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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-14 + 2q-13 - 6q-12 + 9q-11 - 12q-10 + 15q-9 - 14q-8 + 14q-7 - 8q-6 + 7q-5 - 3q-4 + q-3

A2 (sl(3)) Invariant:

- q-44 - 3q-42 - 3q-40 - 4q-38 - 5q-36 + 6q-30 + 8q-28 + 6q-26 + 10q-24 + 3q-22 + 6q-20 + 3q-18 + 2q-14 - 2q-12 + q-10

HOMFLY-PT Polynomial:

a6z2 + 3a6z4 + a6z6 + 3a8z-2 + 13a8 + 21a8z2 + 14a8z4 + 3a8z6 - 8a10z-2 - 24a10 - 23a10z2 - 6a10z4 + 7a12z-2 + 12a12 + 4a12z2 - 2a14z-2 - a14

Kauffman Polynomial:

a6z2 - 3a6z4 + a6z6 + 2a7z3 - 8a7z5 + 3a7z7 - 3a8z-2 + 13a8 - 24a8z2 + 31a8z4 - 23a8z6 + 6a8z8 + 8a9z-1 - 24a9z + 27a9z3 - 7a9z5 - 8a9z7 + 4a9z9 - 8a10z-2 + 28a10 - 51a10z2 + 62a10z4 - 38a10z6 + 8a10z8 + a10z10 + 15a11z-1 - 45a11z + 46a11z3 - 8a11z5 - 11a11z7 + 6a11z9 - 7a12z-2 + 22a12 - 35a12z2 + 33a12z4 - 18a12z6 + 5a12z8 + a12z10 + 7a13z-1 - 21a13z + 22a13z3 - 13a13z5 + 3a13z7 + 2a13z9 - 2a14z-2 + 7a14 - 9a14z2 + 2a14z4 - 2a14z6 + 3a14z8 - a15z-1 + 3a15z - 2a15z3 - 3a15z5 + 3a15z7 + a16 - 3a16z4 + 2a16z6 - a17z-1 + 3a17z - 3a17z3 + a17z5

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 = -5                       1 j = -7                     3 1 j = -9                   4     j = -11                 4 3     j = -13               10 4       j = -15             7 7         j = -17           8 7           j = -19         4 7             j = -21       5 8               j = -23     1 4                 j = -25   1 5                   j = -27   1                     j = -29 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, Alternating, 391]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 391]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 7, 17, 8], X[8, 15, 5, 16], > X[18, 11, 19, 12], X[22, 17, 9, 18], X[20, 13, 21, 14], X[12, 19, 13, 20], > X[14, 21, 15, 22], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -4}, > {11, -2, 5, -8, 7, -9, 4, -3, 6, -5, 8, -7, 9, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-14 2 6 9 12 15 14 14 8 7 3 -3 -q + --- - --- + --- - --- + -- - -- + -- - -- + -- - -- + q 13 12 11 10 9 8 7 6 5 4 q q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-44 3 3 4 5 6 8 6 10 3 6 3 2 -q - --- - --- - --- - --- + --- + --- + --- + --- + --- + --- + --- + --- - 42 40 38 36 30 28 26 24 22 20 18 14 q q q q q q q q q q q q 2 -10 > --- + q 12 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 391]][a, z]
Out[8]=	8 10 12 14 8 10 12 14 3 a 8 a 7 a 2 a 6 2 13 a - 24 a + 12 a - a + ---- - ----- + ----- - ----- + a z + 2 2 2 2 z z z z 8 2 10 2 12 2 6 4 8 4 10 4 6 6 > 21 a z - 23 a z + 4 a z + 3 a z + 14 a z - 6 a z + a z + 8 6 > 3 a z
In[9]:=	Kauffman[Link[11, Alternating, 391]][a, z]
Out[9]=	8 10 12 14 9 8 10 12 14 16 3 a 8 a 7 a 2 a 8 a 13 a + 28 a + 22 a + 7 a + a - ---- - ----- - ----- - ----- + ---- + 2 2 2 2 z z z z z 11 13 15 17 15 a 7 a a a 9 11 13 15 > ------ + ----- - --- - --- - 24 a z - 45 a z - 21 a z + 3 a z + z z z z 17 6 2 8 2 10 2 12 2 14 2 7 3 > 3 a z + a z - 24 a z - 51 a z - 35 a z - 9 a z + 2 a z + 9 3 11 3 13 3 15 3 17 3 6 4 > 27 a z + 46 a z + 22 a z - 2 a z - 3 a z - 3 a z + 8 4 10 4 12 4 14 4 16 4 7 5 > 31 a z + 62 a z + 33 a z + 2 a z - 3 a z - 8 a z - 9 5 11 5 13 5 15 5 17 5 6 6 8 6 > 7 a z - 8 a z - 13 a z - 3 a z + a z + a z - 23 a z - 10 6 12 6 14 6 16 6 7 7 9 7 > 38 a z - 18 a z - 2 a z + 2 a z + 3 a z - 8 a z - 11 7 13 7 15 7 8 8 10 8 12 8 > 11 a z + 3 a z + 3 a z + 6 a z + 8 a z + 5 a z + 14 8 9 9 11 9 13 9 10 10 12 10 > 3 a z + 4 a z + 6 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-7 -5 1 1 1 5 1 4 5 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 29 11 27 10 25 10 25 9 23 9 23 8 21 8 q t q t q t q t q t q t q t 8 4 7 8 7 7 7 10 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 21 7 19 7 19 6 17 6 17 5 15 5 15 4 13 4 q t q t q t q t q t q t q t q t 4 4 3 4 3 > ------ + ------ + ------ + ----- + ---- 13 3 11 3 11 2 9 2 7 q t q t q t q t q t

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