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DrawMorseLink

PD Presentation:

X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X9,18,10,19 X17,11,18,22 X11,21,12,20 X21,17,22,16 X4,15,1,16 X19,10,20,5

Gauss Code:

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

Jones Polynomial:

- q-3 + 5q-2 - 6q-1 + 11 - 10q + 11q2 - 9q3 + 6q4 - 4q5 + q6

A2 (sl(3)) Invariant:

- q-10 + 2q-8 + 5q-6 + 4q-4 + 10q-2 + 7 + 6q2 + 4q4 - 2q6 - 5q10 - 2q12 - 2q16 + q18

HOMFLY-PT Polynomial:

- a-4z-2 - a-4 + a-4z2 + a-4z4 + 4a-2z-2 + 4a-2 - 2a-2z2 - 3a-2z4 - a-2z6 - 5z-2 - 4 + 2z2 + 2z4 + 2a2z-2 + a2 - a2z2

Kauffman Polynomial:

- 2a-6z4 + a-6z6 - a-5z-1 + a-5z + 7a-5z3 - 12a-5z5 + 4a-5z7 + a-4z-2 - 4a-4z2 + 11a-4z4 - 15a-4z6 + 5a-4z8 - 5a-3z-1 + 5a-3z + 13a-3z3 - 19a-3z5 + 2a-3z7 + 2a-3z9 + 4a-2z-2 - 2a-2 - 12a-2z2 + 29a-2z4 - 28a-2z6 + 9a-2z8 - 9a-1z-1 + 9a-1z + 6a-1z3 - 8a-1z5 + 2a-1z9 + 5z-2 - 4 - 11z2 + 21z4 - 12z6 + 4z8 - 5az-1 + 5az + az3 - az5 + 2az7 + 2a2z-2 - 3a2 - 3a2z2 + 5a2z4 + a3z3

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                   1 j = 11                 3   j = 9               3 1   j = 7             6 3     j = 5           5 3       j = 3         5 6         j = 1       6 5           j = -1     2 7             j = -3   3 4               j = -5   4                 j = -7 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 283]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 283]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 6, 13, 5], X[8, 4, 9, 3], X[2, 14, 3, 13], > X[14, 7, 15, 8], X[9, 18, 10, 19], X[17, 11, 18, 22], X[11, 21, 12, 20], > X[21, 17, 22, 16], X[4, 15, 1, 16], X[19, 10, 20, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -10}, {2, -1, 5, -3, -6, 11}, > {-8, -2, 4, -5, 10, 9, -7, 6, -11, 8, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-3 5 6 2 3 4 5 6 11 - q + -- - - - 10 q + 11 q - 9 q + 6 q - 4 q + q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 5 4 10 2 4 6 10 12 16 18 7 - q + -- + -- + -- + -- + 6 q + 4 q - 2 q - 5 q - 2 q - 2 q + q 8 6 4 2 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 283]][a, z]
Out[8]=	2 2 2 -4 4 2 5 1 4 2 a 2 z 2 z 2 2 -4 - a + -- + a - -- - ----- + ----- + ---- + 2 z + -- - ---- - a z + 2 2 4 2 2 2 2 4 2 a z a z a z z a a 4 4 6 4 z 3 z z > 2 z + -- - ---- - -- 4 2 2 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 283]][a, z]
Out[9]=	2 2 2 5 1 4 2 a 1 5 9 5 a z -4 - -- - 3 a + -- + ----- + ----- + ---- - ---- - ---- - --- - --- + -- + 2 2 4 2 2 2 2 5 3 a z z 5 a z a z a z z a z a z a 2 2 3 3 3 5 z 9 z 2 4 z 12 z 2 2 7 z 13 z 6 z > --- + --- + 5 a z - 11 z - ---- - ----- - 3 a z + ---- + ----- + ---- + 3 a 4 2 5 3 a a a a a a 4 4 4 5 5 3 3 3 4 2 z 11 z 29 z 2 4 12 z 19 z > a z + a z + 21 z - ---- + ----- + ----- + 5 a z - ----- - ----- - 6 4 2 5 3 a a a a a 5 6 6 6 7 7 8 z 5 6 z 15 z 28 z 4 z 2 z 7 8 > ---- - a z - 12 z + -- - ----- - ----- + ---- + ---- + 2 a z + 4 z + a 6 4 2 5 3 a a a a a 8 8 9 9 5 z 9 z 2 z 2 z > ---- + ---- + ---- + ---- 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	7 1 4 3 4 2 3 3 2 - + 6 q + ----- + ----- + ----- + ---- + --- + 5 q t + 5 q t + 6 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 5 3 7 3 7 4 9 4 9 5 11 5 13 6 > 5 q t + 3 q t + 6 q t + 3 q t + 3 q t + q t + 3 q t + q t

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