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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X16,8,17,7 X11,20,12,21 X17,22,18,5 X21,18,22,19 X19,10,20,11 X14,10,15,9 X8,16,9,15 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

- q-9/2 + 4q-7/2 - 7q-5/2 + 8q-3/2 - 10q-1/2 + 9q1/2 - 8q3/2 + 5q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-14 - 2q-12 + 2q-8 - q-6 + 3q-4 + 2 + 2q2 + 3q6 - q8 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 - a-1z-1 - 2a-1z - 2a-1z3 - a-1z5 + az-1 + az - az3 - az5 + a3z3

Kauffman Polynomial:

- 2a-4z2 + 3a-4z4 - a-4z6 + 2a-3z - 9a-3z3 + 10a-3z5 - 3a-3z7 - 2a-2z2 - a-2z4 + 7a-2z6 - 3a-2z8 - a-1z-1 + 4a-1z - 13a-1z3 + 17a-1z5 - 4a-1z7 - a-1z9 + 1 + z2 - 4z4 + 10z6 - 5z8 - az-1 + 2az - az3 + 4az5 - 2az7 - az9 + 2a2z2 - 4a2z4 + 2a2z6 - 2a2z8 + 2a3z3 - 3a3z5 - a3z7 + a4z2 - 4a4z4 - 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               3 1   j = 4             5 2     j = 2           4 3       j = 0         6 5         j = -2       4 6           j = -4     3 4             j = -6   1 4               j = -8   3                 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, 93]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 93]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[16, 8, 17, 7], X[11, 20, 12, 21], > X[17, 22, 18, 5], X[21, 18, 22, 19], X[19, 10, 20, 11], X[14, 10, 15, 9], > X[8, 16, 9, 15], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 8, 7, -4, -2, 11, -8, 9, -3, -5, 6, > -7, 4, -6, 5}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 4 7 8 10 3/2 5/2 -q + ---- - ---- + ---- - ------- + 9 Sqrt[q] - 8 q + 5 q - 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 2 2 -6 3 2 6 8 12 14 2 + q - --- + -- - q + -- + 2 q + 3 q - q + q - q 12 8 4 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 93]][a, z]
Out[8]=	3 3 5 1 a z 2 z z 2 z 3 3 3 z 5 -(---) + - + -- - --- + a z + -- - ---- - a z + a z - -- - a z a z z 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 93]][a, z]
Out[9]=	2 2 3 1 a 2 z 4 z 2 2 z 2 z 2 2 4 2 9 z 1 - --- - - + --- + --- + 2 a z + z - ---- - ---- + 2 a z + a z - ---- - a z z 3 a 4 2 3 a a a a 3 4 4 13 z 3 3 3 5 3 4 3 z z 2 4 4 4 > ----- - a z + 2 a z - a z - 4 z + ---- - -- - 4 a z - 4 a z + a 4 2 a a 5 5 6 6 7 10 z 17 z 5 3 5 6 z 7 z 2 6 3 z > ----- + ----- + 4 a z - 3 a z + 10 z - -- + ---- + 2 a z - ---- - 3 a 4 2 3 a a a a 7 8 9 4 z 7 3 7 8 3 z 2 8 z 9 > ---- - 2 a z - a z - 5 z - ---- - 2 a z - -- - a z a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 3 1 4 3 4 4 2 6 + -- + ------ + ----- + ----- + ----- + ----- + ---- + ---- + 5 t + 4 q t + 2 10 4 8 3 6 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 > 3 q t + 5 q t + 2 q t + 3 q t + q t + 2 q t + q t

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