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DrawMorseLink

PD Presentation:

X8192 X12,4,13,3 X18,6,19,5 X22,16,7,15 X16,11,17,12 X20,17,21,18 X10,21,11,22 X14,10,15,9 X4,20,5,19 X2738 X6,14,1,13

Gauss Code:

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

Jones Polynomial:

q-9/2 - 5q-7/2 + 11q-5/2 - 18q-3/2 + 24q-1/2 - 28q1/2 + 27q3/2 - 24q5/2 + 17q7/2 - 10q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-14 + 2q-12 + q-10 - 4q-8 + 5q-6 - 2q-4 + 5 - 4q2 + 5q4 - 4q6 + 2q8 + 3q10 - 4q12 + 5q14 - q18 + q20

HOMFLY-PT Polynomial:

- a-5z-1 - a-5z - a-5z3 + 2a-3z-1 + 5a-3z + 4a-3z3 + 2a-3z5 - 2a-1z-1 - 5a-1z - 6a-1z3 - 3a-1z5 - a-1z7 + az-1 + 2az + 3az3 + 2az5 - a3z3

Kauffman Polynomial:

a-7z3 - a-7z5 - a-6z2 + 4a-6z4 - 4a-6z6 - a-5z-1 + 4a-5z - 10a-5z3 + 13a-5z5 - 9a-5z7 + 3a-4z2 - 11a-4z4 + 17a-4z6 - 12a-4z8 - 2a-3z-1 + 12a-3z - 27a-3z3 + 24a-3z5 - 9a-3z9 - a-2 + 10a-2z2 - 38a-2z4 + 53a-2z6 - 21a-2z8 - 3a-2z10 - 2a-1z-1 + 12a-1z - 25a-1z3 + 13a-1z5 + 23a-1z7 - 18a-1z9 + 7z2 - 36z4 + 54z6 - 19z8 - 3z10 - az-1 + 4az - 13az3 + 12az5 + 9az7 - 9az9 + a2z2 - 12a2z4 + 21a2z6 - 10a2z8 - 4a3z3 + 9a3z5 - 5a3z7 + a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14                       1 j = 12                     3   j = 10                   7 1   j = 8                 10 3     j = 6               14 7       j = 4             14 11         j = 2           14 13           j = 0         11 15             j = -2       7 13               j = -4     4 11                 j = -6   1 7                   j = -8   4                     j = -10 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, 232]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 232]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 4, 13, 3], X[18, 6, 19, 5], X[22, 16, 7, 15], > X[16, 11, 17, 12], X[20, 17, 21, 18], X[10, 21, 11, 22], X[14, 10, 15, 9], > X[4, 20, 5, 19], X[2, 7, 3, 8], X[6, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -9, 3, -11}, > {10, -1, 8, -7, 5, -2, 11, -8, 4, -5, 6, -3, 9, -6, 7, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 5 11 18 24 3/2 5/2 q - ---- + ---- - ---- + ------- - 28 Sqrt[q] + 27 q - 24 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 17 q - 10 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 2 -10 4 5 2 2 4 6 8 10 5 - q + --- + q - -- + -- - -- - 4 q + 5 q - 4 q + 2 q + 3 q - 12 8 6 4 q q q q 12 14 18 20 > 4 q + 5 q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 232]][a, z]
Out[8]=	3 3 3 1 2 2 a z 5 z 5 z z 4 z 6 z 3 -(----) + ---- - --- + - - -- + --- - --- + 2 a z - -- + ---- - ---- + 3 a z - 5 3 a z z 5 3 a 5 3 a a z a z a a a a 5 5 7 3 3 2 z 3 z 5 z > a z + ---- - ---- + 2 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 232]][a, z]
Out[9]=	2 2 -2 1 2 2 a 4 z 12 z 12 z 2 z 3 z -a - ---- - ---- - --- - - + --- + ---- + ---- + 4 a z + 7 z - -- + ---- + 5 3 a z z 5 3 a 6 4 a z a z a a a a 2 3 3 3 3 10 z 2 2 z 10 z 27 z 25 z 3 3 3 4 > ----- + a z + -- - ----- - ----- - ----- - 13 a z - 4 a z - 36 z + 2 7 5 3 a a a a a 4 4 4 5 5 5 5 4 z 11 z 38 z 2 4 4 4 z 13 z 24 z 13 z > ---- - ----- - ----- - 12 a z + a z - -- + ----- + ----- + ----- + 6 4 2 7 5 3 a a a a a a a 6 6 6 5 3 5 6 4 z 17 z 53 z 2 6 4 6 > 12 a z + 9 a z + 54 z - ---- + ----- + ----- + 21 a z - a z - 6 4 2 a a a 7 7 8 8 9 9 z 23 z 7 3 7 8 12 z 21 z 2 8 9 z > ---- + ----- + 9 a z - 5 a z - 19 z - ----- - ----- - 10 a z - ---- - 5 a 4 2 3 a a a a 9 10 18 z 9 10 3 z > ----- - 9 a z - 3 z - ----- a 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 4 1 7 4 11 7 11 15 + 14 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + -- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t q t q t q t q t q t q t q t 13 2 4 4 2 6 2 6 3 8 3 > ---- + 13 q t + 14 q t + 11 q t + 14 q t + 7 q t + 10 q t + 2 q t 8 4 10 4 10 5 12 5 14 6 > 3 q t + 7 q t + q t + 3 q t + q t

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