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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X7,16,8,17 X17,22,18,5 X13,18,14,19 X21,14,22,15 X9,20,10,21 X15,8,16,9 X19,10,20,11 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-25/2 - q-23/2 + q-21/2 + q-19/2 - 3q-17/2 + 3q-15/2 - 5q-13/2 + 4q-11/2 - 4q-9/2 + 2q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-40 - 2q-38 - q-36 - 2q-34 + q-28 + 3q-26 + q-24 + 4q-22 + q-20 + 2q-18 + 2q-16 + q-12 - q-10 + q-8

HOMFLY-PT Polynomial:

- a5z - 3a5z3 - a5z5 - 2a7z-1 - 7a7z - 8a7z3 - 2a7z5 + 2a9z-1 + 4a9z + a9z3 + a11z-1 + 2a11z - a13z-1

Kauffman Polynomial:

- a5z + 3a5z3 - a5z5 - a6z2 + 5a6z4 - 2a6z6 - 2a7z-1 + 9a7z - 13a7z3 + 11a7z5 - 3a7z7 + 2a8 - 5a8z2 + 4a8z4 + a8z6 - a8z8 - 2a9z-1 + 12a9z - 21a9z3 + 13a9z5 - 3a9z7 - 4a10 + 17a10z2 - 27a10z4 + 13a10z6 - 2a10z8 + a11z-1 + a11z + 2a11z3 - 12a11z5 + 7a11z7 - a11z9 - 9a12 + 34a12z2 - 41a12z4 + 17a12z6 - 2a12z8 + a13z-1 - a13z + 7a13z3 - 13a13z5 + 7a13z7 - a13z9 - 4a14 + 13a14z2 - 15a14z4 + 7a14z6 - a14z8

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                     2 1 j = -8                   2     j = -10                 2 2     j = -12             1 4 2       j = -14             1 3         j = -16         1 4 3           j = -18       1   1             j = -20       1 3               j = -22   1 1                   j = -24                         j = -26 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 103]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 103]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[7, 16, 8, 17], X[17, 22, 18, 5], > X[13, 18, 14, 19], X[21, 14, 22, 15], X[9, 20, 10, 21], X[15, 8, 16, 9], > X[19, 10, 20, 11], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 8, -7, 9, 11, -2, -5, 6, -8, 3, -4, 5, > -9, 7, -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) -(23/2) -(21/2) -(19/2) 3 3 5 4 q - q + q + q - ----- + ----- - ----- + ----- - 17/2 15/2 13/2 11/2 q q q q 4 2 -(5/2) > ---- + ---- - q 9/2 7/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 2 -36 2 -28 3 -24 4 -20 2 2 -12 -q - --- - q - --- + q + --- + q + --- + q + --- + --- + q - 38 34 26 22 18 16 q q q q q q -10 -8 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 103]][a, z]
Out[8]=	7 9 11 13 -2 a 2 a a a 5 7 9 11 5 3 ----- + ---- + --- - --- - a z - 7 a z + 4 a z + 2 a z - 3 a z - z z z z 7 3 9 3 5 5 7 5 > 8 a z + a z - a z - 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 103]][a, z]
Out[9]=	7 9 11 13 8 10 12 14 2 a 2 a a a 5 7 2 a - 4 a - 9 a - 4 a - ---- - ---- + --- + --- - a z + 9 a z + z z z z 9 11 13 6 2 8 2 10 2 12 2 > 12 a z + a z - a z - a z - 5 a z + 17 a z + 34 a z + 14 2 5 3 7 3 9 3 11 3 13 3 6 4 > 13 a z + 3 a z - 13 a z - 21 a z + 2 a z + 7 a z + 5 a z + 8 4 10 4 12 4 14 4 5 5 7 5 9 5 > 4 a z - 27 a z - 41 a z - 15 a z - a z + 11 a z + 13 a z - 11 5 13 5 6 6 8 6 10 6 12 6 > 12 a z - 13 a z - 2 a z + a z + 13 a z + 17 a z + 14 6 7 7 9 7 11 7 13 7 8 8 10 8 > 7 a z - 3 a z - 3 a z + 7 a z + 7 a z - a z - 2 a z - 12 8 14 8 11 9 13 9 > 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 1 1 1 3 1 q + q + ------- + ------- + ------ + ------ + ------ + ------ + ------ + 26 11 22 10 22 9 20 8 18 8 20 7 16 7 q t q t q t q t q t q t q t 1 4 3 1 1 3 4 2 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 6 16 6 16 5 14 5 12 5 14 4 12 4 12 3 q t q t q t q t q t q t q t q t 2 2 2 2 > ------ + ------ + ----- + ---- 10 3 10 2 8 2 6 q t q t q t q t

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