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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 8q-11/2 - 15q-9/2 + 21q-7/2 - 26q-5/2 + 26q-3/2 - 24q-1/2 + 18q1/2 - 12q3/2 + 5q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-24 - q-22 + q-20 - 3q-18 + 5q-14 - 3q-12 + 5q-10 - q-6 + 4q-4 - 4q-2 + 7 - 2q2 + 4q6 - 3q8 + q10

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-3z5 + 3a-2z4 - 5a-2z6 - a-1z-1 + 2a-1z - 6a-1z3 + 16a-1z5 - 12a-1z7 + 3z2 - 9z4 + 22z6 - 15z8 - 2az-1 + 9az - 24az3 + 32az5 - 4az7 - 9az9 - 2a2 + 11a2z2 - 39a2z4 + 58a2z6 - 25a2z8 - 2a2z10 - 3a3z-1 + 16a3z - 33a3z3 + 23a3z5 + 12a3z7 - 14a3z9 + 8a4z2 - 32a4z4 + 40a4z6 - 15a4z8 - 2a4z10 - 3a5z-1 + 12a5z - 21a5z3 + 15a5z5 + a5z7 - 5a5z9 + 2a6 - 3a6z2 - 2a6z4 + 8a6z6 - 5a6z8 - a7z-1 + 3a7z - 6a7z3 + 7a7z5 - 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                   8 1   j = 2                 10 4     j = 0               14 8       j = -2             14 12         j = -4           12 12           j = -6         9 14             j = -8       6 12               j = -10     2 9                 j = -12   1 6                   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, 41]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 41]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[22, 15, 5, 16], X[16, 7, 17, 8], > X[20, 18, 21, 17], X[14, 10, 15, 9], X[12, 19, 13, 20], X[18, 13, 19, 14], > X[8, 21, 9, 22], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -9, 6, -2, 11, -7, 8, -6, 3, -4, 5, -8, > 7, -5, 9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 8 15 21 26 26 24 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 18 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 > 12 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -20 3 5 3 5 -6 4 4 2 6 7 - q - q + q - --- + --- - --- + --- - q + -- - -- - 2 q + 4 q - 18 14 12 10 4 2 q q q q q q 8 10 > 3 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 41]][a, z]
Out[8]=	3 5 7 3 1 2 a 3 a 3 a a z 3 5 7 z -(---) + --- - ---- + ---- - -- - - + 4 a z - 7 a z + 5 a z - a z - -- + a z z z z z a a 5 3 3 3 5 3 z 5 3 5 7 > 5 a z - 7 a z + 3 a z - -- + 3 a z - 3 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 41]][a, z]
Out[9]=	3 5 7 2 6 8 1 2 a 3 a 3 a a 2 z 3 -2 a + 2 a + a - --- - --- - ---- - ---- - -- + --- + 9 a z + 16 a z + a z z z z z a 3 5 7 2 2 2 4 2 6 2 8 2 6 z > 12 a z + 3 a z + 3 z + 11 a z + 8 a z - 3 a z - 3 a z - ---- - a 4 3 3 3 5 3 7 3 4 3 z 2 4 > 24 a z - 33 a z - 21 a z - 6 a z - 9 z + ---- - 39 a z - 2 a 5 5 4 4 6 4 8 4 z 16 z 5 3 5 5 5 > 32 a z - 2 a z + 3 a z - -- + ----- + 32 a z + 23 a z + 15 a z + 3 a a 6 7 7 5 6 5 z 2 6 4 6 6 6 8 6 12 z > 7 a z + 22 z - ---- + 58 a z + 40 a z + 8 a z - a z - ----- - 2 a a 7 3 7 5 7 7 7 8 2 8 4 8 > 4 a z + 12 a z + a z - 3 a z - 15 z - 25 a z - 15 a z - 6 8 9 3 9 5 9 2 10 4 10 > 5 a z - 9 a z - 14 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	12 1 2 1 6 2 9 6 12 14 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 9 14 12 12 14 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 8 t + 10 q t + 4 q t + 8 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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