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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X3849 X17,22,18,19 X11,20,12,21 X19,12,20,13 X21,18,22,5 X9,16,10,17 X13,2,14,3

Gauss Code:

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

Jones Polynomial:

- q-12 + 2q-11 - 3q-10 + 5q-9 - 5q-8 + 6q-7 - 3q-6 + 4q-5 - 2q-4 + q-3

A2 (sl(3)) Invariant:

- q-42 - 2q-40 - q-38 - 2q-36 + 2q-32 + 4q-30 + 6q-28 + 4q-26 + 6q-24 + 3q-22 + 3q-20 + 3q-18 + q-16 + q-14 - q-12 + q-10

HOMFLY-PT Polynomial:

a6 + 3a6z2 + 4a6z4 + a6z6 + 2a8z-2 + 8a8 + 14a8z2 + 10a8z4 + 2a8z6 - 5a10z-2 - 15a10 - 14a10z2 - 3a10z4 + 4a12z-2 + 6a12 + a12z2 - a14z-2

Kauffman Polynomial:

- a6 + 3a6z2 - 4a6z4 + a6z6 + 3a7z3 - 7a7z5 + 2a7z7 - 2a8z-2 + 9a8 - 16a8z2 + 20a8z4 - 14a8z6 + 3a8z8 + 5a9z-1 - 20a9z + 30a9z3 - 16a9z5 + a9z9 - 5a10z-2 + 21a10 - 40a10z2 + 47a10z4 - 26a10z6 + 5a10z8 + 9a11z-1 - 33a11z + 40a11z3 - 14a11z5 - a11z7 + a11z9 - 4a12z-2 + 14a12 - 25a12z2 + 25a12z4 - 11a12z6 + 2a12z8 + 5a13z-1 - 15a13z + 14a13z3 - 5a13z5 + a13z7 - a14z-2 + 2a14 - 4a14z2 + 2a14z4 + a15z-1 - 2a15z + a15z3

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -5                   1 j = -7                 2 1 j = -9               2     j = -11           1 2 2     j = -13         1 6 2       j = -15         2 3         j = -17       4 4           j = -19   1 1 2             j = -21   2 3               j = -23   1                 j = -25 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 375]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 375]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[5, 10, 6, 11], > X[3, 8, 4, 9], X[17, 22, 18, 19], X[11, 20, 12, 21], X[19, 12, 20, 13], > X[21, 18, 22, 5], X[9, 16, 10, 17], X[13, 2, 14, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -3}, {-8, 7, -9, 6}, > {-4, -1, 2, 5, -10, 4, -7, 8, -11, -2, 3, 10, -6, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-12 2 3 5 5 6 3 4 2 -3 -q + --- - --- + -- - -- + -- - -- + -- - -- + q 11 10 9 8 7 6 5 4 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 2 -38 2 2 4 6 4 6 3 3 3 -q - --- - q - --- + --- + --- + --- + --- + --- + --- + --- + --- + 40 36 32 30 28 26 24 22 20 18 q q q q q q q q q q -16 -14 -12 -10 > q + q - q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 375]][a, z]
Out[8]=	8 10 12 14 6 8 10 12 2 a 5 a 4 a a 6 2 8 2 a + 8 a - 15 a + 6 a + ---- - ----- + ----- - --- + 3 a z + 14 a z - 2 2 2 2 z z z z 10 2 12 2 6 4 8 4 10 4 6 6 8 6 > 14 a z + a z + 4 a z + 10 a z - 3 a z + a z + 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 375]][a, z]
Out[9]=	8 10 12 14 9 6 8 10 12 14 2 a 5 a 4 a a 5 a -a + 9 a + 21 a + 14 a + 2 a - ---- - ----- - ----- - --- + ---- + 2 2 2 2 z z z z z 11 13 15 9 a 5 a a 9 11 13 15 6 2 > ----- + ----- + --- - 20 a z - 33 a z - 15 a z - 2 a z + 3 a z - z z z 8 2 10 2 12 2 14 2 7 3 9 3 > 16 a z - 40 a z - 25 a z - 4 a z + 3 a z + 30 a z + 11 3 13 3 15 3 6 4 8 4 10 4 > 40 a z + 14 a z + a z - 4 a z + 20 a z + 47 a z + 12 4 14 4 7 5 9 5 11 5 13 5 6 6 > 25 a z + 2 a z - 7 a z - 16 a z - 14 a z - 5 a z + a z - 8 6 10 6 12 6 7 7 11 7 13 7 8 8 > 14 a z - 26 a z - 11 a z + 2 a z - a z + a z + 3 a z + 10 8 12 8 9 9 11 9 > 5 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-7 -5 1 1 2 1 3 1 2 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 25 9 23 8 21 8 19 8 21 7 19 7 19 6 q t q t q t q t q t q t q t 4 4 2 1 3 6 1 2 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 17 6 17 5 15 5 13 5 15 4 13 4 11 4 13 3 q t q t q t q t q t q t q t q t 2 2 2 2 > ------ + ------ + ----- + ---- 11 3 11 2 9 2 7 q t q t q t q t

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