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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-21/2 - 3q-19/2 + 8q-17/2 - 14q-15/2 + 18q-13/2 - 22q-11/2 + 21q-9/2 - 19q-7/2 + 14q-5/2 - 8q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-34 - 2q-32 + q-30 - 3q-26 + 5q-24 + q-20 + 5q-18 - 2q-16 + 4q-14 - 3q-12 + q-10 + 3q-8 - 4q-6 + 4q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - a3z-1 - 3a3z - a3z3 + a3z5 + 2a5z-1 + 7a5z + 7a5z3 + 3a5z5 - 3a7z-1 - 9a7z - 6a7z3 + 3a9z-1 + 4a9z - a11z-1

Kauffman Polynomial:

- az + 2az3 - az5 - a2z2 + 4a2z4 - 3a2z6 - a3z-1 + 5a3z - 8a3z3 + 9a3z5 - 6a3z7 + 6a4z2 - 13a4z4 + 13a4z6 - 8a4z8 - 2a5z-1 + 13a5z - 31a5z3 + 27a5z5 - 5a5z7 - 5a5z9 - 2a6 + 18a6z2 - 45a6z4 + 48a6z6 - 19a6z8 - a6z10 - 3a7z-1 + 15a7z - 36a7z3 + 31a7z5 + 2a7z7 - 9a7z9 + 12a8z2 - 34a8z4 + 42a8z6 - 16a8z8 - a8z10 - 3a9z-1 + 11a9z - 21a9z3 + 21a9z5 - 2a9z7 - 4a9z9 + 2a10 - 2a10z2 - 3a10z4 + 9a10z6 - 5a10z8 - a11z-1 + 3a11z - 6a11z3 + 7a11z5 - 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                   6 1   j = -4                 9 3     j = -6               10 5       j = -8             11 9         j = -10           11 10           j = -12         8 12             j = -14       6 10               j = -16     2 8                 j = -18   1 6                   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, 24]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 24]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 8, 15, 7], X[18, 11, 19, 12], > X[22, 19, 5, 20], X[20, 15, 21, 16], X[16, 21, 17, 22], X[12, 17, 13, 18], > X[8, 14, 9, 13], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 4, -8, 9, -3, 6, -7, 8, -4, > 5, -6, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 8 14 18 22 21 19 14 8 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 3 5 -20 5 2 4 3 -10 3 -1 - q - --- + q - --- + --- + q + --- - --- + --- - --- + q + -- - 32 26 24 18 16 14 12 8 q q q q q q q q 4 4 2 > -- + -- + q 6 4 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 24]][a, z]
Out[8]=	3 5 7 9 11 a 2 a 3 a 3 a a 3 5 7 9 -(--) + ---- - ---- + ---- - --- - a z - 3 a z + 7 a z - 9 a z + 4 a z - z z z z z 3 3 3 5 3 7 3 3 5 5 5 > a z - a z + 7 a z - 6 a z + a z + 3 a z
In[9]:=	Kauffman[Link[11, Alternating, 24]][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 + 5 a z + 13 a z + z z z z z 7 9 11 2 2 4 2 6 2 8 2 > 15 a z + 11 a z + 3 a z - a z + 6 a z + 18 a z + 12 a z - 10 2 12 2 3 3 3 5 3 7 3 9 3 > 2 a z - 3 a z + 2 a z - 8 a z - 31 a z - 36 a z - 21 a z - 11 3 2 4 4 4 6 4 8 4 10 4 12 4 > 6 a z + 4 a z - 13 a z - 45 a z - 34 a z - 3 a z + 3 a z - 5 3 5 5 5 7 5 9 5 11 5 2 6 > a z + 9 a z + 27 a z + 31 a z + 21 a z + 7 a z - 3 a z + 4 6 6 6 8 6 10 6 12 6 3 7 5 7 > 13 a z + 48 a z + 42 a z + 9 a z - a z - 6 a z - 5 a z + 7 7 9 7 11 7 4 8 6 8 8 8 10 8 > 2 a z - 2 a z - 3 a z - 8 a z - 19 a z - 16 a z - 5 a z - 5 9 7 9 9 9 6 10 8 10 > 5 a z - 9 a z - 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 6 1 2 1 6 2 8 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 10 8 12 11 10 11 9 10 5 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 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 9 t 2 2 > ---- + 2 t + -- + q t 4 2 q t q

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