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DrawMorseLink

PD Presentation:

X12,1,13,2 X14,3,15,4 X9,18,10,19 X5,16,6,17 X22,7,11,8 X6,21,7,22 X15,21,16,20 X17,8,18,9 X19,4,20,5 X2,11,3,12 X10,13,1,14

Gauss Code:

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

Jones Polynomial:

- q-23/2 - q-17/2 + 2q-15/2 - 2q-13/2 + 2q-11/2 - 2q-9/2 + q-7/2 - q-5/2

A2 (sl(3)) Invariant:

q-36 + q-34 + 2q-32 + 2q-30 + q-28 + q-26 - q-24 - q-20 + q-16 + q-12 + q-8

HOMFLY-PT Polynomial:

- 3a5z - 4a5z3 - a5z5 - 3a7z - 4a7z3 - a7z5 - a9z-1 + a11z-1 + a11z

Kauffman Polynomial:

- 3a5z + 4a5z3 - a5z5 - a6z2 + 3a6z4 - a6z6 + 3a7z - 5a7z3 + 4a7z5 - a7z7 + 2a8z4 - a8z6 + a9z-1 - 2a9z + 2a9z3 - a9z5 - a10 - a10z2 + a11z-1 - 3a11z3 + a11z5 - 2a12z2 + a12z4 + 8a13z - 14a13z3 + 7a13z5 - a13z7

Khovanov Homology:

trqj r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                     1 j = -6                   1 1 j = -8                 1     j = -10               1 1     j = -12           1 2 1       j = -14           1 1         j = -16       1 2 2           j = -18         1             j = -20     1 1               j = -22 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, 252]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 252]]
Out[4]=	PD[X[12, 1, 13, 2], X[14, 3, 15, 4], X[9, 18, 10, 19], X[5, 16, 6, 17], > X[22, 7, 11, 8], X[6, 21, 7, 22], X[15, 21, 16, 20], X[17, 8, 18, 9], > X[19, 4, 20, 5], X[2, 11, 3, 12], X[10, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, 9, -4, -6, 5, 8, -3, -11}, > {10, -1, 11, -2, -7, 4, -8, 3, -9, 7, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) -(17/2) 2 2 2 2 -(7/2) -(5/2) -q - q + ----- - ----- + ----- - ---- + q - q 15/2 13/2 11/2 9/2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-36 -34 2 2 -28 -26 -24 -20 -16 -12 -8 q + q + --- + --- + q + q - q - q + q + q + q 32 30 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 252]][a, z]
Out[8]=	9 11 a a 5 7 11 5 3 7 3 5 5 7 5 -(--) + --- - 3 a z - 3 a z + a z - 4 a z - 4 a z - a z - a z z z
In[9]:=	Kauffman[Link[11, NonAlternating, 252]][a, z]
Out[9]=	9 11 10 a a 5 7 9 13 6 2 10 2 -a + -- + --- - 3 a z + 3 a z - 2 a z + 8 a z - a z - a z - z z 12 2 5 3 7 3 9 3 11 3 13 3 6 4 > 2 a z + 4 a z - 5 a z + 2 a z - 3 a z - 14 a z + 3 a z + 8 4 12 4 5 5 7 5 9 5 11 5 13 5 6 6 > 2 a z + a z - a z + 4 a z - a z + a z + 7 a z - a z - 8 6 7 7 13 7 > a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 1 1 1 1 2 q + q + ------- + ------- + ------ + ------ + ------ + ------ + ------ + 24 10 22 10 20 8 20 7 16 7 18 6 16 6 q t q t q t q t q t q t q t 2 1 1 1 2 1 1 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 16 5 14 5 12 5 14 4 12 4 12 3 10 3 10 2 q t q t q t q t q t q t q t q t 1 1 > ----- + ---- 8 2 6 q t q t

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