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DrawMorseLink

PD Presentation:

X6172 X14,4,15,3 X16,5,17,6 X18,14,19,13 X12,15,5,16 X10,19,11,20 X8,21,9,22 X20,7,21,8 X22,9,13,10 X2,11,3,12 X4,18,1,17

Gauss Code:

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

Jones Polynomial:

q-10 - 4q-9 + 10q-8 - 14q-7 + 19q-6 - 21q-5 + 21q-4 - 17q-3 + 13q-2 - 7q-1 + 4 - q

A2 (sl(3)) Invariant:

q-30 - q-28 + q-26 + 4q-24 + 7q-20 + 2q-18 + 3q-16 + 4q-14 - 3q-12 + 5q-10 - 2q-8 + 3q-6 + 3q-4 - q-2 + 2 - q2

HOMFLY-PT Polynomial:

2a2 - a2z2 - 3a2z4 - a2z6 + a4z-2 + a4 + 6a4z2 + 9a4z4 + 5a4z6 + a4z8 - 2a6z-2 - 5a6 - 8a6z2 - 7a6z4 - 2a6z6 + a8z-2 + 2a8 + 2a8z2 + a8z4

Kauffman Polynomial:

2az3 - 3az5 + az7 - 2a2 - 4a2z2 + 17a2z4 - 15a2z6 + 4a2z8 + a3z + 11a3z5 - 15a3z7 + 5a3z9 - a4z-2 + 3a4 - 14a4z2 + 42a4z4 - 38a4z6 + 7a4z8 + 2a4z10 + 2a5z-1 - 8a5z + 14a5z3 + 4a5z5 - 26a5z7 + 12a5z9 - 2a6z-2 + 9a6 - 22a6z2 + 42a6z4 - 50a6z6 + 16a6z8 + 2a6z10 + 2a7z-1 - 8a7z + 22a7z3 - 32a7z5 + 4a7z7 + 7a7z9 - a8z-2 + 3a8 - 6a8z2 + 6a8z4 - 17a8z6 + 13a8z8 + a9z + 6a9z3 - 18a9z5 + 14a9z7 - 2a10 + 6a10z2 - 10a10z4 + 10a10z6 + 4a11z5 + a12z4

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 3                       1 j = 1                     3   j = -1                   4 1   j = -3                 9 3     j = -5               10 6       j = -7             11 7         j = -9           10 10           j = -11         9 11             j = -13       6 11               j = -15     4 8                 j = -17     6                   j = -19 1 4                     j = -21 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, 462]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 462]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[16, 5, 17, 6], X[18, 14, 19, 13], > X[12, 15, 5, 16], X[10, 19, 11, 20], X[8, 21, 9, 22], X[20, 7, 21, 8], > X[22, 9, 13, 10], X[2, 11, 3, 12], X[4, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {3, -1, 8, -7, 9, -6, 10, -5}, > {4, -2, 5, -3, 11, -4, 6, -8, 7, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-10 4 10 14 19 21 21 17 13 7 4 + q - -- + -- - -- + -- - -- + -- - -- + -- - - - q 9 8 7 6 5 4 3 2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 -26 4 7 2 3 4 3 5 2 3 2 + q - q + q + --- + --- + --- + --- + --- - --- + --- - -- + -- + 24 20 18 16 14 12 10 8 6 q q q q q q q q q 3 -2 2 > -- - q - q 4 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 462]][a, z]
Out[8]=	4 6 8 2 4 6 8 a 2 a a 2 2 4 2 6 2 2 a + a - 5 a + 2 a + -- - ---- + -- - a z + 6 a z - 8 a z + 2 2 2 z z z 8 2 2 4 4 4 6 4 8 4 2 6 4 6 6 6 > 2 a z - 3 a z + 9 a z - 7 a z + a z - a z + 5 a z - 2 a z + 4 8 > a z
In[9]:=	Kauffman[Link[11, Alternating, 462]][a, z]
Out[9]=	4 6 8 5 7 2 4 6 8 10 a 2 a a 2 a 2 a 3 -2 a + 3 a + 9 a + 3 a - 2 a - -- - ---- - -- + ---- + ---- + a z - 2 2 2 z z z z z 5 7 9 2 2 4 2 6 2 8 2 > 8 a z - 8 a z + a z - 4 a z - 14 a z - 22 a z - 6 a z + 10 2 3 5 3 7 3 9 3 2 4 4 4 > 6 a z + 2 a z + 14 a z + 22 a z + 6 a z + 17 a z + 42 a z + 6 4 8 4 10 4 12 4 5 3 5 5 5 > 42 a z + 6 a z - 10 a z + a z - 3 a z + 11 a z + 4 a z - 7 5 9 5 11 5 2 6 4 6 6 6 > 32 a z - 18 a z + 4 a z - 15 a z - 38 a z - 50 a z - 8 6 10 6 7 3 7 5 7 7 7 9 7 > 17 a z + 10 a z + a z - 15 a z - 26 a z + 4 a z + 14 a z + 2 8 4 8 6 8 8 8 3 9 5 9 7 9 > 4 a z + 7 a z + 16 a z + 13 a z + 5 a z + 12 a z + 7 a z + 4 10 6 10 > 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 9 1 1 4 6 4 8 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 8 19 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 11 9 11 10 10 11 7 10 3 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + --- + 13 4 11 4 11 3 9 3 9 2 7 2 7 5 3 q t q t q t q t q t q t q t q t q 2 4 t t 2 3 3 > --- + -- + 3 q t + q t q q

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