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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-21/2 - 3q-19/2 + 7q-17/2 - 14q-15/2 + 17q-13/2 - 22q-11/2 + 19q-9/2 - 20q-7/2 + 13q-5/2 - 8q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-34 - 2q-32 + q-30 + q-28 + 10q-24 + 7q-22 + 9q-20 + 14q-18 + 8q-16 + 14q-14 + 6q-12 + 7q-10 + 6q-8 - 3q-6 + 4q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - a3z-3 - 4a3z-1 - 4a3z - a3z3 + a3z5 + 3a5z-3 + 10a5z-1 + 12a5z + 8a5z3 + 3a5z5 - 3a7z-3 - 9a7z-1 - 11a7z - 6a7z3 + a9z-3 + 4a9z-1 + 4a9z - a11z-1

Kauffman Polynomial:

- az + 2az3 - az5 - a2z2 + 4a2z4 - 3a2z6 + a3z-3 - 5a3z-1 + 11a3z - 12a3z3 + 10a3z5 - 6a3z7 - 3a4z-2 + 9a4 - 12a4z2 + 2a4z4 + 7a4z6 - 7a4z8 + 3a5z-3 - 14a5z-1 + 40a5z - 59a5z3 + 44a5z5 - 11a5z7 - 4a5z9 - 6a6z-2 + 21a6 - 27a6z2 + a6z4 + 26a6z6 - 15a6z8 - a6z10 + 3a7z-3 - 18a7z-1 + 52a7z - 79a7z3 + 59a7z5 - 9a7z7 - 7a7z9 - 3a8z-2 + 18a8 - 27a8z2 + 8a8z4 + 21a8z6 - 12a8z8 - a8z10 + a9z-3 - 11a9z-1 + 30a9z - 43a9z3 + 34a9z5 - 7a9z7 - 3a9z9 + 6a10 - 14a10z2 + 8a10z4 + 4a10z6 - 4a10z8 - 2a11z-1 + 6a11z - 9a11z3 + 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                   6 1   j = -4                 9 4     j = -6               11 4       j = -8             8 9         j = -10           14 11           j = -12         9 14             j = -14       5 8               j = -16     2 9                 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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 538]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 538]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 7, 15, 8], X[8, 13, 5, 14], > X[18, 12, 19, 11], X[22, 19, 17, 20], X[16, 21, 9, 22], X[20, 15, 21, 16], > X[12, 18, 13, 17], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -4}, {9, -5, 6, -8, 7, -6}, > {11, -2, 5, -9, 4, -3, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 7 14 17 22 19 20 13 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 -28 10 7 9 14 8 14 6 7 -1 - q - --- + q + q + --- + --- + --- + --- + --- + --- + --- + --- + 32 24 22 20 18 16 14 12 10 q q q q q q q q q 6 3 4 2 > -- - -- + -- + q 8 6 4 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 538]][a, z]
Out[8]=	3 5 7 9 3 5 7 9 11 a 3 a 3 a a 4 a 10 a 9 a 4 a a 3 -(--) + ---- - ---- + -- - ---- + ----- - ---- + ---- - --- - a z - 4 a z + 3 3 3 3 z z z z z z z z z 5 7 9 3 3 3 5 3 7 3 3 5 > 12 a z - 11 a z + 4 a z - a z - a z + 8 a z - 6 a z + a z + 5 5 > 3 a z
In[9]:=	Kauffman[Link[11, Alternating, 538]][a, z]
Out[9]=	3 5 7 9 4 6 4 6 8 10 12 a 3 a 3 a a 3 a 6 a 9 a + 21 a + 18 a + 6 a + a + -- + ---- + ---- + -- - ---- - ---- - 3 3 3 3 2 2 z z z z z z 8 3 5 7 9 11 3 a 5 a 14 a 18 a 11 a 2 a 3 5 > ---- - ---- - ----- - ----- - ----- - ----- - a z + 11 a z + 40 a z + 2 z z z z z z 7 9 11 2 2 4 2 6 2 8 2 > 52 a z + 30 a z + 6 a z - a z - 12 a z - 27 a z - 27 a z - 10 2 12 2 3 3 3 5 3 7 3 9 3 > 14 a z - 3 a z + 2 a z - 12 a z - 59 a z - 79 a z - 43 a z - 11 3 2 4 4 4 6 4 8 4 10 4 12 4 > 9 a z + 4 a z + 2 a z + a z + 8 a z + 8 a z + 3 a z - 5 3 5 5 5 7 5 9 5 11 5 2 6 > a z + 10 a z + 44 a z + 59 a z + 34 a z + 8 a z - 3 a z + 4 6 6 6 8 6 10 6 12 6 3 7 5 7 > 7 a z + 26 a z + 21 a z + 4 a z - a z - 6 a z - 11 a z - 7 7 9 7 11 7 4 8 6 8 8 8 10 8 > 9 a z - 7 a z - 3 a z - 7 a z - 15 a z - 12 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]=	4 6 1 2 1 5 2 9 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 9 14 14 11 8 9 11 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 9 t 2 2 > ---- + 2 t + -- + q t 4 2 q t q

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