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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X16,8,17,7 X22,18,5,17 X18,14,19,13 X14,22,15,21 X20,10,21,9 X8,16,9,15 X10,20,11,19 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 3q-5/2 - 8q-3/2 + 13q-1/2 - 20q1/2 + 22q3/2 - 23q5/2 + 20q7/2 - 15q9/2 + 10q11/2 - 4q13/2 + q15/2

A2 (sl(3)) Invariant:

q-12 - 2q-8 + 4q-6 + q-4 - q-2 + 8 + 4q4 + 2q6 - 2q8 + 2q10 - 7q12 + q14 - 4q18 + 3q20 - q24

HOMFLY-PT Polynomial:

a-7z + a-5z-1 - a-5z - 3a-5z3 - a-3z-1 + a-3z + 2a-3z3 + 2a-3z5 - 2a-1z-1 - 5a-1z - 3a-1z3 + a-1z5 + 2az-1 + az - 2az3 + a3z

Kauffman Polynomial:

- a-8z2 + 2a-8z4 - a-8z6 + a-7z - 4a-7z3 + 8a-7z5 - 4a-7z7 - 4a-6 + 10a-6z2 - 15a-6z4 + 19a-6z6 - 8a-6z8 + a-5z-1 + 2a-5z - 9a-5z3 + 7a-5z5 + 9a-5z7 - 7a-5z9 - 9a-4 + 34a-4z2 - 59a-4z4 + 54a-4z6 - 16a-4z8 - 2a-4z10 + a-3z-1 + 2a-3z - 9a-3z3 - 2a-3z5 + 21a-3z7 - 13a-3z9 - 4a-2 + 20a-2z2 - 46a-2z4 + 44a-2z6 - 15a-2z8 - 2a-2z10 - 2a-1z-1 + 9a-1z - 16a-1z3 + 10a-1z5 + 2a-1z7 - 6a-1z9 + 2 - 4z2 + 7z6 - 7z8 - 2az-1 + 7az - 10az3 + 10az5 - 6az7 - a2z2 + 4a2z4 - 3a2z6 - a3z + 2a3z3 - a3z5

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 16                       1 j = 14                     3   j = 12                   7 1   j = 10                 8 3     j = 8               12 7       j = 6             11 8         j = 4           11 12           j = 2         9 11             j = 0       5 12               j = -2     3 8                 j = -4     5                   j = -6 1 3                     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, Alternating, 79]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 79]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 8, 17, 7], X[22, 18, 5, 17], > X[18, 14, 19, 13], X[14, 22, 15, 21], X[20, 10, 21, 9], X[8, 16, 9, 15], > X[10, 20, 11, 19], 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, 7, -9, 11, -2, 5, -6, 8, -3, 4, -5, > 9, -7, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 3 8 13 3/2 5/2 7/2 -q + ---- - ---- + ------- - 20 Sqrt[q] + 22 q - 23 q + 20 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 15 q + 10 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 2 4 -4 -2 4 6 8 10 12 14 8 + q - -- + -- + q - q + 4 q + 2 q - 2 q + 2 q - 7 q + q - 8 6 q q 18 20 24 > 4 q + 3 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 79]][a, z]
Out[8]=	3 3 1 1 2 2 a z z z 5 z 3 3 z 2 z ---- - ---- - --- + --- + -- - -- + -- - --- + a z + a z - ---- + ---- - 5 3 a z z 7 5 3 a 5 3 a z a z a a a a a 3 5 5 3 z 3 2 z z > ---- - 2 a z + ---- + -- a 3 a a
In[9]:=	Kauffman[Link[11, Alternating, 79]][a, z]
Out[9]=	4 9 4 1 1 2 2 a z 2 z 2 z 9 z 2 - -- - -- - -- + ---- + ---- - --- - --- + -- + --- + --- + --- + 7 a z - 6 4 2 5 3 a z z 7 5 3 a a a a a z a z a a a 2 2 2 2 3 3 3 3 2 z 10 z 34 z 20 z 2 2 4 z 9 z 9 z > a z - 4 z - -- + ----- + ----- + ----- - a z - ---- - ---- - ---- - 8 6 4 2 7 5 3 a a a a a a a 3 4 4 4 4 5 16 z 3 3 3 2 z 15 z 59 z 46 z 2 4 8 z > ----- - 10 a z + 2 a z + ---- - ----- - ----- - ----- + 4 a z + ---- + a 8 6 4 2 7 a a a a a 5 5 5 6 6 6 6 7 z 2 z 10 z 5 3 5 6 z 19 z 54 z 44 z > ---- - ---- + ----- + 10 a z - a z + 7 z - -- + ----- + ----- + ----- - 5 3 a 8 6 4 2 a a a a a a 7 7 7 7 8 8 2 6 4 z 9 z 21 z 2 z 7 8 8 z 16 z > 3 a z - ---- + ---- + ----- + ---- - 6 a z - 7 z - ---- - ----- - 7 5 3 a 6 4 a a a a a 8 9 9 9 10 10 15 z 7 z 13 z 6 z 2 z 2 z > ----- - ---- - ----- - ---- - ----- - ----- 2 5 3 a 4 2 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 3 5 3 5 8 2 12 + 9 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 11 q t + 8 4 6 4 6 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 4 2 6 2 6 3 8 3 8 4 10 4 > 11 q t + 12 q t + 11 q t + 8 q t + 12 q t + 7 q t + 8 q t + 10 5 12 5 12 6 14 6 16 7 > 3 q t + 7 q t + q t + 3 q t + q t

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