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DrawMorseLink

PD Presentation:

X6172 X12,7,13,8 X4,13,1,14 X9,18,10,19 X8493 X5,14,6,15 X15,22,16,5 X17,20,18,21 X21,16,22,17 X19,10,20,11 X2,12,3,11

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-28 - 3q-26 - q-22 - q-20 + 3q-18 + 3q-14 + q-12 + q-10 + 3q-8 - q-6 + 3q-4 + q-2 + q2

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

az-1 - 3az + 3az3 - az5 - a2 - a2z2 + 4a2z4 - 2a2z6 + a3z-1 - 5a3z + 7a3z3 - 2a3z7 + a4z4 + a4z6 - 2a4z8 + a5z-1 - 5a5z + 6a5z3 - a5z5 - a5z7 - a5z9 - 4a6 + 13a6z2 - 14a6z4 + 8a6z6 - 4a6z8 + 2a7z-1 - 4a7z + 5a7z3 - 3a7z5 - a7z9 - 7a8 + 18a8z2 - 14a8z4 + 5a8z6 - 2a8z8 + a9z-1 - a9z + 3a9z3 - a9z5 - a9z7 - 3a10 + 6a10z2 - 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             5 3     j = -6           5 2       j = -8         4 5         j = -10       5 5           j = -12     2 4             j = -14   2 5               j = -16   2                 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, 43]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 43]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 7, 13, 8], X[4, 13, 1, 14], X[9, 18, 10, 19], > X[8, 4, 9, 3], X[5, 14, 6, 15], X[15, 22, 16, 5], X[17, 20, 18, 21], > X[21, 16, 22, 17], X[19, 10, 20, 11], X[2, 12, 3, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-6, -1, 2, -5, -4, 10, 11, -2, 3, 6, -7, 9, -8, 4, > -10, 8, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	2 4 7 9 9 10 7 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 -22 -20 3 3 -12 -10 3 -6 3 -2 2 -q - --- - q - q + --- + --- + q + q + -- - q + -- + q + q 26 18 14 8 4 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 43]][a, z]
Out[8]=	3 5 7 9 a a a 2 a a 3 5 7 3 3 3 -(-) + -- - -- + ---- - -- - 2 a z + 2 a z - a z + a z - a z + 2 a z + z z z z z 5 3 7 3 3 5 5 5 > a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 43]][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 - 5 a z - 5 a z - z z z z z 7 9 2 2 6 2 8 2 10 2 3 3 3 > 4 a z - a z - a z + 13 a z + 18 a z + 6 a z + 3 a z + 7 a z + 5 3 7 3 9 3 2 4 4 4 6 4 8 4 > 6 a z + 5 a z + 3 a z + 4 a z + a z - 14 a z - 14 a z - 10 4 5 5 5 7 5 9 5 2 6 4 6 6 6 > 3 a z - a z - a z - 3 a z - a z - 2 a z + a z + 8 a z + 8 6 3 7 5 7 9 7 4 8 6 8 8 8 5 9 > 5 a z - 2 a z - a z - a z - 2 a z - 4 a z - 2 a z - a z - 7 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 2 2 2 5 2 4 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 5 4 5 5 2 5 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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