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DrawMorseLink

PD Presentation:

X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X16,12,17,11 X18,9,11,10 X10,17,5,18 X4,15,1,16

Gauss Code:

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

Jones Polynomial:

- q-5 + 4q-4 - 6q-3 + 10q-2 - 10q-1 + 12 - 9q + 7q2 - 4q3 + q4

A2 (sl(3)) Invariant:

- q-16 + q-14 + 3q-12 + q-10 + 6q-8 + 4q-6 + 4q-4 + 6q-2 + 1 + 4q2 - 2q4 + q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2z2 + a-2z4 + z-2 + 1 - 3z2 - 3z4 - z6 - 2a2z-2 - a2 + 3a2z2 + 2a2z4 + a4z-2 - a4z2

Kauffman Polynomial:

a-4z4 - 3a-3z3 + 4a-3z5 + 4a-2z2 - 9a-2z4 + 7a-2z6 - 2a-1z3 - 4a-1z5 + 6a-1z7 - z-2 + 1 + 12z2 - 25z4 + 11z6 + 2z8 + 2az-1 - az + 4az3 - 17az5 + 11az7 - 2a2z-2 + a2 + 12a2z2 - 23a2z4 + 8a2z6 + 2a2z8 + 2a3z-1 - a3z + 2a3z3 - 8a3z5 + 5a3z7 - a4z-2 + a4 + 4a4z2 - 8a4z4 + 4a4z6 - a5z3 + a5z5

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                   1 j = 7                 3   j = 5               4 1   j = 3             5 3     j = 1           7 4       j = -1         5 7         j = -3       5 5           j = -5     3 7             j = -7   1 3               j = -9   3                 j = -11 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[9, Alternating, 46]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 46]]
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[16, 12, 17, 11], X[18, 9, 11, 10], X[10, 17, 5, 18], > X[4, 15, 1, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -9}, {2, -1, 5, -3, 7, -8}, {6, -2, 4, -5, 9, -6, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-5 4 6 10 10 2 3 4 12 - q + -- - -- + -- - -- - 9 q + 7 q - 4 q + q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 3 -10 6 4 4 6 2 4 8 10 1 - q + q + --- + q + -- + -- + -- + -- + 4 q - 2 q + q - 2 q + 12 8 6 4 2 q q q q q 12 > q
In[8]:=	HOMFLYPT[Link[9, Alternating, 46]][a, z]
Out[8]=	2 4 2 4 2 -2 2 a a 2 z 2 2 4 2 4 z 2 4 1 - a + z - ---- + -- - 3 z + -- + 3 a z - a z - 3 z + -- + 2 a z - 2 2 2 2 z z a a 6 > z
In[9]:=	Kauffman[Link[9, Alternating, 46]][a, z]
Out[9]=	2 4 3 2 2 4 -2 2 a a 2 a 2 a 3 2 4 z 1 + a + a - z - ---- - -- + --- + ---- - a z - a z + 12 z + ---- + 2 2 z z 2 z z a 3 3 4 2 2 4 2 3 z 2 z 3 3 3 5 3 4 z > 12 a z + 4 a z - ---- - ---- + 4 a z + 2 a z - a z - 25 z + -- - 3 a 4 a a 4 5 5 9 z 2 4 4 4 4 z 4 z 5 3 5 5 5 > ---- - 23 a z - 8 a z + ---- - ---- - 17 a z - 8 a z + a z + 2 3 a a a 6 7 6 7 z 2 6 4 6 6 z 7 3 7 8 2 8 > 11 z + ---- + 8 a z + 4 a z + ---- + 11 a z + 5 a z + 2 z + 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	7 1 3 1 3 3 7 5 5 5 - + 7 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 9 4 > 4 q t + 5 q t + 3 q t + 4 q t + q t + 3 q t + q t

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