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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X22,10,19,9 X8493 X16,21,17,22 X18,11,5,12 X20,5,21,6 X10,17,11,18 X12,20,13,19 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

- q-8 + 5q-7 - 13q-6 + 20q-5 - 27q-4 + 32q-3 - 30q-2 + 28q-1 - 18 + 12q - 5q2 + q3

A2 (sl(3)) Invariant:

- q-24 + 2q-22 - q-20 - 5q-18 + 2q-16 - 8q-14 + q-12 + 3q-10 + 2q-8 + 14q-6 + 2q-4 + 12q-2 + 3 - q2 + 4q4 - 3q6 + q8

HOMFLY-PT Polynomial:

2z-2 + 2 + z2 + 2z4 + z6 - 5a2z-2 - 5a2 - 4a2z2 - 6a2z4 - 4a2z6 - a2z8 + 4a4z-2 + 4a4 + 4a4z2 + 5a4z4 + 2a4z6 - a6z-2 - a6 - a6z2 - a6z4

Kauffman Polynomial:

- a-2z4 + a-2z6 + 2a-1z3 - 8a-1z5 + 5a-1z7 - 2z-2 + 3 - 4z2 + 17z4 - 24z6 + 11z8 + 5az-1 - 5az + 5az3 + az5 - 16az7 + 11az9 - 5a2z-2 + 5a2 - 13a2z2 + 51a2z4 - 65a2z6 + 21a2z8 + 4a2z10 + 9a3z-1 - 12a3z + 14a3z3 + 3a3z5 - 38a3z7 + 24a3z9 - 4a4z-2 + 4a4 - 13a4z2 + 49a4z4 - 72a4z6 + 28a4z8 + 4a4z10 + 5a5z-1 - 10a5z + 18a5z3 - 23a5z5 - 4a5z7 + 13a5z9 - a6z-2 + a6 - 4a6z2 + 14a6z4 - 27a6z6 + 18a6z8 + a7z-1 - 3a7z + 7a7z3 - 16a7z5 + 13a7z7 - 2a8z4 + 5a8z6 + a9z5

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     4   j = 3                   8 1   j = 1                 10 4     j = -1               18 8       j = -3             16 14         j = -5           16 14           j = -7         11 16             j = -9       9 16               j = -11     4 11                 j = -13   1 9                   j = -15   4                     j = -17 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 487]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 487]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[22, 10, 19, 9], > X[8, 4, 9, 3], X[16, 21, 17, 22], X[18, 11, 5, 12], X[20, 5, 21, 6], > X[10, 17, 11, 18], X[12, 20, 13, 19], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {10, -8, 6, -4}, > {8, -1, 2, -5, 4, -9, 7, -10, 11, -2, 3, -6, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-8 5 13 20 27 32 30 28 2 3 -18 - q + -- - -- + -- - -- + -- - -- + -- + 12 q - 5 q + q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 -20 5 2 8 -12 3 2 14 2 12 2 3 - q + --- - q - --- + --- - --- + q + --- + -- + -- + -- + -- - q + 22 18 16 14 10 8 6 4 2 q q q q q q q q q 4 6 8 > 4 q - 3 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 487]][a, z]
Out[8]=	2 4 6 2 4 6 2 5 a 4 a a 2 2 2 4 2 6 2 2 - 5 a + 4 a - a + -- - ---- + ---- - -- + z - 4 a z + 4 a z - a z + 2 2 2 2 z z z z 4 2 4 4 4 6 4 6 2 6 4 6 2 8 > 2 z - 6 a z + 5 a z - a z + z - 4 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 487]][a, z]
Out[9]=	2 4 6 3 5 7 2 4 6 2 5 a 4 a a 5 a 9 a 5 a a 3 + 5 a + 4 a + a - -- - ---- - ---- - -- + --- + ---- + ---- + -- - 5 a z - 2 2 2 2 z z z z z z z z 3 3 5 7 2 2 2 4 2 6 2 2 z > 12 a z - 10 a z - 3 a z - 4 z - 13 a z - 13 a z - 4 a z + ---- + a 4 3 3 3 5 3 7 3 4 z 2 4 4 4 > 5 a z + 14 a z + 18 a z + 7 a z + 17 z - -- + 51 a z + 49 a z + 2 a 5 6 4 8 4 8 z 5 3 5 5 5 7 5 9 5 > 14 a z - 2 a z - ---- + a z + 3 a z - 23 a z - 16 a z + a z - a 6 7 6 z 2 6 4 6 6 6 8 6 5 z 7 > 24 z + -- - 65 a z - 72 a z - 27 a z + 5 a z + ---- - 16 a z - 2 a a 3 7 5 7 7 7 8 2 8 4 8 6 8 > 38 a z - 4 a z + 13 a z + 11 z + 21 a z + 28 a z + 18 a z + 9 3 9 5 9 2 10 4 10 > 11 a z + 24 a z + 13 a z + 4 a z + 4 a z
In[10]:=	Kh[L][q, t]
Out[10]=	14 18 1 4 1 9 4 11 9 16 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 11 16 16 14 16 8 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 10 q t + 4 q t + 8 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 4 q t + q t

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