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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-9/2 + q-7/2 - 4q-5/2 + 5q-3/2 - 6q-1/2 + 6q1/2 - 5q3/2 + 4q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-16 + 2q-14 + q-12 + 2q-10 + 3q-8 + q-4 - q-2 - 1 - q4 + 2q6 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 - 3a-1z - 3a-1z3 - a-1z5 + 2az + 2az3 - a3z-1 - 2a3z + a5z-1

Kauffman Polynomial:

- a-4z2 + 3a-4z4 - a-4z6 + a-3z - 8a-3z3 + 11a-3z5 - 3a-3z7 + 2a-2z2 - 9a-2z4 + 11a-2z6 - 3a-2z8 + 2a-1z - 11a-1z3 + 10a-1z5 - a-1z9 + 8z2 - 20z4 + 16z6 - 4z8 - az - 3az5 + 3az7 - az9 + 5a2z2 - 9a2z4 + 4a2z6 - a2z8 - a3z-1 + 2a3z3 - 2a3z5 + a4 - a4z4 - a5z-1 + 2a5z - a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                   1 j = 8                 2   j = 6               2 1   j = 4             3 2     j = 2           3 2       j = 0         3 3         j = -2       3 4           j = -4     1 2             j = -6     3               j = -8 1 1                 j = -10 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, 84]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 84]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[14, 8, 15, 7], X[9, 18, 10, 19], > X[19, 5, 20, 22], X[15, 21, 16, 20], X[21, 17, 22, 16], X[17, 8, 18, 9], > 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, 8, -4, -9, 11, -2, 9, -3, -6, 7, -8, 4, > -5, 6, -7, 5}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) -(7/2) 4 5 6 3/2 5/2 -q + q - ---- + ---- - ------- + 6 Sqrt[q] - 5 q + 4 q - 5/2 3/2 Sqrt[q] q q 7/2 9/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 -12 2 3 -4 -2 4 6 12 14 -1 + q + --- + q + --- + -- + q - q - q + 2 q + q - q 14 10 8 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 84]][a, z]
Out[8]=	3 5 3 3 5 a a z 3 z 3 z 3 z 3 z -(--) + -- + -- - --- + 2 a z - 2 a z + -- - ---- + 2 a z - -- z z 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 84]][a, z]
Out[9]=	3 5 2 2 3 4 a a z 2 z 5 2 z 2 z 2 2 8 z a - -- - -- + -- + --- - a z + 2 a z + 8 z - -- + ---- + 5 a z - ---- - z z 3 a 4 2 3 a a a a 3 4 4 5 11 z 3 3 5 3 4 3 z 9 z 2 4 4 4 11 z > ----- + 2 a z - a z - 20 z + ---- - ---- - 9 a z - a z + ----- + a 4 2 3 a a a 5 6 6 7 10 z 5 3 5 6 z 11 z 2 6 3 z 7 > ----- - 3 a z - 2 a z + 16 z - -- + ----- + 4 a z - ---- + 3 a z - a 4 2 3 a a a 8 9 8 3 z 2 8 z 9 > 4 z - ---- - a z - -- - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 1 1 3 1 2 3 2 3 + -- + ------ + ----- + ----- + ----- + ----- + ---- + ---- + 3 t + 3 q t + 2 10 4 8 4 8 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t 2 2 4 2 4 3 6 3 6 4 8 4 10 5 > 2 q t + 3 q t + 2 q t + 2 q t + q t + 2 q t + q t

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