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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X15,2,16,3 X16,7,17,8 X19,22,20,15 X21,14,22,11 X13,20,14,21 X9,18,10,19 X11,10,12,5 X4,17,1,18

Gauss Code:

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

Jones Polynomial:

q-25/2 - 3q-23/2 + 3q-21/2 - 6q-19/2 + 4q-17/2 - 6q-15/2 + 3q-13/2 - 4q-11/2 + q-9/2 - q-7/2

A2 (sl(3)) Invariant:

- q-42 + q-40 + q-38 + 3q-36 + 6q-34 + 8q-32 + 12q-30 + 10q-28 + 12q-26 + 9q-24 + 7q-22 + 6q-20 + 3q-18 + 3q-16 + q-12

HOMFLY-PT Polynomial:

- a7z-3 - 5a7z-1 - 11a7z - 12a7z3 - 6a7z5 - a7z7 + 3a9z-3 + 10a9z-1 + 11a9z - 4a9z5 - a9z7 - 3a11z-3 - 5a11z-1 + a11z + 4a11z3 + a11z5 + a13z-3 - a13z

Kauffman Polynomial:

a7z-3 - 5a7z-1 + 11a7z - 12a7z3 + 6a7z5 - a7z7 - 3a8z-2 + 10a8 - 12a8z2 + 3a8z4 + 3a8z6 - a8z8 + 3a9z-3 - 12a9z-1 + 23a9z - 22a9z3 + 8a9z5 + 2a9z7 - a9z9 - 6a10z-2 + 19a10 - 15a10z2 - 10a10z4 + 16a10z6 - 4a10z8 + 3a11z-3 - 12a11z-1 + 21a11z - 25a11z3 + 14a11z5 - a11z9 - 3a12z-2 + 10a12 - 3a12z2 - 12a12z4 + 12a12z6 - 3a12z8 + a13z-3 - 5a13z-1 + 12a13z - 18a13z3 + 12a13z5 - 3a13z7 - a14z2 + a14z4 - a14z6 + 3a15z - 3a15z3 - a16z2

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -6                   1 j = -8                 1 1 j = -10               3     j = -12           1 1 1     j = -14           6 3       j = -16       2 2 2         j = -18       6 4           j = -20   1 2 4             j = -22   2 2               j = -24   2                 j = -26 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, NonAlternating, 453]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 453]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[15, 2, 16, 3], > X[16, 7, 17, 8], X[19, 22, 20, 15], X[21, 14, 22, 11], X[13, 20, 14, 21], > X[9, 18, 10, 19], X[11, 10, 12, 5], X[4, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -11}, {-10, 2, -8, 7}, {-2, -1, 5, 3, -9, 10}, > {-4, -5, 11, 9, -6, 8, -7, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) 3 3 6 4 6 3 4 -(9/2) q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + q - 23/2 21/2 19/2 17/2 15/2 13/2 11/2 q q q q q q q -(7/2) > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -40 -38 3 6 8 12 10 12 9 7 6 -q + q + q + --- + --- + --- + --- + --- + --- + --- + --- + --- + 36 34 32 30 28 26 24 22 20 q q q q q q q q q 3 3 -12 > --- + --- + q 18 16 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 453]][a, z]
Out[8]=	7 9 11 13 7 9 11 a 3 a 3 a a 5 a 10 a 5 a 7 9 11 -(--) + ---- - ----- + --- - ---- + ----- - ----- - 11 a z + 11 a z + a z - 3 3 3 3 z z z z z z z 13 7 3 11 3 7 5 9 5 11 5 7 7 9 7 > a z - 12 a z + 4 a z - 6 a z - 4 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 453]][a, z]
Out[9]=	7 9 11 13 8 10 12 8 10 12 a 3 a 3 a a 3 a 6 a 3 a 10 a + 19 a + 10 a + -- + ---- + ----- + --- - ---- - ----- - ----- - 3 3 3 3 2 2 2 z z z z z z z 7 9 11 13 5 a 12 a 12 a 5 a 7 9 11 13 > ---- - ----- - ------ - ----- + 11 a z + 23 a z + 21 a z + 12 a z + z z z z 15 8 2 10 2 12 2 14 2 16 2 7 3 > 3 a z - 12 a z - 15 a z - 3 a z - a z - a z - 12 a z - 9 3 11 3 13 3 15 3 8 4 10 4 > 22 a z - 25 a z - 18 a z - 3 a z + 3 a z - 10 a z - 12 4 14 4 7 5 9 5 11 5 13 5 8 6 > 12 a z + a z + 6 a z + 8 a z + 14 a z + 12 a z + 3 a z + 10 6 12 6 14 6 7 7 9 7 13 7 8 8 > 16 a z + 12 a z - a z - a z + 2 a z - 3 a z - a z - 10 8 12 8 9 9 11 9 > 4 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 2 2 1 2 2 4 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 26 9 24 8 22 8 20 8 22 7 20 7 20 6 q t q t q t q t q t q t q t 6 2 4 2 2 6 1 3 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 6 16 6 18 5 16 5 16 4 14 4 12 4 14 3 q t q t q t q t q t q t q t q t 1 1 3 1 > ------ + ------ + ------ + ---- 12 3 12 2 10 2 8 q t q t q t q t

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