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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 2q-9/2 - 7q-7/2 + 10q-5/2 - 17q-3/2 + 17q-1/2 - 21q1/2 + 17q3/2 - 14q5/2 + 9q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

q-18 + 2q-16 + 2q-14 + 7q-12 + 8q-10 + 7q-8 + 14q-6 + 9q-4 + 12q-2 + 11 + 4q2 + 7q4 - 3q6 + 2q8 - 3q12 + 2q14 - q16

HOMFLY-PT Polynomial:

a-3z-1 + 2a-3z + 2a-3z3 + a-3z5 - a-1z-3 - 6a-1z-1 - 9a-1z - 8a-1z3 - 4a-1z5 - a-1z7 + 3az-3 + 11az-1 + 14az + 10az3 + 3az5 - 3a3z-3 - 8a3z-1 - 8a3z - 3a3z3 + a5z-3 + 2a5z-1 + a5z

Kauffman Polynomial:

- a-6z4 + a-5z3 - 4a-5z5 + a-4 - 3a-4z2 + 8a-4z4 - 9a-4z6 - 2a-3z-1 + 6a-3z - 13a-3z3 + 19a-3z5 - 13a-3z7 + 6a-2 - 14a-2z2 + 10a-2z4 + 10a-2z6 - 11a-2z8 + a-1z-3 - 11a-1z-1 + 30a-1z - 53a-1z3 + 52a-1z5 - 12a-1z7 - 5a-1z9 - 3z-2 + 18 - 28z2 - 2z4 + 36z6 - 16z8 - z10 + 3az-3 - 18az-1 + 49az - 74az3 + 48az5 + az7 - 7az9 - 6a2z-2 + 21a2 - 24a2z2 - 6a2z4 + 23a2z6 - 7a2z8 - a2z10 + 3a3z-3 - 14a3z-1 + 35a3z - 45a3z3 + 24a3z5 - a3z7 - 2a3z9 - 3a4z-2 + 9a4 - 7a4z2 - 3a4z4 + 6a4z6 - 2a4z8 + a5z-3 - 5a5z-1 + 10a5z - 10a5z3 + 5a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                       1 j = 10                     3   j = 8                   6 1   j = 6                 8 3     j = 4               10 7       j = 2             11 7         j = 0           9 13           j = -2         8 8             j = -4       5 12               j = -6     2 5                 j = -8   1 6                   j = -10   1                     j = -12 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, 537]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 537]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[18, 12, 19, 11], X[10, 3, 11, 4], > X[4, 9, 1, 10], X[8, 18, 5, 17], X[16, 8, 17, 7], X[14, 16, 9, 15], > X[22, 20, 15, 19], X[20, 13, 21, 14], X[12, 21, 13, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 4, -5}, {2, -1, 7, -6}, {5, -4, 3, -11, 10, -8}, > {8, -7, 6, -3, 9, -10, 11, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 7 10 17 17 3/2 -q + ---- - ---- + ---- - ---- + ------- - 21 Sqrt[q] + 17 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 14 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 2 7 8 7 14 9 12 2 4 6 11 + q + --- + --- + --- + --- + -- + -- + -- + -- + 4 q + 7 q - 3 q + 16 14 12 10 8 6 4 2 q q q q q q q q 8 12 14 16 > 2 q - 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 537]][a, z]
Out[8]=	3 5 3 5 1 3 a 3 a a 1 6 11 a 8 a 2 a 2 z 9 z -(----) + --- - ---- + -- + ---- - --- + ---- - ---- + ---- + --- - --- + 3 3 3 3 3 a z z z z 3 a a z z z z a z a 3 3 5 5 3 5 2 z 8 z 3 3 3 z 4 z > 14 a z - 8 a z + a z + ---- - ---- + 10 a z - 3 a z + -- - ---- + 3 a 3 a a a 7 5 z > 3 a z - -- a
In[9]:=	Kauffman[Link[11, Alternating, 537]][a, z]
Out[9]=	3 5 2 4 -4 6 2 4 1 3 a 3 a a 3 6 a 3 a 18 + a + -- + 21 a + 9 a + ---- + --- + ---- + -- - -- - ---- - ---- - 2 3 3 3 3 2 2 2 a a z z z z z z z 3 5 2 11 18 a 14 a 5 a 6 z 30 z 3 > ---- - --- - ---- - ----- - ---- + --- + ---- + 49 a z + 35 a z + 3 a z z z z 3 a a z a 2 2 3 3 3 5 2 3 z 14 z 2 2 4 2 z 13 z 53 z > 10 a z - 28 z - ---- - ----- - 24 a z - 7 a z + -- - ----- - ----- - 4 2 5 3 a a a a a 4 4 4 3 3 3 5 3 4 z 8 z 10 z 2 4 > 74 a z - 45 a z - 10 a z - 2 z - -- + ---- + ----- - 6 a z - 6 4 2 a a a 5 5 5 4 4 4 z 19 z 52 z 5 3 5 5 5 6 > 3 a z - ---- + ----- + ----- + 48 a z + 24 a z + 5 a z + 36 z - 5 3 a a a 6 6 7 7 9 z 10 z 2 6 4 6 13 z 12 z 7 3 7 5 7 > ---- + ----- + 23 a z + 6 a z - ----- - ----- + a z - a z - a z - 4 2 3 a a a a 8 9 8 11 z 2 8 4 8 5 z 9 3 9 10 2 10 > 16 z - ----- - 7 a z - 2 a z - ---- - 7 a z - 2 a z - z - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 6 2 5 5 12 13 + 11 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 8 9 8 2 4 4 2 6 2 6 3 > ----- + - + ---- + 7 q t + 10 q t + 7 q t + 8 q t + 3 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 6 q t + q t + 3 q t + q t

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