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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,8,13,7 X9,16,10,17 X13,20,14,21 X17,5,18,22 X21,19,22,18 X19,14,20,15 X15,8,16,9 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-12 + 2q-8 + 2q-4 + q-2 + 1 + 2q2 + 2q6 + q12 - q14

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-4z2 + 3a-4z4 - a-4z6 + 2a-3z - 8a-3z3 + 11a-3z5 - 3a-3z7 + a-2z2 - 8a-2z4 + 11a-2z6 - 3a-2z8 - a-1z-1 + 6a-1z - 16a-1z3 + 12a-1z5 - a-1z9 + 1 + 5z2 - 20z4 + 17z6 - 4z8 - az-1 + 6az - 10az3 + az5 + 3az7 - az9 + 3a2z2 - 9a2z4 + 5a2z6 - a2z8 + 2a3z - 2a3z3

Khovanov Homology:

trqj 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       1 3 2       j = 0       4 3         j = -2     2 4           j = -4   1 2             j = -6   2               j = -8 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, 53]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 53]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[9, 16, 10, 17], > X[13, 20, 14, 21], X[17, 5, 18, 22], X[21, 19, 22, 18], X[19, 14, 20, 15], > X[15, 8, 16, 9], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, 9, -4, -2, 11, -3, -5, 8, -9, 4, -6, 7, > -8, 5, -7, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 3 4 6 3/2 5/2 7/2 9/2 q - ---- + ---- - ------- + 5 Sqrt[q] - 5 q + 4 q - 3 q + q 5/2 3/2 Sqrt[q] q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 2 2 -2 2 6 12 14 1 - q + -- + -- + q + 2 q + 2 q + q - q 8 4 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 53]][a, z]
Out[8]=	3 3 5 1 a z 3 z 3 z 3 z 3 z -(---) + - + -- - --- + 3 a z - a z + -- - ---- + 2 a z - -- a z z 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 53]][a, z]
Out[9]=	2 2 3 1 a 2 z 6 z 3 2 z z 2 2 8 z 1 - --- - - + --- + --- + 6 a z + 2 a z + 5 z - -- + -- + 3 a z - ---- - a z z 3 a 4 2 3 a a a a 3 4 4 5 5 16 z 3 3 3 4 3 z 8 z 2 4 11 z 12 z > ----- - 10 a z - 2 a z - 20 z + ---- - ---- - 9 a z + ----- + ----- + a 4 2 3 a a a a 6 6 7 8 5 6 z 11 z 2 6 3 z 7 8 3 z 2 8 > a z + 17 z - -- + ----- + 5 a z - ---- + 3 a z - 4 z - ---- - a z - 4 2 3 2 a a a a 9 z 9 > -- - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	4 2 1 2 1 2 2 2 2 2 4 + -- + q + ----- + ----- + ----- + ---- + ---- + 3 t + 3 q t + 2 q t + 2 8 3 6 2 4 2 4 2 q q t q t q t q t q t 4 2 4 3 6 3 6 4 8 4 10 5 > 3 q t + 2 q t + 2 q t + q t + 2 q t + q t

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