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DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X20,9,21,10 X8,19,9,20 X4,21,1,22 X14,6,15,5 X12,4,13,3 X22,14,5,13 X16,11,17,12 X2,16,3,15 X10,17,11,18

Gauss Code:

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

Jones Polynomial:

q-15/2 - 4q-13/2 + 8q-11/2 - 13q-9/2 + 17q-7/2 - 19q-5/2 + 19q-3/2 - 17q-1/2 + 11q1/2 - 7q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 + 2q-20 - q-18 + q-16 + 3q-14 - 3q-12 + 3q-10 - 3q-8 - q-6 + q-4 - 2q-2 + 6 + 2q4 + 2q6 - q8 + q10

HOMFLY-PT Polynomial:

- 2a-1z-1 - 3a-1z - 3a-1z3 - a-1z5 + 4az-1 + 7az + 7az3 + 4az5 + az7 - 3a3z-1 - 4a3z + a3z3 + 3a3z5 + a3z7 + a5z-1 - 2a5z3 - a5z5

Kauffman Polynomial:

2a-3z3 - a-3z5 + 5a-2z4 - 3a-2z6 - 2a-1z-1 + 7a-1z - 14a-1z3 + 14a-1z5 - 6a-1z7 + 2 + z2 - 14z4 + 15z6 - 7z8 - 4az-1 + 15az - 25az3 + 9az5 + 4az7 - 5az9 + 3a2 - 4a2z2 - 12a2z4 + 17a2z6 - 6a2z8 - 2a2z10 - 3a3z-1 + 7a3z + 3a3z3 - 17a3z5 + 21a3z7 - 10a3z9 + 3a4 - 5a4z2 + 13a4z6 - 5a4z8 - 2a4z10 - a5z-1 - 2a5z + 6a5z3 - a5z5 + 7a5z7 - 5a5z9 + a6 - a6z2 - 5a6z4 + 13a6z6 - 6a6z8 - a7z - 6a7z3 + 10a7z5 - 4a7z7 - a8z2 + 2a8z4 - a8z6

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 = 8                       1 j = 6                     2   j = 4                   5 1   j = 2                 6 2     j = 0               11 5       j = -2             10 8         j = -4           9 9           j = -6         8 10             j = -8       5 9               j = -10     3 8                 j = -12   1 5                   j = -14   3                     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[11, Alternating, 102]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 102]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[20, 9, 21, 10], X[8, 19, 9, 20], > X[4, 21, 1, 22], X[14, 6, 15, 5], X[12, 4, 13, 3], X[22, 14, 5, 13], > X[16, 11, 17, 12], X[2, 16, 3, 15], X[10, 17, 11, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 7, -5}, {6, -1, 2, -4, 3, -11, 9, -7, 8, -6, 10, -9, 11, -2, > 4, -3, 5, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 4 8 13 17 19 19 17 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 11 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 7 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 -18 -16 3 3 3 3 -6 -4 2 4 6 - q + --- - q + q + --- - --- + --- - -- - q + q - -- + 2 q + 20 14 12 10 8 2 q q q q q q 6 8 10 > 2 q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 102]][a, z]
Out[8]=	3 5 3 -2 4 a 3 a a 3 z 3 3 z 3 3 3 --- + --- - ---- + -- - --- + 7 a z - 4 a z - ---- + 7 a z + a z - a z z z z a a 5 5 3 z 5 3 5 5 5 7 3 7 > 2 a z - -- + 4 a z + 3 a z - a z + a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 102]][a, z]
Out[9]=	3 5 2 4 6 2 4 a 3 a a 7 z 3 5 2 + 3 a + 3 a + a - --- - --- - ---- - -- + --- + 15 a z + 7 a z - 2 a z - a z z z z a 3 3 7 2 2 2 4 2 6 2 8 2 2 z 14 z 3 > a z + z - 4 a z - 5 a z - a z - a z + ---- - ----- - 25 a z + 3 a a 4 3 3 5 3 7 3 4 5 z 2 4 6 4 8 4 > 3 a z + 6 a z - 6 a z - 14 z + ---- - 12 a z - 5 a z + 2 a z - 2 a 5 5 6 z 14 z 5 3 5 5 5 7 5 6 3 z > -- + ----- + 9 a z - 17 a z - a z + 10 a z + 15 z - ---- + 3 a 2 a a 7 2 6 4 6 6 6 8 6 6 z 7 3 7 > 17 a z + 13 a z + 13 a z - a z - ---- + 4 a z + 21 a z + a 5 7 7 7 8 2 8 4 8 6 8 9 > 7 a z - 4 a z - 7 z - 6 a z - 5 a z - 6 a z - 5 a z - 3 9 5 9 2 10 4 10 > 10 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 1 3 1 5 3 8 5 9 11 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 8 10 9 9 10 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 5 t + 6 q t + 2 q t + 5 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 2 q t + q t

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