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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,8,13,7 X9,16,10,17 X20,17,5,18 X18,13,19,14 X14,19,15,20 X15,8,16,9 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

q-15/2 - 2q-13/2 + 4q-11/2 - 5q-9/2 + 5q-7/2 - 6q-5/2 + 4q-3/2 - 4q-1/2 + q1/2

A2 (sl(3)) Invariant:

- q-24 - q-22 - 2q-18 + 3q-10 + 2q-8 + 4q-6 + 2q-4 + q-2 + 2 - q2

HOMFLY-PT Polynomial:

- az-1 + az3 - 3a3z - 3a3z3 - a3z5 + 2a5z-1 + 4a5z + 2a5z3 - a7z-1 - a7z

Kauffman Polynomial:

- z2 + az-1 + az - 4az3 - a2 - 3a2z2 + 3a2z4 - 2a2z6 + 6a3z - 12a3z3 + 9a3z5 - 3a3z7 + 3a4 - 11a4z2 + 12a4z4 - 2a4z6 - a4z8 - 2a5z-1 + 7a5z - 14a5z3 + 16a5z5 - 5a5z7 + 5a6 - 14a6z2 + 13a6z4 - a6z6 - a6z8 - a7z-1 + 2a7z - 6a7z3 + 7a7z5 - 2a7z7 + 2a8 - 5a8z2 + 4a8z4 - a8z6

Khovanov Homology:

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

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