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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X20,14,15,13 X7,16,8,17 X15,8,16,9 X18,12,19,11 X12,20,13,19 X14,18,5,17 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

2q-4 - 3q-3 + 6q-2 - 6q-1 + 8 - 6q + 5q2 - 3q3 + q4

A2 (sl(3)) Invariant:

q-14 + 3q-12 + 2q-10 + 5q-8 + 4q-6 + 3q-4 + 5q-2 + 1 + 3q2 - q4 + q8 - q10 + q12

HOMFLY-PT Polynomial:

a-2 + 2a-2z2 + a-2z4 + z-2 - 5z2 - 4z4 - z6 - 2a2z-2 - 2a2 + a2z2 + a2z4 + a4z-2 + a4

Kauffman Polynomial:

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

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                 1 j = 7               2   j = 5             3 1   j = 3           4 3     j = 1         4 2       j = -1       3 5         j = -3     3 3           j = -5   1 4             j = -7 1 2               j = -9 2

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, NonAlternating, 85]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 85]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[20, 14, 15, 13], X[7, 16, 8, 17], > X[15, 8, 16, 9], X[18, 12, 19, 11], X[12, 20, 13, 19], X[14, 18, 5, 17], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {-5, 4, 8, -6, 7, -3}, > {9, -1, -4, 5, 10, -2, 6, -7, 3, -8}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 6 6 2 3 4 8 + -- - -- + -- - - - 6 q + 5 q - 3 q + q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 3 2 5 4 3 5 2 4 8 10 12 1 + q + --- + --- + -- + -- + -- + -- + 3 q - q + q - q + q 12 10 8 6 4 2 q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 85]][a, z]
Out[8]=	2 4 2 4 -2 2 4 -2 2 a a 2 2 z 2 2 4 z 2 4 6 a - 2 a + a + z - ---- + -- - 5 z + ---- + a z - 4 z + -- + a z - z 2 2 2 2 z z a a
In[9]:=	Kauffman[Link[10, NonAlternating, 85]][a, z]
Out[9]=	2 4 3 -2 2 4 -2 2 a a 2 a 2 a 3 2 2 - a + 6 a + 4 a - z - ---- - -- + --- + ---- - 5 a z - 5 a z + 3 z - 2 2 z z z z 2 2 3 3 4 z 3 z 2 2 4 2 4 z z 3 3 3 4 z > -- + ---- - 8 a z - 7 a z - ---- + -- + 7 a z + 2 a z - 6 z + -- - 4 2 3 a 4 a a a a 4 5 5 6 6 z 2 4 4 4 3 z 3 z 5 3 5 6 4 z > ---- + 4 a z + 3 a z + ---- - ---- - 7 a z - a z + 3 z + ---- - 2 3 a 2 a a a 7 2 6 3 z 7 3 7 8 2 8 > a z + ---- + 4 a z + a z + z + a z a
In[10]:=	Kh[L][q, t]
Out[10]=	5 2 1 2 1 4 3 3 3 - + 4 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 2 q t + q 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 3 3 2 5 2 5 3 7 3 9 4 > 4 q t + 3 q t + 3 q t + q t + 2 q t + q t

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