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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 4q-13/2 + 9q-11/2 - 18q-9/2 + 24q-7/2 - 29q-5/2 + 30q-3/2 - 26q-1/2 + 20q1/2 - 13q3/2 + 5q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-24 + 3q-20 - 2q-18 + 2q-16 + 7q-14 - 3q-12 + 5q-10 - 2q-8 - 3q-6 + 2q-4 - 6q-2 + 7 - 3q2 + 5q6 - 3q8 + q10

HOMFLY-PT Polynomial:

- a-1z-1 - 2a-1z - a-1z3 - a-1z5 + 3az-1 + 5az + 6az3 + 3az5 + az7 - 4a3z-1 - 7a3z - 6a3z3 - 3a3z5 + 2a5z-1 + 3a5z + 3a5z3 - a7z

Kauffman Polynomial:

- a-3z5 + 2a-2z4 - 5a-2z6 - a-1z-1 + 4a-1z - 7a-1z3 + 16a-1z5 - 13a-1z7 - 1 + 4z2 - 14z4 + 28z6 - 18z8 - 3az-1 + 15az - 29az3 + 31az5 + az7 - 12az9 - 3a2 + 11a2z2 - 45a2z4 + 70a2z6 - 29a2z8 - 3a2z10 - 4a3z-1 + 24a3z - 45a3z3 + 27a3z5 + 20a3z7 - 19a3z9 - 2a4 + 9a4z2 - 37a4z4 + 50a4z6 - 18a4z8 - 3a4z10 - 2a5z-1 + 16a5z - 31a5z3 + 22a5z5 + 2a5z7 - 7a5z9 - a6 + a6z2 - 6a6z4 + 12a6z6 - 7a6z8 + 3a7z - 8a7z3 + 9a7z5 - 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                     4   j = 4                   9 1   j = 2                 11 4     j = 0               15 9       j = -2             16 12         j = -4           13 14           j = -6         11 16             j = -8       7 13               j = -10     3 12                 j = -12   1 6                   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, 67]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 67]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 8, 17, 7], X[22, 18, 5, 17], > X[14, 9, 15, 10], X[10, 20, 11, 19], X[8, 21, 9, 22], X[18, 14, 19, 13], > X[20, 15, 21, 16], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -7, 5, -6, 11, -2, 8, -5, 9, -3, 4, -8, > 6, -9, 7, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 4 9 18 24 29 30 26 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 20 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 > 13 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 3 2 2 7 3 5 2 3 2 6 2 7 - q + --- - --- + --- + --- - --- + --- - -- - -- + -- - -- - 3 q + 20 18 16 14 12 10 8 6 4 2 q q q q q q q q q q 6 8 10 > 5 q - 3 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 67]][a, z]
Out[8]=	3 5 3 1 3 a 4 a 2 a 2 z 3 5 7 z -(---) + --- - ---- + ---- - --- + 5 a z - 7 a z + 3 a z - a z - -- + a z z z z a a 5 3 3 3 5 3 z 5 3 5 7 > 6 a z - 6 a z + 3 a z - -- + 3 a z - 3 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 67]][a, z]
Out[9]=	3 5 2 4 6 1 3 a 4 a 2 a 4 z 3 -1 - 3 a - 2 a - a - --- - --- - ---- - ---- + --- + 15 a z + 24 a z + a z z z z a 3 5 7 2 2 2 4 2 6 2 8 2 7 z > 16 a z + 3 a z + 4 z + 11 a z + 9 a z + a z - a z - ---- - a 4 3 3 3 5 3 7 3 4 2 z 2 4 > 29 a z - 45 a z - 31 a z - 8 a z - 14 z + ---- - 45 a z - 2 a 5 5 4 4 6 4 8 4 z 16 z 5 3 5 5 5 > 37 a z - 6 a z + 2 a z - -- + ----- + 31 a z + 27 a z + 22 a z + 3 a a 6 7 7 5 6 5 z 2 6 4 6 6 6 8 6 13 z > 9 a z + 28 z - ---- + 70 a z + 50 a z + 12 a z - a z - ----- + 2 a a 7 3 7 5 7 7 7 8 2 8 4 8 > a z + 20 a z + 2 a z - 4 a z - 18 z - 29 a z - 18 a z - 6 8 9 3 9 5 9 2 10 4 10 > 7 a z - 12 a z - 19 a z - 7 a z - 3 a z - 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	12 1 3 1 6 3 12 7 13 15 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 11 16 13 14 16 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 9 t + 11 q t + 4 q t + 9 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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