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DrawMorseLink

PD Presentation:

X6172 X12,6,13,5 X8493 X2,16,3,15 X16,7,17,8 X22,20,15,19 X14,22,11,21 X20,14,21,13 X18,10,19,9 X10,12,5,11 X4,17,1,18

Gauss Code:

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

Jones Polynomial:

q-3/2 - 4q-1/2 + 8q1/2 - 17q3/2 + 20q5/2 - 27q7/2 + 25q9/2 - 24q11/2 + 17q13/2 - 11q15/2 + 5q17/2 - q19/2

A2 (sl(3)) Invariant:

- q-4 + 2q-2 - 1 + 2q2 + 9q4 + 4q6 + 16q8 + 11q10 + 12q12 + 14q14 + 4q16 + 10q18 - q20 + 2q24 - 3q26 + q28

HOMFLY-PT Polynomial:

- a-7z-3 + a-7z - a-7z3 - a-7z5 + 3a-5z-3 + 2a-5z-1 - 3a-5z - a-5z3 + 2a-5z5 + a-5z7 - 3a-3z-3 - 4a-3z-1 + 2a-3z + 4a-3z3 + 3a-3z5 + a-3z7 + a-1z-3 + 2a-1z-1 - 2a-1z3 - a-1z5

Kauffman Polynomial:

- a-11z5 + 4a-10z4 - 5a-10z6 - 6a-9z3 + 15a-9z5 - 11a-9z7 - a-8z2 - 3a-8z4 + 16a-8z6 - 13a-8z8 - a-7z-3 + 2a-7z-1 + 3a-7z - 20a-7z3 + 31a-7z5 - 6a-7z7 - 8a-7z9 + 3a-6z-2 - 4a-6 - 3a-6z2 - 14a-6z4 + 43a-6z6 - 22a-6z8 - 2a-6z10 - 3a-5z-3 + 3a-5z-1 + 11a-5z - 36a-5z3 + 35a-5z5 + 5a-5z7 - 13a-5z9 + 6a-4z-2 - 7a-4 - 4a-4z2 - 9a-4z4 + 32a-4z6 - 15a-4z8 - 2a-4z10 - 3a-3z-3 + 3a-3z-1 + 11a-3z - 32a-3z3 + 30a-3z5 - 4a-3z7 - 5a-3z9 + 3a-2z-2 - 4a-2 - 3a-2z2 + 9a-2z6 - 6a-2z8 - a-1z-3 + 2a-1z-1 + 3a-1z - 10a-1z3 + 10a-1z5 - 4a-1z7 - z2 + 2z4 - z6

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 20                       1 j = 18                     4   j = 16                   7 1   j = 14                 10 4     j = 12               14 7       j = 10             13 12         j = 8           14 12           j = 6         10 17             j = 4       7 10               j = 2     3 12                 j = 0   1 5                   j = -2   3                     j = -4 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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 542]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 542]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 6, 13, 5], X[8, 4, 9, 3], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[22, 20, 15, 19], X[14, 22, 11, 21], X[20, 14, 21, 13], > X[18, 10, 19, 9], X[10, 12, 5, 11], X[4, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11}, {10, -2, 8, -7}, {2, -1, 5, -3, 9, -10}, > {4, -5, 11, -9, 6, -8, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 4 3/2 5/2 7/2 9/2 q - ------- + 8 Sqrt[q] - 17 q + 20 q - 27 q + 25 q - Sqrt[q] 11/2 13/2 15/2 17/2 19/2 > 24 q + 17 q - 11 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 2 2 4 6 8 10 12 14 16 -1 - q + -- + 2 q + 9 q + 4 q + 16 q + 11 q + 12 q + 14 q + 4 q + 2 q 18 20 24 26 28 > 10 q - q + 2 q - 3 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 542]][a, z]
Out[8]=	3 1 3 3 1 2 4 2 z 3 z 2 z z -(-----) + ----- - ----- + ---- + ---- - ---- + --- + -- - --- + --- - -- - 7 3 5 3 3 3 3 5 3 a z 7 5 3 7 a z a z a z a z a z a z a a a a 3 3 3 5 5 5 5 7 7 z 4 z 2 z z 2 z 3 z z z z > -- + ---- - ---- - -- + ---- + ---- - -- + -- + -- 5 3 a 7 5 3 a 5 3 a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 542]][a, z]
Out[9]=	-4 7 4 1 3 3 1 3 6 3 2 -- - -- - -- - ----- - ----- - ----- - ---- + ----- + ----- + ----- + ---- + 6 4 2 7 3 5 3 3 3 3 6 2 4 2 2 2 7 a a a a z a z a z a z a z a z a z a z 2 2 2 3 3 2 3 z 11 z 11 z 3 z 2 z 3 z 4 z > ---- + ---- + --- + --- + ---- + ---- + --- - z - -- - ---- - ---- - 5 3 a z 7 5 3 a 8 6 4 a z a z a a a a a a 2 3 3 3 3 3 4 4 4 3 z 6 z 20 z 36 z 32 z 10 z 4 4 z 3 z 14 z > ---- - ---- - ----- - ----- - ----- - ----- + 2 z + ---- - ---- - ----- - 2 9 7 5 3 a 10 8 6 a a a a a a a a 4 5 5 5 5 5 5 6 6 9 z z 15 z 31 z 35 z 30 z 10 z 6 5 z 16 z > ---- - --- + ----- + ----- + ----- + ----- + ----- - z - ---- + ----- + 4 11 9 7 5 3 a 10 8 a a a a a a a a 6 6 6 7 7 7 7 7 8 8 43 z 32 z 9 z 11 z 6 z 5 z 4 z 4 z 13 z 22 z > ----- + ----- + ---- - ----- - ---- + ---- - ---- - ---- - ----- - ----- - 6 4 2 9 7 5 3 a 8 6 a a a a a a a a a 8 8 9 9 9 10 10 15 z 6 z 8 z 13 z 5 z 2 z 2 z > ----- - ---- - ---- - ----- - ---- - ----- - ----- 4 2 7 5 3 6 4 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 -2 3 5 3 q 4 6 6 2 12 q + 7 q + ----- + t + ----- + - + ---- + 10 q t + 10 q t + 17 q t + 4 3 2 2 t t q t q t 8 2 8 3 10 3 10 4 12 4 12 5 > 14 q t + 12 q t + 13 q t + 12 q t + 14 q t + 7 q t + 14 5 14 6 16 6 16 7 18 7 20 8 > 10 q t + 4 q t + 7 q t + q t + 4 q t + q t

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