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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X18,9,19,10 X8,17,9,18 X19,1,20,4 X5,12,6,13 X3,10,4,11 X13,22,14,5 X21,14,22,15 X11,20,12,21 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-32 - 2q-28 - 4q-26 - q-24 - 2q-22 + 2q-20 + 4q-18 + 3q-16 + 4q-14 + 2q-10 + 2q-8 + 2q-4

HOMFLY-PT Polynomial:

- a3z-1 - 4a3z - 2a3z3 - a5z-1 + a5z + 2a5z3 + a5z5 + 4a7z-1 + 5a7z + 3a7z3 + a7z5 - 2a9z-1 - 2a9z - a9z3

Kauffman Polynomial:

- a3z-1 + 5a3z - 3a3z3 + a4 - a4z2 - a4z6 + a5z-1 + a5z - 7a5z3 + 4a5z5 - 2a5z7 - 5a6 + 12a6z2 - 12a6z4 + 5a6z6 - 2a6z8 + 4a7z-1 - 9a7z + 3a7z3 + a7z5 - a7z9 - 6a8 + 15a8z2 - 14a8z4 + 11a8z6 - 4a8z8 + 2a9z-1 - 6a9z + 4a9z3 + 3a9z5 - a9z9 + a10 - 3a10z2 + 2a10z4 + 4a10z6 - 2a10z8 - a11z - 3a11z3 + 6a11z5 - 2a11z7 + 2a12 - 5a12z2 + 4a12z4 - 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 2 j = -6               3     j = -8             4 3     j = -10           5 3       j = -12         2 4         j = -14       4 5           j = -16     1 2             j = -18   1 4               j = -20   1                 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, 78]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 78]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[18, 9, 19, 10], X[8, 17, 9, 18], > X[19, 1, 20, 4], X[5, 12, 6, 13], X[3, 10, 4, 11], X[13, 22, 14, 5], > X[21, 14, 22, 15], X[11, 20, 12, 21], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -7, 5}, {-6, -1, 2, -4, 3, 7, -10, 6, -8, 9, 11, -2, 4, -3, > -5, 10, -9, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 2 5 6 7 9 7 6 3 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 4 -24 2 2 4 3 4 2 2 2 -q - --- - --- - q - --- + --- + --- + --- + --- + --- + -- + -- 28 26 22 20 18 16 14 10 8 4 q q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 78]][a, z]
Out[8]=	3 5 7 9 a a 4 a 2 a 3 5 7 9 3 3 -(--) - -- + ---- - ---- - 4 a z + a z + 5 a z - 2 a z - 2 a z + z z z z 5 3 7 3 9 3 5 5 7 5 > 2 a z + 3 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 78]][a, z]
Out[9]=	3 5 7 9 4 6 8 10 12 a a 4 a 2 a 3 5 a - 5 a - 6 a + a + 2 a - -- + -- + ---- + ---- + 5 a z + a z - z z z z 7 9 11 4 2 6 2 8 2 10 2 > 9 a z - 6 a z - a z - a z + 12 a z + 15 a z - 3 a z - 12 2 3 3 5 3 7 3 9 3 11 3 6 4 > 5 a z - 3 a z - 7 a z + 3 a z + 4 a z - 3 a z - 12 a z - 8 4 10 4 12 4 5 5 7 5 9 5 11 5 > 14 a z + 2 a z + 4 a z + 4 a z + a z + 3 a z + 6 a z - 4 6 6 6 8 6 10 6 12 6 5 7 11 7 > a z + 5 a z + 11 a z + 4 a z - a z - 2 a z - 2 a z - 6 8 8 8 10 8 7 9 9 9 > 2 a z - 4 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 4 1 2 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 5 2 4 5 3 4 3 3 3 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 4 q t q t q t q t q t q t q t q t q t

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