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DrawMorseLink

PD Presentation:

X10,1,11,2 X7,16,8,17 X11,18,12,19 X2,19,3,20 X3,12,4,13 X20,13,21,14 X14,5,15,6 X6,9,7,10 X15,22,16,9 X17,8,18,1 X21,4,22,5

Gauss Code:

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

Jones Polynomial:

- q-23/2 + q-21/2 - q-13/2 - q-9/2

A2 (sl(3)) Invariant:

- q-44 + q-38 + q-36 - q-30 + q-26 + 2q-24 + 2q-22 + 2q-20 + q-18 + q-16

HOMFLY-PT Polynomial:

- 2a9z-1 - 18a9z - 36a9z3 - 28a9z5 - 9a9z7 - a9z9 + 3a11z-1 + 17a11z + 20a11z3 + 8a11z5 + a11z7 - a13z-1 - 3a13z - a13z3

Kauffman Polynomial:

2a9z-1 - 18a9z + 36a9z3 - 28a9z5 + 9a9z7 - a9z9 - 3a10 + 17a10z2 - 20a10z4 + 8a10z6 - a10z8 + 3a11z-1 - 20a11z + 37a11z3 - 28a11z5 + 9a11z7 - a11z9 - 3a12 + 17a12z2 - 20a12z4 + 8a12z6 - a12z8 + a13z-1 - 3a13z + a13z3 - a14 - a15z

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -8                 1 j = -10                 1 j = -12             1     j = -14         1         j = -16         1 1       j = -18     1 1           j = -20       1           j = -22 1 1               j = -24 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, 204]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 204]]
Out[4]=	PD[X[10, 1, 11, 2], X[7, 16, 8, 17], X[11, 18, 12, 19], X[2, 19, 3, 20], > X[3, 12, 4, 13], X[20, 13, 21, 14], X[14, 5, 15, 6], X[6, 9, 7, 10], > X[15, 22, 16, 9], X[17, 8, 18, 1], X[21, 4, 22, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -5, 11, 7, -8, -2, 10}, > {8, -1, -3, 5, 6, -7, -9, 2, -10, 3, 4, -6, -11, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) -(21/2) -(13/2) -(9/2) -q + q - q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-44 -38 -36 -30 -26 2 2 2 -18 -16 -q + q + q - q + q + --- + --- + --- + q + q 24 22 20 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 204]][a, z]
Out[8]=	9 11 13 -2 a 3 a a 9 11 13 9 3 11 3 ----- + ----- - --- - 18 a z + 17 a z - 3 a z - 36 a z + 20 a z - z z z 13 3 9 5 11 5 9 7 11 7 9 9 > a z - 28 a z + 8 a z - 9 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 204]][a, z]
Out[9]=	9 11 13 10 12 14 2 a 3 a a 9 11 13 -3 a - 3 a - a + ---- + ----- + --- - 18 a z - 20 a z - 3 a z - z z z 15 10 2 12 2 9 3 11 3 13 3 10 4 > a z + 17 a z + 17 a z + 36 a z + 37 a z + a z - 20 a z - 12 4 9 5 11 5 10 6 12 6 9 7 > 20 a z - 28 a z - 28 a z + 8 a z + 8 a z + 9 a z + 11 7 10 8 12 8 9 9 11 9 > 9 a z - a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-10 -8 1 1 1 1 1 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 24 8 22 8 22 7 18 6 20 5 18 5 16 4 q t q t q t q t q t q t q t 1 1 1 > ------ + ------ + ------ 14 4 16 3 12 2 q t q t q t

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