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DrawMorseLink

PD Presentation:

X6172 X20,7,21,8 X4,21,1,22 X5,14,6,15 X10,4,11,3 X11,16,12,17 X15,12,16,13 X13,22,14,5 X18,9,19,10 X2,18,3,17 X8,19,9,20

Gauss Code:

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

Jones Polynomial:

2q-17/2 - 3q-15/2 + 5q-13/2 - 8q-11/2 + 8q-9/2 - 8q-7/2 + 6q-5/2 - 5q-3/2 + 2q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-28 - 3q-26 - q-24 - q-22 + 4q-18 + q-16 + 2q-14 + 3q-8 + 3q-4 + q-2 + q2

HOMFLY-PT Polynomial:

- az-1 - 2az - az3 + a3z-1 + a3z + 2a3z3 + a3z5 - a5z-1 + a5z + 2a5z3 + a5z5 + 2a7z-1 - a7z3 - a9z-1

Kauffman Polynomial:

az-1 - 3az + 3az3 - az5 - a2 + 4a2z4 - 2a2z6 + a3z-1 - 3a3z + 6a3z3 + a3z5 - 2a3z7 + 2a4z2 - 2a4z4 + 3a4z6 - 2a4z8 + a5z-1 - a5z - 3a5z3 + 2a5z5 - a5z9 - 4a6 + 14a6z2 - 22a6z4 + 13a6z6 - 4a6z8 + 2a7z-1 - 2a7z - 5a7z3 + a7z5 + a7z7 - a7z9 - 7a8 + 20a8z2 - 19a8z4 + 8a8z6 - 2a8z8 + a9z-1 - a9z + a9z3 + a9z5 - a9z7 - 3a10 + 8a10z2 - 3a10z4

Khovanov Homology:

trqj 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                 1   j = -2               4 1   j = -4             4 3     j = -6           4 2       j = -8         4 4         j = -10       4 4           j = -12     1 4             j = -14   2 4               j = -16   1                 j = -18 2

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, 83]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 83]]
Out[4]=	PD[X[6, 1, 7, 2], X[20, 7, 21, 8], X[4, 21, 1, 22], X[5, 14, 6, 15], > X[10, 4, 11, 3], X[11, 16, 12, 17], X[15, 12, 16, 13], X[13, 22, 14, 5], > X[18, 9, 19, 10], X[2, 18, 3, 17], X[8, 19, 9, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {-4, -1, 2, -11, 9, -5, -6, 7, -8, 4, -7, 6, 10, -9, > 11, -2, 3, 8}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 5 8 8 8 6 5 2 ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - Sqrt[q] 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 3 -24 -22 4 -16 2 3 3 -2 2 -q - --- - q - q + --- + q + --- + -- + -- + q + q 26 18 14 8 4 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 83]][a, z]
Out[8]=	3 5 7 9 a a a 2 a a 3 5 3 3 3 5 3 -(-) + -- - -- + ---- - -- - 2 a z + a z + a z - a z + 2 a z + 2 a z - z z z z z 7 3 3 5 5 5 > a z + a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 83]][a, z]
Out[9]=	3 5 7 9 2 6 8 10 a a a 2 a a 3 5 -a - 4 a - 7 a - 3 a + - + -- + -- + ---- + -- - 3 a z - 3 a z - a z - z z z z z 7 9 4 2 6 2 8 2 10 2 3 > 2 a z - a z + 2 a z + 14 a z + 20 a z + 8 a z + 3 a z + 3 3 5 3 7 3 9 3 2 4 4 4 6 4 > 6 a z - 3 a z - 5 a z + a z + 4 a z - 2 a z - 22 a z - 8 4 10 4 5 3 5 5 5 7 5 9 5 2 6 > 19 a z - 3 a z - a z + a z + 2 a z + a z + a z - 2 a z + 4 6 6 6 8 6 3 7 7 7 9 7 4 8 > 3 a z + 13 a z + 8 a z - 2 a z + a z - a z - 2 a z - 6 8 8 8 5 9 7 9 > 4 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 2 1 2 4 1 4 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 4 4 4 4 2 4 t 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + t + -- + 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

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