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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-27/2 - 4q-25/2 + 9q-23/2 - 17q-21/2 + 21q-19/2 - 27q-17/2 + 25q-15/2 - 24q-13/2 + 16q-11/2 - 11q-9/2 + 4q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-42 + 4q-38 + 6q-34 + 12q-32 + 6q-30 + 15q-28 + 9q-26 + 9q-24 + 10q-22 + q-20 + 9q-18 - q-16 + 4q-12 - 3q-10 + q-8

HOMFLY-PT Polynomial:

- a5z3 - a5z5 - a7z-3 - 5a7z-1 - 11a7z - 11a7z3 - 4a7z5 + 3a9z-3 + 10a9z-1 + 11a9z + a9z3 - 2a9z5 - 3a11z-3 - 5a11z-1 + a11z + 3a11z3 + a13z-3 - a13z

Kauffman Polynomial:

a5z3 - a5z5 + 3a6z4 - 4a6z6 + a7z-3 - 5a7z-1 + 11a7z - 17a7z3 + 17a7z5 - 10a7z7 - 3a8z-2 + 10a8 - 12a8z2 - a8z4 + 15a8z6 - 12a8z8 + 3a9z-3 - 12a9z-1 + 23a9z - 32a9z3 + 30a9z5 - 4a9z7 - 8a9z9 - 6a10z-2 + 19a10 - 15a10z2 - 22a10z4 + 50a10z6 - 23a10z8 - 2a10z10 + 3a11z-3 - 12a11z-1 + 21a11z - 31a11z3 + 26a11z5 + 10a11z7 - 14a11z9 - 3a12z-2 + 10a12 - 3a12z2 - 26a12z4 + 45a12z6 - 18a12z8 - 2a12z10 + a13z-3 - 5a13z-1 + 12a13z - 25a13z3 + 23a13z5 - 6a13z9 - a14z2 - 6a14z4 + 13a14z6 - 7a14z8 + 3a15z - 8a15z3 + 9a15z5 - 4a15z7 - a16z2 + 2a16z4 - a16z6

Khovanov Homology:

trqj r = -11 r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                       1 j = -6                     4 1 j = -8                   7     j = -10                 9 4     j = -12               15 7       j = -14             12 11         j = -16           15 13           j = -18         10 16             j = -20       7 11               j = -22     3 11                 j = -24   1 6                   j = -26   3                     j = -28 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, 547]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 547]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[20, 7, 21, 8], X[10, 19, 5, 20], > X[22, 13, 19, 14], X[18, 21, 11, 22], X[14, 17, 15, 18], X[16, 9, 17, 10], > X[8, 15, 9, 16], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {4, -3, 6, -5}, {10, -1, 3, -9, 8, -4}, > {11, -2, 5, -7, 9, -8, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) 4 9 17 21 27 25 24 16 q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ----- - 25/2 23/2 21/2 19/2 17/2 15/2 13/2 11/2 q q q q q q q q 11 4 -(5/2) > ---- + ---- - q 9/2 7/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 4 6 12 6 15 9 9 10 -20 9 -16 -q + --- + --- + --- + --- + --- + --- + --- + --- + q + --- - q + 38 34 32 30 28 26 24 22 18 q q q q q q q q q 4 3 -8 > --- - --- + q 12 10 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 547]][a, z]
Out[8]=	7 9 11 13 7 9 11 a 3 a 3 a a 5 a 10 a 5 a 7 9 11 -(--) + ---- - ----- + --- - ---- + ----- - ----- - 11 a z + 11 a z + a z - 3 3 3 3 z z z z z z z 13 5 3 7 3 9 3 11 3 5 5 7 5 9 5 > a z - a z - 11 a z + a z + 3 a z - a z - 4 a z - 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 547]][a, z]
Out[9]=	7 9 11 13 8 10 12 8 10 12 a 3 a 3 a a 3 a 6 a 3 a 10 a + 19 a + 10 a + -- + ---- + ----- + --- - ---- - ----- - ----- - 3 3 3 3 2 2 2 z z z z z z z 7 9 11 13 5 a 12 a 12 a 5 a 7 9 11 13 > ---- - ----- - ------ - ----- + 11 a z + 23 a z + 21 a z + 12 a z + z z z z 15 8 2 10 2 12 2 14 2 16 2 5 3 > 3 a z - 12 a z - 15 a z - 3 a z - a z - a z + a z - 7 3 9 3 11 3 13 3 15 3 6 4 8 4 > 17 a z - 32 a z - 31 a z - 25 a z - 8 a z + 3 a z - a z - 10 4 12 4 14 4 16 4 5 5 7 5 9 5 > 22 a z - 26 a z - 6 a z + 2 a z - a z + 17 a z + 30 a z + 11 5 13 5 15 5 6 6 8 6 10 6 > 26 a z + 23 a z + 9 a z - 4 a z + 15 a z + 50 a z + 12 6 14 6 16 6 7 7 9 7 11 7 > 45 a z + 13 a z - a z - 10 a z - 4 a z + 10 a z - 15 7 8 8 10 8 12 8 14 8 9 9 > 4 a z - 12 a z - 23 a z - 18 a z - 7 a z - 8 a z - 11 9 13 9 10 10 12 10 > 14 a z - 6 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 3 1 6 3 11 7 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 28 11 26 10 24 10 24 9 22 9 22 8 20 8 q t q t q t q t q t q t q t 11 10 16 15 13 12 11 15 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 20 7 18 7 18 6 16 6 16 5 14 5 14 4 12 4 q t q t q t q t q t q t q t q t 7 9 4 7 4 > ------ + ------ + ------ + ----- + ---- 12 3 10 3 10 2 8 2 6 q t q t q t q t q t

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