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DrawMorseLink

PD Presentation:

X8192 X9,21,10,20 X14,5,15,6 X18,12,19,11 X3,10,4,11 X12,7,13,8 X16,13,17,14 X22,17,7,18 X6,15,1,16 X4,21,5,22 X19,2,20,3

Gauss Code:

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

Jones Polynomial:

q-21/2 - 4q-19/2 + 6q-17/2 - 9q-15/2 + 11q-13/2 - 11q-11/2 + 10q-9/2 - 8q-7/2 + 4q-5/2 - 2q-3/2

A2 (sl(3)) Invariant:

- q-32 + 2q-30 + q-28 + 4q-24 - q-22 - q-18 - 2q-16 + 2q-14 - q-12 + 3q-10 + 2q-8 - q-6 + 2q-4

HOMFLY-PT Polynomial:

- a3z-1 - 3a3z - 2a3z3 + 2a5z-1 + a5z + a5z3 + a5z5 - 2a7z-1 - a7z + a7z3 + a7z5 + a9z-1 - a9z3

Kauffman Polynomial:

- a3z-1 + 4a3z - 3a3z3 + a4z2 - 2a4z4 - a4z6 - 2a5z-1 + 7a5z - 10a5z3 + 5a5z5 - 3a5z7 - a6 + 9a6z2 - 16a6z4 + 10a6z6 - 4a6z8 - 2a7z-1 + 9a7z - 17a7z3 + 12a7z5 - a7z7 - 2a7z9 + 11a8z2 - 27a8z4 + 27a8z6 - 9a8z8 - a9z-1 + 5a9z - 17a9z3 + 19a9z5 - 2a9z7 - 2a9z9 + 3a10z2 - 11a10z4 + 15a10z6 - 5a10z8 - a11z - 7a11z3 + 12a11z5 - 4a11z7 + 2a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                   2 j = -4                 3 1 j = -6               5 1   j = -8             5 3     j = -10           6 5       j = -12         5 5         j = -14       4 6           j = -16     3 6             j = -18   1 3               j = -20   3                 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, NonAlternating, 175]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 175]]
Out[4]=	PD[X[8, 1, 9, 2], X[9, 21, 10, 20], X[14, 5, 15, 6], X[18, 12, 19, 11], > X[3, 10, 4, 11], X[12, 7, 13, 8], X[16, 13, 17, 14], X[22, 17, 7, 18], > X[6, 15, 1, 16], X[4, 21, 5, 22], X[19, 2, 20, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -10, 3, -9}, > {6, -1, -2, 5, 4, -6, 7, -3, 9, -7, 8, -4, -11, 2, 10, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 4 6 9 11 11 10 8 4 2 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
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 2 -28 4 -22 -18 2 2 -12 3 2 -6 2 -q + --- + q + --- - q - q - --- + --- - q + --- + -- - q + -- 30 24 16 14 10 8 4 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 175]][a, z]
Out[8]=	3 5 7 9 a 2 a 2 a a 3 5 7 3 3 5 3 7 3 -(--) + ---- - ---- + -- - 3 a z + a z - a z - 2 a z + a z + a z - z z z z 9 3 5 5 7 5 > a z + a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 175]][a, z]
Out[9]=	3 5 7 9 6 a 2 a 2 a a 3 5 7 9 11 -a - -- - ---- - ---- - -- + 4 a z + 7 a z + 9 a z + 5 a z - a z + z z z z 4 2 6 2 8 2 10 2 3 3 5 3 7 3 > a z + 9 a z + 11 a z + 3 a z - 3 a z - 10 a z - 17 a z - 9 3 11 3 4 4 6 4 8 4 10 4 > 17 a z - 7 a z - 2 a z - 16 a z - 27 a z - 11 a z + 12 4 5 5 7 5 9 5 11 5 4 6 6 6 > 2 a z + 5 a z + 12 a z + 19 a z + 12 a z - a z + 10 a z + 8 6 10 6 12 6 5 7 7 7 9 7 11 7 > 27 a z + 15 a z - a z - 3 a z - a z - 2 a z - 4 a z - 6 8 8 8 10 8 7 9 9 9 > 4 a z - 9 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 1 3 1 3 3 6 4 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q t q t q t q t q t q t q t 6 5 5 6 5 5 3 5 1 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 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 3 > ---- 4 q t

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