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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 2q-17/2 - 6q-15/2 + 9q-13/2 - 16q-11/2 + 17q-9/2 - 19q-7/2 + 15q-5/2 - 14q-3/2 + 8q-1/2 - 4q1/2 + q3/2

A2 (sl(3)) Invariant:

q-30 + 2q-28 + 2q-26 + 6q-24 + 7q-22 + 6q-20 + 13q-18 + 8q-16 + 10q-14 + 9q-12 + 4q-10 + 9q-8 + 4q-4 + q-2 - 2 + 2q2 - q4

HOMFLY-PT Polynomial:

az + 2az3 + az5 - a3z-3 - 5a3z-1 - 10a3z - 8a3z3 - 4a3z5 - a3z7 + 3a5z-3 + 12a5z-1 + 17a5z + 11a5z3 + 3a5z5 - 3a7z-3 - 9a7z-1 - 9a7z - 3a7z3 + a9z-3 + 2a9z-1 + a9z

Kauffman Polynomial:

- z2 + 2z4 - z6 + 3az - 8az3 + 10az5 - 4az7 - a2z2 - 2a2z4 + 12a2z6 - 6a2z8 + a3z-3 - 6a3z-1 + 18a3z - 31a3z3 + 28a3z5 - 2a3z7 - 4a3z9 - 3a4z-2 + 13a4 - 12a4z2 - 16a4z4 + 32a4z6 - 12a4z8 - a4z10 + 3a5z-3 - 14a5z-1 + 29a5z - 39a5z3 + 23a5z5 + 4a5z7 - 7a5z9 - 6a6z-2 + 24a6 - 28a6z2 - 3a6z4 + 19a6z6 - 9a6z8 - a6z10 + 3a7z-3 - 12a7z-1 + 20a7z - 20a7z3 + 9a7z5 - a7z7 - 3a7z9 - 3a8z-2 + 11a8 - 16a8z2 + 12a8z4 - 2a8z6 - 3a8z8 + a9z-3 - 3a9z-1 + 3a9z - a9z3 + 3a9z5 - 3a9z7 - a10 + 3a10z4 - 2a10z6 + a11z-1 - 3a11z + 3a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                       1 j = 2                     3   j = 0                   5 1   j = -2                 9 3     j = -4               9 8       j = -6             10 6         j = -8           7 9           j = -10         9 10             j = -12       4 11               j = -14     2 5                 j = -16     4                   j = -18 1 2                     j = -20 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, 540]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 540]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 12, 9, 11], X[8, 16, 5, 15], > X[14, 8, 15, 7], X[22, 17, 19, 18], X[20, 13, 21, 14], X[12, 19, 13, 20], > X[16, 21, 17, 22], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 5, -4}, {8, -7, 9, -6}, > {11, -2, 3, -8, 7, -5, 4, -9, 6, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 2 6 9 16 17 19 15 14 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 8 3/2 > ------- - 4 Sqrt[q] + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 2 2 6 7 6 13 8 10 9 4 9 -2 + q + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + -- + 28 26 24 22 20 18 16 14 12 10 8 q q q q q q q q q q q 4 -2 2 4 > -- + q + 2 q - q 4 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 540]][a, z]
Out[8]=	3 5 7 9 3 5 7 9 a 3 a 3 a a 5 a 12 a 9 a 2 a 3 -(--) + ---- - ---- + -- - ---- + ----- - ---- + ---- + a z - 10 a z + 3 3 3 3 z z z z z z z z 5 7 9 3 3 3 5 3 7 3 5 > 17 a z - 9 a z + a z + 2 a z - 8 a z + 11 a z - 3 a z + a z - 3 5 5 5 3 7 > 4 a z + 3 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 540]][a, z]
Out[9]=	3 5 7 9 4 6 8 4 6 8 10 a 3 a 3 a a 3 a 6 a 3 a 13 a + 24 a + 11 a - a + -- + ---- + ---- + -- - ---- - ---- - ---- - 3 3 3 3 2 2 2 z z z z z z z 3 5 7 9 11 6 a 14 a 12 a 3 a a 3 5 7 > ---- - ----- - ----- - ---- + --- + 3 a z + 18 a z + 29 a z + 20 a z + z z z z z 9 11 2 2 2 4 2 6 2 8 2 3 > 3 a z - 3 a z - z - a z - 12 a z - 28 a z - 16 a z - 8 a z - 3 3 5 3 7 3 9 3 11 3 4 2 4 > 31 a z - 39 a z - 20 a z - a z + 3 a z + 2 z - 2 a z - 4 4 6 4 8 4 10 4 5 3 5 5 5 > 16 a z - 3 a z + 12 a z + 3 a z + 10 a z + 28 a z + 23 a z + 7 5 9 5 11 5 6 2 6 4 6 6 6 > 9 a z + 3 a z - a z - z + 12 a z + 32 a z + 19 a z - 8 6 10 6 7 3 7 5 7 7 7 9 7 > 2 a z - 2 a z - 4 a z - 2 a z + 4 a z - a z - 3 a z - 2 8 4 8 6 8 8 8 3 9 5 9 7 9 > 6 a z - 12 a z - 9 a z - 3 a z - 4 a z - 7 a z - 3 a z - 4 10 6 10 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 9 1 1 2 4 2 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 8 18 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 11 9 10 7 9 10 6 9 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 5 t + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t 3 t 2 2 2 4 3 > --- + t + 3 q t + q t 2 q

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