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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X14,8,15,7 X15,20,16,21 X9,19,10,18 X19,9,20,8 X17,22,18,5 X21,16,22,17 X10,14,11,13 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- 2q-7/2 + q-5/2 - 2q-3/2 + 2q-1/2 - 2q1/2 + 2q3/2 - q5/2

A2 (sl(3)) Invariant:

q-16 + q-14 + 2q-12 + 2q-10 + 2q-8 + 2q-6 - 1 - q2 + q8

HOMFLY-PT Polynomial:

- 2a-1z - a-1z3 + 3az + 4az3 + az5 - a3z-1 - 3a3z - a3z3 + a5z-1

Kauffman Polynomial:

2a-1z - 6a-1z3 + 5a-1z5 - a-1z7 + 7z2 - 16z4 + 11z6 - 2z8 + az - 6az3 + 4az7 - az9 + 10a2z2 - 25a2z4 + 17a2z6 - 3a2z8 - a3z-1 + a3z3 - 5a3z5 + 5a3z7 - a3z9 + a4 + 4a4z2 - 9a4z4 + 6a4z6 - a4z8 - a5z-1 + a5z + a5z3 + a6z2

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 6                 1 j = 4               1   j = 2             1 1   j = 0         1 2 1     j = -2       1 2 1       j = -4       1 2         j = -6     2 1           j = -8 1   1             j = -10 1 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, 87]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 87]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[14, 8, 15, 7], X[15, 20, 16, 21], > X[9, 19, 10, 18], X[19, 9, 20, 8], X[17, 22, 18, 5], X[21, 16, 22, 17], > X[10, 14, 11, 13], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, 6, -5, -9, 11, -2, 9, -3, -4, 8, -7, 5, > -6, 4, -8, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-2 -(5/2) 2 2 3/2 5/2 ---- + q - ---- + ------- - 2 Sqrt[q] + 2 q - q 7/2 3/2 Sqrt[q] q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 2 2 2 2 2 8 -1 + q + q + --- + --- + -- + -- - q + q 12 10 8 6 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 87]][a, z]
Out[8]=	3 5 3 a a 2 z 3 z 3 3 3 5 -(--) + -- - --- + 3 a z - 3 a z - -- + 4 a z - a z + a z z z a a
In[9]:=	Kauffman[Link[11, NonAlternating, 87]][a, z]
Out[9]=	3 5 3 4 a a 2 z 5 2 2 2 4 2 6 2 6 z a - -- - -- + --- + a z + a z + 7 z + 10 a z + 4 a z + a z - ---- - z z a a 5 3 3 3 5 3 4 2 4 4 4 5 z 3 5 > 6 a z + a z + a z - 16 z - 25 a z - 9 a z + ---- - 5 a z + a 7 6 2 6 4 6 z 7 3 7 8 2 8 > 11 z + 17 a z + 6 a z - -- + 4 a z + 5 a z - 2 z - 3 a z - a 4 8 9 3 9 > a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 1 2 1 1 1 1 + -- + -- + ------ + ----- + ------ + ----- + ----- + ---- + ---- + ---- + 4 2 10 4 8 4 10 3 8 2 6 2 6 4 2 q q q t q t q t q t q t q t q t q t t 2 2 2 2 3 4 3 6 4 > 2 t + -- + t + q t + q t + q t + q t 2 q

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