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DrawMorseLink

PD Presentation:

X8192 X14,3,15,4 X12,15,7,16 X10,19,11,20 X16,9,17,10 X20,11,13,12 X18,5,19,6 X2738 X4,13,5,14 X6,17,1,18

Gauss Code:

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

Jones Polynomial:

q-12 - 3q-11 + 7q-10 - 9q-9 + 13q-8 - 13q-7 + 13q-6 - 10q-5 + 7q-4 - 3q-3 + q-2

A2 (sl(3)) Invariant:

q-38 + 2q-36 + 4q-32 + 4q-30 + q-28 + 6q-26 + 2q-24 + 3q-22 + 2q-20 - q-18 + 3q-16 - 2q-14 + q-12 + 2q-10 - 2q-8 + q-6

HOMFLY-PT Polynomial:

a4z2 + a4z4 + a6 + 5a6z2 + 3a6z4 + a8z-2 + 4a8 + 6a8z2 + 3a8z4 - 2a10z-2 - 6a10 - 4a10z2 + a12z-2 + a12

Kauffman Polynomial:

- a4z2 + a4z4 - 2a5z3 + 3a5z5 - a6 + 6a6z2 - 8a6z4 + 6a6z6 + 5a7z3 - 9a7z5 + 7a7z7 - a8z-2 + 5a8 - 10a8z2 + 10a8z4 - 9a8z6 + 6a8z8 + 2a9z-1 - 9a9z + 19a9z3 - 22a9z5 + 7a9z7 + 2a9z9 - 2a10z-2 + 11a10 - 23a10z2 + 26a10z4 - 25a10z6 + 10a10z8 + 2a11z-1 - 9a11z + 17a11z3 - 18a11z5 + 3a11z7 + 2a11z9 - a12z-2 + 5a12 - 3a12z2 + 4a12z4 - 9a12z6 + 4a12z8 + 5a13z3 - 8a13z5 + 3a13z7 - a14 + 3a14z2 - 3a14z4 + a14z6

Khovanov Homology:

trqj r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -3                     1 j = -5                   3 1 j = -7                 4     j = -9               6 3     j = -11             7 4       j = -13           6 6         j = -15         7 7           j = -17       4 8             j = -19     3 5               j = -21   1 5                 j = -23   2                   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[10, Alternating, 164]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 164]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 3, 15, 4], X[12, 15, 7, 16], X[10, 19, 11, 20], > X[16, 9, 17, 10], X[20, 11, 13, 12], X[18, 5, 19, 6], X[2, 7, 3, 8], > X[4, 13, 5, 14], X[6, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9, 7, -10}, {8, -1, 5, -4, 6, -3}, > {9, -2, 3, -5, 10, -7, 4, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-12 3 7 9 13 13 13 10 7 3 -2 q - --- + --- - -- + -- - -- + -- - -- + -- - -- + q 11 10 9 8 7 6 5 4 3 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-38 2 4 4 -28 6 2 3 2 -18 3 2 q + --- + --- + --- + q + --- + --- + --- + --- - q + --- - --- + 36 32 30 26 24 22 20 16 14 q q q q q q q q q -12 2 2 -6 > q + --- - -- + q 10 8 q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 164]][a, z]
Out[8]=	8 10 12 6 8 10 12 a 2 a a 4 2 6 2 8 2 a + 4 a - 6 a + a + -- - ----- + --- + a z + 5 a z + 6 a z - 2 2 2 z z z 10 2 4 4 6 4 8 4 > 4 a z + a z + 3 a z + 3 a z
In[9]:=	Kauffman[Link[10, Alternating, 164]][a, z]
Out[9]=	8 10 12 9 11 6 8 10 12 14 a 2 a a 2 a 2 a 9 -a + 5 a + 11 a + 5 a - a - -- - ----- - --- + ---- + ----- - 9 a z - 2 2 2 z z z z z 11 4 2 6 2 8 2 10 2 12 2 14 2 > 9 a z - a z + 6 a z - 10 a z - 23 a z - 3 a z + 3 a z - 5 3 7 3 9 3 11 3 13 3 4 4 6 4 > 2 a z + 5 a z + 19 a z + 17 a z + 5 a z + a z - 8 a z + 8 4 10 4 12 4 14 4 5 5 7 5 9 5 > 10 a z + 26 a z + 4 a z - 3 a z + 3 a z - 9 a z - 22 a z - 11 5 13 5 6 6 8 6 10 6 12 6 14 6 > 18 a z - 8 a z + 6 a z - 9 a z - 25 a z - 9 a z + a z + 7 7 9 7 11 7 13 7 8 8 10 8 12 8 > 7 a z + 7 a z + 3 a z + 3 a z + 6 a z + 10 a z + 4 a z + 9 9 11 9 > 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-5 -3 1 2 1 5 3 5 4 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 25 10 23 9 21 9 21 8 19 8 19 7 17 7 q t q t q t q t q t q t q t 8 7 7 6 6 7 4 6 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 17 6 15 6 15 5 13 5 13 4 11 4 11 3 9 3 q t q t q t q t q t q t q t q t 3 4 3 > ----- + ----- + ---- 9 2 7 2 5 q t q t q t

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