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DrawMorseLink

PD Presentation:

X8192 X14,3,15,4 X12,13,7,14 X16,7,17,8 X22,17,13,18 X18,5,19,6 X20,9,21,10 X10,19,11,20 X4,11,5,12 X2,15,3,16 X6,21,1,22

Gauss Code:

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

Jones Polynomial:

- q-13 + 5q-12 - 10q-11 + 17q-10 - 22q-9 + 26q-8 - 25q-7 + 23q-6 - 16q-5 + 10q-4 - 4q-3 + q-2

A2 (sl(3)) Invariant:

- q-40 + 3q-38 + 4q-36 - 3q-34 + 5q-32 + 2q-30 - q-28 + 8q-26 + q-24 + 6q-22 + q-20 - q-18 + 5q-16 - 5q-14 + 2q-12 + 3q-10 - 3q-8 + q-6

HOMFLY-PT Polynomial:

a4z4 + 4a6z2 + 4a6z4 + a8z-2 + 7a8 + 11a8z2 + 6a8z4 - 2a10z-2 - 9a10 - 6a10z2 + a10z4 + a12z-2 + 2a12 - a12z2

Kauffman Polynomial:

a4z4 + 4a5z5 + 4a6z2 - 8a6z4 + 10a6z6 + 8a7z3 - 20a7z5 + 16a7z7 - a8z-2 + 7a8 - 19a8z2 + 28a8z4 - 35a8z6 + 19a8z8 + 2a9z-1 - 9a9z + 19a9z3 - 17a9z5 - 13a9z7 + 13a9z9 - 2a10z-2 + 11a10 - 33a10z2 + 70a10z4 - 76a10z6 + 20a10z8 + 4a10z10 + 2a11z-1 - 9a11z + 9a11z3 + 26a11z5 - 53a11z7 + 21a11z9 - a12z-2 + 3a12 - 11a12z2 + 46a12z4 - 46a12z6 + 6a12z8 + 4a12z10 - a13z3 + 17a13z5 - 23a13z7 + 8a13z9 - 2a14 - a14z2 + 13a14z4 - 15a14z6 + 5a14z8 + a15z3 - 2a15z5 + 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                 10 4     j = -11               13 6       j = -13             12 10         j = -15           14 13           j = -17         10 14             j = -19       7 12               j = -21     4 11                 j = -23   1 6                   j = -25   4                     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, 529]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 529]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 3, 15, 4], X[12, 13, 7, 14], X[16, 7, 17, 8], > X[22, 17, 13, 18], X[18, 5, 19, 6], X[20, 9, 21, 10], X[10, 19, 11, 20], > X[4, 11, 5, 12], X[2, 15, 3, 16], X[6, 21, 1, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -9, 6, -11}, {4, -1, 7, -8, 9, -3}, > {3, -2, 10, -4, 5, -6, 8, -7, 11, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-13 5 10 17 22 26 25 23 16 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]=	-40 3 4 3 5 2 -28 8 -24 6 -20 -18 -q + --- + --- - --- + --- + --- - q + --- + q + --- + q - q + 38 36 34 32 30 26 22 q q q q q q q 5 5 2 3 3 -6 > --- - --- + --- + --- - -- + q 16 14 12 10 8 q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 529]][a, z]
Out[8]=	8 10 12 8 10 12 a 2 a a 6 2 8 2 10 2 7 a - 9 a + 2 a + -- - ----- + --- + 4 a z + 11 a z - 6 a z - 2 2 2 z z z 12 2 4 4 6 4 8 4 10 4 > a z + a z + 4 a z + 6 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 529]][a, z]
Out[9]=	8 10 12 9 11 8 10 12 14 a 2 a a 2 a 2 a 9 7 a + 11 a + 3 a - 2 a - -- - ----- - --- + ---- + ----- - 9 a z - 2 2 2 z z z z z 11 6 2 8 2 10 2 12 2 14 2 7 3 > 9 a z + 4 a z - 19 a z - 33 a z - 11 a z - a z + 8 a z + 9 3 11 3 13 3 15 3 4 4 6 4 8 4 > 19 a z + 9 a z - a z + a z + a z - 8 a z + 28 a z + 10 4 12 4 14 4 5 5 7 5 9 5 > 70 a z + 46 a z + 13 a z + 4 a z - 20 a z - 17 a z + 11 5 13 5 15 5 6 6 8 6 10 6 > 26 a z + 17 a z - 2 a z + 10 a z - 35 a z - 76 a z - 12 6 14 6 7 7 9 7 11 7 13 7 > 46 a z - 15 a z + 16 a z - 13 a z - 53 a z - 23 a z + 15 7 8 8 10 8 12 8 14 8 9 9 > a z + 19 a z + 20 a z + 6 a z + 5 a z + 13 a z + 11 9 13 9 10 10 12 10 > 21 a z + 8 a z + 4 a z + 4 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-5 -3 1 4 1 6 4 11 7 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 12 10 14 14 13 12 10 13 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 10 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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