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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-21/2 - 3q-19/2 + 7q-17/2 - 11q-15/2 + 14q-13/2 - 18q-11/2 + 17q-9/2 - 15q-7/2 + 11q-5/2 - 7q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-34 - 2q-32 + q-30 - 2q-26 + 4q-24 + q-20 + 4q-18 - q-16 + 3q-14 - 2q-12 + q-10 + 3q-8 - 3q-6 + 3q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - a3z-1 - 2a3z + a3z5 + 2a5z-1 + 5a5z + 4a5z3 + 2a5z5 - 3a7z-1 - 8a7z - 5a7z3 + 3a9z-1 + 4a9z - a11z-1

Kauffman Polynomial:

- az + 2az3 - az5 - a2z2 + 5a2z4 - 3a2z6 - a3z-1 + 4a3z - 4a3z3 + 8a3z5 - 5a3z7 + 5a4z2 - 10a4z4 + 11a4z6 - 6a4z8 - 2a5z-1 + 12a5z - 22a5z3 + 12a5z5 + a5z7 - 4a5z9 - 2a6 + 16a6z2 - 40a6z4 + 34a6z6 - 11a6z8 - a6z10 - 3a7z-1 + 16a7z - 32a7z3 + 14a7z5 + 8a7z7 - 7a7z9 + 9a8z2 - 26a8z4 + 28a8z6 - 9a8z8 - a8z10 - 3a9z-1 + 12a9z - 22a9z3 + 19a9z5 - a9z7 - 3a9z9 + 2a10 - 4a10z2 + 2a10z4 + 7a10z6 - 4a10z8 - a11z-1 + 3a11z - 6a11z3 + 8a11z5 - 3a11z7 + a12 - 3a12z2 + 3a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                       1 j = 0                     2   j = -2                   5 1   j = -4                 7 3     j = -6               8 4       j = -8             9 7         j = -10           9 8           j = -12         6 10             j = -14       5 8               j = -16     2 6                 j = -18   1 5                   j = -20   2                     j = -22 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, 89]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 89]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 10, 17, 9], X[22, 13, 5, 14], > X[14, 21, 15, 22], X[10, 16, 11, 15], X[20, 17, 21, 18], X[18, 7, 19, 8], > X[8, 19, 9, 20], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 8, -9, 3, -6, 11, -2, 4, -5, 6, -3, 7, -8, > 9, -7, 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 7 11 14 18 17 15 11 7 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q 3 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 2 -30 2 4 -20 4 -16 3 2 -10 -1 - q - --- + q - --- + --- + q + --- - q + --- - --- + q + 32 26 24 18 14 12 q q q q q q 3 3 3 2 > -- - -- + -- + q 8 6 4 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 89]][a, z]
Out[8]=	3 5 7 9 11 a 2 a 3 a 3 a a 3 5 7 9 -(--) + ---- - ---- + ---- - --- - a z - 2 a z + 5 a z - 8 a z + 4 a z - z z z z z 3 5 3 7 3 3 5 5 5 > a z + 4 a z - 5 a z + a z + 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 89]][a, z]
Out[9]=	3 5 7 9 11 6 10 12 a 2 a 3 a 3 a a 3 5 -2 a + 2 a + a - -- - ---- - ---- - ---- - --- - a z + 4 a z + 12 a z + z z z z z 7 9 11 2 2 4 2 6 2 8 2 > 16 a z + 12 a z + 3 a z - a z + 5 a z + 16 a z + 9 a z - 10 2 12 2 3 3 3 5 3 7 3 9 3 > 4 a z - 3 a z + 2 a z - 4 a z - 22 a z - 32 a z - 22 a z - 11 3 2 4 4 4 6 4 8 4 10 4 12 4 > 6 a z + 5 a z - 10 a z - 40 a z - 26 a z + 2 a z + 3 a z - 5 3 5 5 5 7 5 9 5 11 5 2 6 > a z + 8 a z + 12 a z + 14 a z + 19 a z + 8 a z - 3 a z + 4 6 6 6 8 6 10 6 12 6 3 7 5 7 > 11 a z + 34 a z + 28 a z + 7 a z - a z - 5 a z + a z + 7 7 9 7 11 7 4 8 6 8 8 8 10 8 > 8 a z - a z - 3 a z - 6 a z - 11 a z - 9 a z - 4 a z - 5 9 7 9 9 9 6 10 8 10 > 4 a z - 7 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 2 1 5 2 6 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q q t q t q t q t q t q t q t 8 6 10 9 8 9 7 8 4 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 7 t 2 2 > ---- + 2 t + -- + q t 4 2 q t q

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