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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 X12,21,13,22 X20,13,21,14 X14,19,15,20 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-13 + 2q-12 - 6q-11 + 10q-10 - 14q-9 + 19q-8 - 18q-7 + 18q-6 - 13q-5 + 10q-4 - 4q-3 + q-2

A2 (sl(3)) Invariant:

- q-42 - 3q-40 - 2q-38 - 2q-36 - 5q-34 + 2q-32 + 3q-30 + q-28 + 9q-26 + 5q-24 + 7q-22 + 5q-20 + q-18 + 6q-16 - 2q-14 + 2q-12 + 3q-10 - 3q-8 + q-6

HOMFLY-PT Polynomial:

a4z4 + a6z-2 + 4a6 + 7a6z2 + 4a6z4 - a8z-2 - a8 + 2a8z2 + 3a8z4 - 2a10z-2 - 7a10 - 6a10z2 + 3a12z-2 + 4a12 - a14z-2

Kauffman Polynomial:

a4z4 + 4a5z5 + a6z-2 - 4a6 + 7a6z2 - 11a6z4 + 10a6z6 - 2a7z-1 + 4a7z + 2a7z3 - 16a7z5 + 13a7z7 + a8z-2 - 3a8 + 4a8z2 - 2a8z4 - 12a8z6 + 11a8z8 + 2a9z-1 - 10a9z + 26a9z3 - 35a9z5 + 8a9z7 + 5a9z9 - 2a10z-2 + 4a10 - 8a10z2 + 27a10z4 - 40a10z6 + 15a10z8 + a10z10 + 10a11z-1 - 34a11z + 50a11z3 - 27a11z5 - 7a11z7 + 7a11z9 - 3a12z-2 + 5a12 - 7a12z2 + 24a12z4 - 25a12z6 + 6a12z8 + a12z10 + 8a13z-1 - 27a13z + 35a13z3 - 17a13z5 - a13z7 + 2a13z9 - a14z-2 + a14 - 2a14z2 + 7a14z4 - 7a14z6 + 2a14z8 + 2a15z-1 - 7a15z + 9a15z3 - 5a15z5 + a15z7

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 = -3                       1 j = -5                     4 1 j = -7                   6     j = -9                 7 4     j = -11               11 6       j = -13             10 10         j = -15           9 8           j = -17         5 10             j = -19       5 9               j = -21     1 5                 j = -23   1 5                   j = -25   1                     j = -27 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, 392]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 392]]
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[12, 21, 13, 22], X[20, 13, 21, 14], > X[14, 19, 15, 20], 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, -7, 8, -9, 4, -3, 6, -5, 9, -8, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-13 2 6 10 14 19 18 18 13 10 4 -2 -q + --- - --- + --- - -- + -- - -- + -- - -- + -- - -- + q 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 3 2 2 5 2 3 -28 9 5 7 5 -q - --- - --- - --- - --- + --- + --- + q + --- + --- + --- + --- + 40 38 36 34 32 30 26 24 22 20 q q q q q q q q q q -18 6 2 2 3 3 -6 > q + --- - --- + --- + --- - -- + q 16 14 12 10 8 q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 392]][a, z]
Out[8]=	6 8 10 12 14 6 8 10 12 a a 2 a 3 a a 6 2 8 2 4 a - a - 7 a + 4 a + -- - -- - ----- + ----- - --- + 7 a z + 2 a z - 2 2 2 2 2 z z z z z 10 2 4 4 6 4 8 4 > 6 a z + a z + 4 a z + 3 a z
In[9]:=	Kauffman[Link[11, Alternating, 392]][a, z]
Out[9]=	6 8 10 12 14 7 6 8 10 12 14 a a 2 a 3 a a 2 a -4 a - 3 a + 4 a + 5 a + a + -- + -- - ----- - ----- - --- - ---- + 2 2 2 2 2 z z z z z z 9 11 13 15 2 a 10 a 8 a 2 a 7 9 11 13 > ---- + ------ + ----- + ----- + 4 a z - 10 a z - 34 a z - 27 a z - z z z z 15 6 2 8 2 10 2 12 2 14 2 7 3 > 7 a z + 7 a z + 4 a z - 8 a z - 7 a z - 2 a z + 2 a z + 9 3 11 3 13 3 15 3 4 4 6 4 8 4 > 26 a z + 50 a z + 35 a z + 9 a z + a z - 11 a z - 2 a z + 10 4 12 4 14 4 5 5 7 5 9 5 > 27 a z + 24 a z + 7 a z + 4 a z - 16 a z - 35 a z - 11 5 13 5 15 5 6 6 8 6 10 6 > 27 a z - 17 a z - 5 a z + 10 a z - 12 a z - 40 a z - 12 6 14 6 7 7 9 7 11 7 13 7 15 7 > 25 a z - 7 a z + 13 a z + 8 a z - 7 a z - a z + a z + 8 8 10 8 12 8 14 8 9 9 11 9 > 11 a z + 15 a z + 6 a z + 2 a z + 5 a z + 7 a z + 13 9 10 10 12 10 > 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-5 -3 1 1 1 5 1 5 5 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 27 11 25 10 23 10 23 9 21 9 21 8 19 8 q t q t q t q t q t q t q t 9 5 10 9 8 10 10 11 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 19 7 17 7 17 6 15 6 15 5 13 5 13 4 11 4 q t q t q t q t q t q t q t q t 6 7 4 6 4 > ------ + ----- + ----- + ----- + ---- 11 3 9 3 9 2 7 2 5 q t q t q t q t q t

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