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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X10,14,11,13 X22,17,5,18 X18,7,19,8 X14,20,15,19 X16,10,17,9 X20,16,21,15 X8,21,9,22 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 7q-11/2 - 14q-9/2 + 19q-7/2 - 24q-5/2 + 24q-3/2 - 22q-1/2 + 17q1/2 - 11q3/2 + 5q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-24 - q-22 + q-20 - 2q-18 + q-16 + 6q-14 - q-12 + 6q-10 + q-8 - q-6 + 3q-4 - 5q-2 + 5 - 3q2 - q4 + 3q6 - 3q8 + q10

HOMFLY-PT Polynomial:

- a-1z3 - a-1z5 + az-1 + 2az + 4az3 + 3az5 + az7 - 4a3z-1 - 9a3z - 8a3z3 - 3a3z5 + 4a5z-1 + 6a5z + 3a5z3 - a7z-1 - a7z

Kauffman Polynomial:

- a-3z5 + 4a-2z4 - 5a-2z6 - 4a-1z3 + 15a-1z5 - 11a-1z7 + 1 - z2 - 4z4 + 18z6 - 13z8 - az-1 + 3az - 12az3 + 20az5 - az7 - 8az9 + 4a2 - 8a2z2 - 13a2z4 + 37a2z6 - 18a2z8 - 2a2z10 - 4a3z-1 + 15a3z - 25a3z3 + 14a3z5 + 12a3z7 - 12a3z9 + 7a4 - 15a4z2 - a4z4 + 19a4z6 - 9a4z8 - 2a4z10 - 4a5z-1 + 16a5z - 25a5z3 + 18a5z5 - a5z7 - 4a5z9 + 4a6 - 11a6z2 + 7a6z4 + 4a6z6 - 4a6z8 - a7z-1 + 4a7z - 8a7z3 + 8a7z5 - 3a7z7 + a8 - 3a8z2 + 3a8z4 - 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                     4   j = 4                   7 1   j = 2                 10 4     j = 0               12 7       j = -2             13 11         j = -4           11 11           j = -6         8 13             j = -8       6 11               j = -10     2 9                 j = -12   1 5                   j = -14   2                     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, 87]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 87]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[10, 14, 11, 13], X[22, 17, 5, 18], > X[18, 7, 19, 8], X[14, 20, 15, 19], X[16, 10, 17, 9], X[20, 16, 21, 15], > X[8, 21, 9, 22], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 5, -9, 7, -3, 11, -2, 3, -6, 8, -7, 4, -5, > 6, -8, 9, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 7 14 19 24 24 22 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 17 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 > 11 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -20 2 -16 6 -12 6 -8 -6 3 5 5 - q - q + q - --- + q + --- - q + --- + q - q + -- - -- - 18 14 10 4 2 q q q q q 2 4 6 8 10 > 3 q - q + 3 q - 3 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 87]][a, z]
Out[8]=	3 5 7 3 a 4 a 4 a a 3 5 7 z 3 3 3 - - ---- + ---- - -- + 2 a z - 9 a z + 6 a z - a z - -- + 4 a z - 8 a z + z z z z a 5 5 3 z 5 3 5 7 > 3 a z - -- + 3 a z - 3 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 87]][a, z]
Out[9]=	3 5 7 2 4 6 8 a 4 a 4 a a 3 1 + 4 a + 7 a + 4 a + a - - - ---- - ---- - -- + 3 a z + 15 a z + z z z z 3 5 7 2 2 2 4 2 6 2 8 2 4 z > 16 a z + 4 a z - z - 8 a z - 15 a z - 11 a z - 3 a z - ---- - a 4 3 3 3 5 3 7 3 4 4 z 2 4 4 4 > 12 a z - 25 a z - 25 a z - 8 a z - 4 z + ---- - 13 a z - a z + 2 a 5 5 6 4 8 4 z 15 z 5 3 5 5 5 7 5 > 7 a z + 3 a z - -- + ----- + 20 a z + 14 a z + 18 a z + 8 a z + 3 a a 6 7 6 5 z 2 6 4 6 6 6 8 6 11 z 7 > 18 z - ---- + 37 a z + 19 a z + 4 a z - a z - ----- - a z + 2 a a 3 7 5 7 7 7 8 2 8 4 8 6 8 > 12 a z - a z - 3 a z - 13 z - 18 a z - 9 a z - 4 a z - 9 3 9 5 9 2 10 4 10 > 8 a z - 12 a z - 4 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	11 1 2 1 5 2 9 6 11 12 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 13 11 11 13 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 7 t + 10 q t + 4 q t + 7 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 + 4 q t + q t

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