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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-11/2 + q-9/2 - 5q-7/2 + 6q-5/2 - 12q-3/2 + 12q-1/2 - 15q1/2 + 13q3/2 - 11q5/2 + 7q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

q-18 + 3q-16 + 4q-14 + 7q-12 + 9q-10 + 8q-8 + 12q-6 + 8q-4 + 9q-2 + 9 + 3q2 + 6q4 - q6 + 2q8 + q10 - q12 + 2q14 - q16

HOMFLY-PT Polynomial:

2a-3z3 + a-3z5 - a-1z-3 - 4a-1z-1 - 6a-1z - 6a-1z3 - 4a-1z5 - a-1z7 + 3az-3 + 11az-1 + 16az + 12az3 + 3az5 - 3a3z-3 - 10a3z-1 - 11a3z - 3a3z3 + a5z-3 + 3a5z-1 + a5z

Kauffman Polynomial:

- a-6z4 + 3a-5z3 - 4a-5z5 + 7a-4z4 - 7a-4z6 + a-3z-1 - 3a-3z + 9a-3z5 - 8a-3z7 - a-2 - 4a-2z4 + 10a-2z6 - 7a-2z8 + a-1z-3 - 3a-1z-1 + 3a-1z + 4a-1z7 - 4a-1z9 - 3z-2 + 11 - 16z2 + z4 + 13z6 - 5z8 - z10 + 3az-3 - 12az-1 + 21az - 16az3 - 2az5 + 12az7 - 5az9 - 6a2z-2 + 24a2 - 33a2z2 + 19a2z4 - 2a2z6 + a2z8 - a2z10 + 3a3z-3 - 14a3z-1 + 28a3z - 32a3z3 + 17a3z5 - a3z7 - a3z9 - 3a4z-2 + 13a4 - 17a4z2 + 6a4z4 + 2a4z6 - a4z8 + a5z-3 - 6a5z-1 + 13a5z - 13a5z3 + 6a5z5 - 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                   4 1   j = 6                 7 3     j = 4               9 7       j = 2             6 4         j = 0           7 10           j = -2         5 5             j = -4       2 8               j = -6     3 4                 j = -8   1 5                   j = -10                         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, 535]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 535]]
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[16, 7, 17, 8], X[8, 15, 5, 16], X[22, 18, 15, 17], > X[20, 14, 21, 13], X[12, 20, 13, 19], X[14, 22, 9, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -10, 9, -11}, > {7, -6, 8, -3, 10, -9, 11, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) -(9/2) 5 6 12 12 3/2 -q + q - ---- + ---- - ---- + ------- - 15 Sqrt[q] + 13 q - 7/2 5/2 3/2 Sqrt[q] q q q 5/2 7/2 9/2 11/2 > 11 q + 7 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 3 4 7 9 8 12 8 9 2 4 6 9 + q + --- + --- + --- + --- + -- + -- + -- + -- + 3 q + 6 q - q + 16 14 12 10 8 6 4 2 q q q q q q q q 8 10 12 14 16 > 2 q + q - q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 535]][a, z]
Out[8]=	3 5 3 5 1 3 a 3 a a 4 11 a 10 a 3 a 6 z -(----) + --- - ---- + -- - --- + ---- - ----- + ---- - --- + 16 a z - 3 3 3 3 a z z z z a a z z z z 3 3 5 5 7 3 5 2 z 6 z 3 3 3 z 4 z 5 z > 11 a z + a z + ---- - ---- + 12 a z - 3 a z + -- - ---- + 3 a z - -- 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 535]][a, z]
Out[9]=	3 5 2 4 -2 2 4 1 3 a 3 a a 3 6 a 3 a 1 11 - a + 24 a + 13 a + ---- + --- + ---- + -- - -- - ---- - ---- + ---- - 3 3 3 3 2 2 2 3 a z z z z z z z a z 3 5 3 12 a 14 a 6 a 3 z 3 z 3 5 > --- - ---- - ----- - ---- - --- + --- + 21 a z + 28 a z + 13 a z - a z z z z 3 a a 3 2 2 2 4 2 3 z 3 3 3 5 3 4 > 16 z - 33 a z - 17 a z + ---- - 16 a z - 32 a z - 13 a z + z - 5 a 4 4 4 5 5 z 7 z 4 z 2 4 4 4 4 z 9 z 5 3 5 > -- + ---- - ---- + 19 a z + 6 a z - ---- + ---- - 2 a z + 17 a z + 6 4 2 5 3 a a a a a 6 6 7 7 5 5 6 7 z 10 z 2 6 4 6 8 z 4 z > 6 a z + 13 z - ---- + ----- - 2 a z + 2 a z - ---- + ---- + 4 2 3 a a a a 8 9 7 3 7 5 7 8 7 z 2 8 4 8 4 z 9 > 12 a z - a z - a z - 5 z - ---- + a z - a z - ---- - 5 a z - 2 a a 3 9 10 2 10 > a z - z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 5 3 4 2 8 5 10 + 6 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 8 5 8 4 6 4 6 3 4 3 4 2 2 2 q t q t q t q t q t q t q t q t 7 5 2 4 4 2 6 2 6 3 8 3 > - + ---- + 4 q t + 9 q t + 7 q t + 7 q t + 3 q t + 4 q t + t 2 q t 8 4 10 4 12 5 > q t + 3 q t + q t

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