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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X7,16,8,17 X17,22,18,5 X14,9,15,10 X10,20,11,19 X21,9,22,8 X18,14,19,13 X20,15,21,16 X2536 X4,11,1,12

Gauss Code:

{{1, -10, 2, -11}, {10, -1, -3, 7, 5, -6, 11, -2, 8, -5, 9, 3, -4, -8, 6, -9, -7, 4}}

Jones Polynomial:

- q-17/2 + 2q-15/2 - 4q-13/2 + 5q-11/2 - 6q-9/2 + 6q-7/2 - 5q-5/2 + 4q-3/2 - 3q-1/2

A2 (sl(3)) Invariant:

q-28 + 2q-26 + q-22 + q-20 - 2q-18 - q-14 + q-10 + 2q-6 + q-2 + 3

HOMFLY-PT Polynomial:

- az-1 - 3az + a3z-1 + 2a3z + 2a3z3 + a5z-1 + 2a5z + 2a5z3 - 2a7z-1 - 3a7z + a9z-1

Kauffman Polynomial:

az-1 - 4az - a2 + 3a2z2 - 3a2z4 + a3z-1 - 3a3z - 2a3z3 + 5a3z5 - 2a3z7 - 2a4 + 12a4z2 - 17a4z4 + 12a4z6 - 3a4z8 - a5z-1 + 9a5z - 19a5z3 + 15a5z5 - a5z7 - a5z9 - 3a6 + 13a6z2 - 25a6z4 + 21a6z6 - 5a6z8 - 2a7z-1 + 13a7z - 25a7z3 + 15a7z5 - a7z9 - a8 + 4a8z2 - 11a8z4 + 9a8z6 - 2a8z8 - a9z-1 + 5a9z - 8a9z3 + 5a9z5 - a9z7

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = 0                 3 j = -2               2 1 j = -4             3 2   j = -6           3 2     j = -8         3 3       j = -10       3 4         j = -12     1 2           j = -14   1 3             j = -16   1               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[11, NonAlternating, 90]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 90]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[7, 16, 8, 17], X[17, 22, 18, 5], > X[14, 9, 15, 10], X[10, 20, 11, 19], X[21, 9, 22, 8], X[18, 14, 19, 13], > X[20, 15, 21, 16], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 7, 5, -6, 11, -2, 8, -5, 9, 3, -4, -8, > 6, -9, -7, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 2 4 5 6 6 5 4 3 -q + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 2 -22 -20 2 -14 -10 2 -2 3 + q + --- + q + q - --- - q + q + -- + q 26 18 6 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 90]][a, z]
Out[8]=	3 5 7 9 a a a 2 a a 3 5 7 3 3 -(-) + -- + -- - ---- + -- - 3 a z + 2 a z + 2 a z - 3 a z + 2 a z + z z z z z 5 3 > 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 90]][a, z]
Out[9]=	3 5 7 9 2 4 6 8 a a a 2 a a 3 5 -a - 2 a - 3 a - a + - + -- - -- - ---- - -- - 4 a z - 3 a z + 9 a z + z z z z z 7 9 2 2 4 2 6 2 8 2 3 3 > 13 a z + 5 a z + 3 a z + 12 a z + 13 a z + 4 a z - 2 a z - 5 3 7 3 9 3 2 4 4 4 6 4 8 4 > 19 a z - 25 a z - 8 a z - 3 a z - 17 a z - 25 a z - 11 a z + 3 5 5 5 7 5 9 5 4 6 6 6 8 6 > 5 a z + 15 a z + 15 a z + 5 a z + 12 a z + 21 a z + 9 a z - 3 7 5 7 9 7 4 8 6 8 8 8 5 9 7 9 > 2 a z - a z - a z - 3 a z - 5 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-2 1 1 1 3 1 2 3 3 + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 8 16 7 14 7 14 6 12 6 12 5 10 5 q t q t q t q t q t q t q t 4 3 3 3 2 3 2 2 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- 10 4 8 4 8 3 6 3 6 2 4 2 4 2 q t q t q t q t q t q t q t q t

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