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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 2q-9/2 - 7q-7/2 + 10q-5/2 - 18q-3/2 + 19q-1/2 - 22q1/2 + 18q3/2 - 16q5/2 + 10q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

q-18 + 2q-16 + 2q-14 + 7q-12 + 8q-10 + 7q-8 + 15q-6 + 9q-4 + 11q-2 + 10 + 3q2 + 8q4 - 2q6 + 3q8 - 4q12 + 2q14 - q16

HOMFLY-PT Polynomial:

a-3z-1 + 3a-3z + 2a-3z3 + a-3z5 - a-1z-3 - 6a-1z-1 - 11a-1z - 9a-1z3 - 4a-1z5 - a-1z7 + 3az-3 + 11az-1 + 15az + 10az3 + 3az5 - 3a3z-3 - 8a3z-1 - 8a3z - 3a3z3 + a5z-3 + 2a5z-1 + a5z

Kauffman Polynomial:

- a-6z4 - 4a-5z5 + a-4 - 4a-4z2 + 8a-4z4 - 10a-4z6 - 2a-3z-1 + 9a-3z - 17a-3z3 + 25a-3z5 - 16a-3z7 + 6a-2 - 15a-2z2 + 6a-2z4 + 17a-2z6 - 14a-2z8 + a-1z-3 - 11a-1z-1 + 35a-1z - 61a-1z3 + 59a-1z5 - 13a-1z7 - 6a-1z9 - 3z-2 + 18 - 23z2 - 14z4 + 48z6 - 20z8 - z10 + 3az-3 - 18az-1 + 50az - 77az3 + 48az5 + 3az7 - 8az9 - 6a2z-2 + 21a2 - 19a2z2 - 14a2z4 + 27a2z6 - 8a2z8 - a2z10 + 3a3z-3 - 14a3z-1 + 34a3z - 43a3z3 + 23a3z5 - 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                   7 1   j = 6                 9 3     j = 4               9 7       j = 2             13 9         j = 0           12 15           j = -2         6 7             j = -4       4 12               j = -6     3 6                 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, 539]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 539]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[20, 14, 21, 13], X[16, 12, 17, 11], > X[12, 20, 13, 19], X[8, 16, 5, 15], X[14, 8, 15, 7], X[22, 17, 19, 18], > X[18, 21, 9, 22], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 7, -6}, {5, -3, 9, -8}, > {11, -2, 4, -5, 3, -7, 6, -4, 8, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 7 10 18 19 3/2 -q + ---- - ---- + ---- - ---- + ------- - 22 Sqrt[q] + 18 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 16 q + 10 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 2 7 8 7 15 9 11 2 4 6 10 + q + --- + --- + --- + --- + -- + -- + -- + -- + 3 q + 8 q - 2 q + 16 14 12 10 8 6 4 2 q q q q q q q q 8 12 14 16 > 3 q - 4 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 539]][a, z]
Out[8]=	3 5 3 5 1 3 a 3 a a 1 6 11 a 8 a 2 a 3 z 11 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 9 z 3 3 3 z 4 z > 15 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, 539]][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 9 z 35 z 3 > ---- - --- - ---- - ----- - ---- + --- + ---- + 50 a z + 34 a z + 3 a z z z z 3 a a z a 2 2 3 3 5 2 4 z 15 z 2 2 4 2 17 z 61 z > 10 a z - 23 z - ---- - ----- - 19 a z - 7 a z - ----- - ----- - 4 2 3 a a a a 4 4 4 3 3 3 5 3 4 z 8 z 6 z 2 4 > 77 a z - 43 a z - 10 a z - 14 z - -- + ---- + ---- - 14 a z - 6 4 2 a a a 5 5 5 4 4 4 z 25 z 59 z 5 3 5 5 5 6 > 3 a z - ---- + ----- + ----- + 48 a z + 23 a z + 5 a z + 48 z - 5 3 a a a 6 6 7 7 10 z 17 z 2 6 4 6 16 z 13 z 7 3 7 > ----- + ----- + 27 a z + 6 a z - ----- - ----- + 3 a z - a z - 4 2 3 a a a a 8 9 5 7 8 14 z 2 8 4 8 6 z 9 3 9 10 > a z - 20 z - ----- - 8 a z - 2 a z - ---- - 8 a z - 2 a z - z - 2 a a 2 10 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 6 3 6 4 12 15 + 13 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 6 12 7 2 4 4 2 6 2 6 3 > ----- + -- + ---- + 9 q t + 9 q t + 7 q t + 9 q t + 3 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 7 q t + q t + 3 q t + q t

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