Invariant:

q-18 + 2q-16 + q-14 + 3q-12 + q-10 + 2q-6 - 2q-4 - 1 - q2 + 2q4 - q6 + 2q8 - q12 + q14

HOMFLY-PT Polynomial:

- a-3z - a-3z3 - a-1z-1 - 2a-1z + a-1z3 + a-1z5 + 3az-1 + 8az + 7az3 + 2az5 - 4a3z-1 - 8a3z - 3a3z3 + 2a5z-1 + a5z

Kauffman Polynomial:

- a-5z3 - 3a-4z4 - 2a-3z + 5a-3z3 - 6a-3z5 + a-2 - 6a-2z2 + 12a-2z4 - 8a-2z6 - a-1z-1 + 2a-1z - 3a-1z3 + 11a-1z5 - 7a-1z7 + 3 - 9z2 + 13z4 + 2z6 - 4z8 - 3az-1 + 14az - 33az3 + 33az5 - 8az7 - az9 + 3a2 - 4a2z2 - 10a2z4 + 18a2z6 - 6a2z8 - 4a3z-1 + 17a3z - 33a3z3 + 21a3z5 - 2a3z7 - a3z9 + 2a4 - a4z2 - 8a4z4 + 8a4z6 - 2a4z8 - 2a5z-1 + 7a5z - 9a5z3 + 5a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 10                     1 j = 8                   2   j = 6                 4 1   j = 4               4 2     j = 2             5 4       j = 0           6 5         j = -2         3 4           j = -4       3 6             j = -6     2 3               j = -8   1 4                 j = -10   1                   j = -12 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, 33]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 33]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 8, 17, 7], X[18, 10, 19, 9], > X[20, 13, 5, 14], X[8, 18, 9, 17], X[14, 19, 15, 20], X[10, 16, 11, 15], > X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -6, 4, -8, 10, -2, 5, -7, 8, -3, 6, -4, > 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 5 6 9 10 3/2 5/2 -q + ---- - ---- + ---- - ---- + ------- - 9 Sqrt[q] + 8 q - 6 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 7/2 9/2 > 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 -14 3 -10 2 2 2 4 6 8 12 -1 + q + --- + q + --- + q + -- - -- - q + 2 q - q + 2 q - q + 16 12 6 4 q q q q 14 > q
In[8]:=	HOMFLYPT[Link[10, Alternating, 33]][a, z]
Out[8]=	3 5 3 3 1 3 a 4 a 2 a z 2 z 3 5 z z -(---) + --- - ---- + ---- - -- - --- + 8 a z - 8 a z + a z - -- + -- + a z z z z 3 a 3 a a a 5 3 3 3 z 5 > 7 a z - 3 a z + -- + 2 a z a
In[9]:=	Kauffman[Link[10, Alternating, 33]][a, z]
Out[9]=	3 5 -2 2 4 1 3 a 4 a 2 a 2 z 2 z 3 + a + 3 a + 2 a - --- - --- - ---- - ---- - --- + --- + 14 a z + a z z z z 3 a a 2 3 3 3 3 5 2 6 z 2 2 4 2 z 5 z 3 z > 17 a z + 7 a z - 9 z - ---- - 4 a z - a z - -- + ---- - ---- - 2 5 3 a a a a 4 4 3 3 3 5 3 4 3 z 12 z 2 4 4 4 > 33 a z - 33 a z - 9 a z + 13 z - ---- + ----- - 10 a z - 8 a z - 4 2 a a 5 5 6 6 z 11 z 5 3 5 5 5 6 8 z 2 6 > ---- + ----- + 33 a z + 21 a z + 5 a z + 2 z - ---- + 18 a z + 3 a 2 a a 7 4 6 7 z 7 3 7 5 7 8 2 8 4 8 > 8 a z - ---- - 8 a z - 2 a z - a z - 4 z - 6 a z - 2 a z - a 9 3 9 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 4 2 3 3 6 5 + 5 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 3 6 4 2 4 4 2 6 2 6 3 8 3 > ----- + - + ---- + 4 q t + 4 q t + 2 q t + 4 q t + q t + 2 q t + 2 2 t 2 q t q t 10 4 > q t

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