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DrawMorseLink

PD Presentation:

X8192 X14,5,15,6 X20,9,13,10 X2,20,3,19 X10,4,11,3 X18,12,19,11 X16,8,17,7 X12,18,7,17 X6,13,1,14 X4,15,5,16

Gauss Code:

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

Jones Polynomial:

- q-5 + 3q-4 - 5q-3 + 9q-2 - 10q-1 + 12 - 10q + 9q2 - 5q3 + 3q4 - q5

A2 (sl(3)) Invariant:

- q-16 + q-14 + q-12 - 2q-10 + 3q-8 + 2q-6 + 2q-4 + 6q-2 + 3 + 6q2 + 2q4 + 2q6 + 3q8 - 2q10 + q12 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 + a-2z-2 + a-2 + a-2z4 - 2z-2 - 2 + z2 + 2z4 + a2z-2 + a2 + a2z4 - a4z2

Kauffman Polynomial:

- 2a-5z3 + a-5z5 + 3a-4z2 - 7a-4z4 + 3a-4z6 + 3a-3z3 - 8a-3z5 + 4a-3z7 + a-2z-2 - 2a-2 - 3a-2z2 + 9a-2z4 - 9a-2z6 + 4a-2z8 - 2a-1z-1 + 2a-1z + 3a-1z3 - 2a-1z5 - a-1z7 + 2a-1z9 + 2z-2 - 3 - 12z2 + 32z4 - 24z6 + 8z8 - 2az-1 + 2az + 3az3 - 2az5 - az7 + 2az9 + a2z-2 - 2a2 - 3a2z2 + 9a2z4 - 9a2z6 + 4a2z8 + 3a3z3 - 8a3z5 + 4a3z7 + 3a4z2 - 7a4z4 + 3a4z6 - 2a5z3 + a5z5

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                     1 j = 9                   2   j = 7                 3 1   j = 5               6 2     j = 3             5 4       j = 1           7 5         j = -1         5 7           j = -3       4 5             j = -5     2 6               j = -7   1 3                 j = -9   2                   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[10, Alternating, 158]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 158]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 5, 15, 6], X[20, 9, 13, 10], X[2, 20, 3, 19], > X[10, 4, 11, 3], X[18, 12, 19, 11], X[16, 8, 17, 7], X[12, 18, 7, 17], > X[6, 13, 1, 14], X[4, 15, 5, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 5, -10, 2, -9}, {7, -1, 3, -5, 6, -8}, > {9, -2, 10, -7, 8, -6, 4, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-5 3 5 9 10 2 3 4 5 12 - q + -- - -- + -- - -- - 10 q + 9 q - 5 q + 3 q - q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 -12 2 3 2 2 6 2 4 6 8 3 - q + q + q - --- + -- + -- + -- + -- + 6 q + 2 q + 2 q + 3 q - 10 8 6 4 2 q q q q q 10 12 14 16 > 2 q + q + q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 158]][a, z]
Out[8]=	2 2 4 -2 2 2 1 a 2 z 4 2 4 z 2 4 -2 + a + a - -- + ----- + -- + z - -- - a z + 2 z + -- + a z 2 2 2 2 4 2 z a z z a a
In[9]:=	Kauffman[Link[10, Alternating, 158]][a, z]
Out[9]=	2 2 2 2 2 1 a 2 2 a 2 z 2 3 z -3 - -- - 2 a + -- + ----- + -- - --- - --- + --- + 2 a z - 12 z + ---- - 2 2 2 2 2 a z z a 4 a z a z z a 2 3 3 3 3 z 2 2 4 2 2 z 3 z 3 z 3 3 3 > ---- - 3 a z + 3 a z - ---- + ---- + ---- + 3 a z + 3 a z - 2 5 3 a a a a 4 4 5 5 5 5 3 4 7 z 9 z 2 4 4 4 z 8 z 2 z > 2 a z + 32 z - ---- + ---- + 9 a z - 7 a z + -- - ---- - ---- - 4 2 5 3 a a a a a 6 6 7 5 3 5 5 5 6 3 z 9 z 2 6 4 6 4 z > 2 a z - 8 a z + a z - 24 z + ---- - ---- - 9 a z + 3 a z + ---- - 4 2 3 a a a 7 8 9 z 7 3 7 8 4 z 2 8 2 z 9 > -- - a z + 4 a z + 8 z + ---- + 4 a z + ---- + 2 a z a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	7 1 2 1 3 2 6 4 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 7 4 9 4 > 5 q t + 5 q t + 4 q t + 6 q t + 2 q t + 3 q t + q t + 2 q t + 11 5 > q t

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