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DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X14,17,15,18 X16,5,17,6 X4,15,5,16 X18,13,19,14 X22,19,7,20 X20,9,21,10 X10,21,11,22 X2738 X6,11,1,12

Gauss Code:

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

Jones Polynomial:

q-27/2 - 4q-25/2 + 8q-23/2 - 13q-21/2 + 18q-19/2 - 20q-17/2 + 20q-15/2 - 18q-13/2 + 12q-11/2 - 8q-9/2 + 3q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-42 + 3q-38 - 2q-36 + q-32 - 5q-30 + 2q-28 - q-26 + 2q-24 + 4q-22 - q-20 + 6q-18 - q-16 + 3q-12 - 2q-10 + q-8

HOMFLY-PT Polynomial:

- a5z - 2a5z3 - a5z5 - 2a7z-1 - 7a7z - 8a7z3 - 3a7z5 + 3a9z-1 + 3a9z - 2a9z3 - 2a9z5 - a11z-1 + 3a11z + 3a11z3 - a13z

Kauffman Polynomial:

- a5z + 2a5z3 - a5z5 - a6z2 + 4a6z4 - 3a6z6 - 2a7z-1 + 8a7z - 12a7z3 + 11a7z5 - 6a7z7 + 3a8 - 8a8z2 + 5a8z4 + 4a8z6 - 6a8z8 - 3a9z-1 + 10a9z - 19a9z3 + 19a9z5 - 5a9z7 - 4a9z9 + 3a10 - a10z2 - 17a10z4 + 29a10z6 - 14a10z8 - a10z10 - a11z-1 + 3a11z - 16a11z3 + 19a11z5 + 3a11z7 - 8a11z9 + a12 + 8a12z2 - 27a12z4 + 36a12z6 - 14a12z8 - a12z10 + 3a13z - 18a13z3 + 22a13z5 - 2a13z7 - 4a13z9 + a14z2 - 7a14z4 + 13a14z6 - 6a14z8 + a15z - 7a15z3 + 10a15z5 - 4a15z7 - a16z2 + 2a16z4 - a16z6

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 = -4                       1 j = -6                     3 1 j = -8                   5     j = -10                 7 3     j = -12               11 5       j = -14             9 7         j = -16           11 11           j = -18         8 10             j = -20       5 10               j = -22     3 8                 j = -24   1 5                   j = -26   3                     j = -28 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 216]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 216]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[14, 17, 15, 18], X[16, 5, 17, 6], > X[4, 15, 5, 16], X[18, 13, 19, 14], X[22, 19, 7, 20], X[20, 9, 21, 10], > X[10, 21, 11, 22], X[2, 7, 3, 8], X[6, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -5, 4, -11}, > {10, -1, 8, -9, 11, -2, 6, -3, 5, -4, 3, -6, 7, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) 4 8 13 18 20 20 18 12 q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ----- - 25/2 23/2 21/2 19/2 17/2 15/2 13/2 11/2 q q q q q q q q 8 3 -(5/2) > ---- + ---- - q 9/2 7/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 3 2 -32 5 2 -26 2 4 -20 6 -16 -q + --- - --- + q - --- + --- - q + --- + --- - q + --- - q + 38 36 30 28 24 22 18 q q q q q q q 3 2 -8 > --- - --- + q 12 10 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 216]][a, z]
Out[8]=	7 9 11 -2 a 3 a a 5 7 9 11 13 5 3 ----- + ---- - --- - a z - 7 a z + 3 a z + 3 a z - a z - 2 a z - z z z 7 3 9 3 11 3 5 5 7 5 9 5 > 8 a z - 2 a z + 3 a z - a z - 3 a z - 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 216]][a, z]
Out[9]=	7 9 11 8 10 12 2 a 3 a a 5 7 9 11 3 a + 3 a + a - ---- - ---- - --- - a z + 8 a z + 10 a z + 3 a z + z z z 13 15 6 2 8 2 10 2 12 2 14 2 16 2 > 3 a z + a z - a z - 8 a z - a z + 8 a z + a z - a z + 5 3 7 3 9 3 11 3 13 3 15 3 > 2 a z - 12 a z - 19 a z - 16 a z - 18 a z - 7 a z + 6 4 8 4 10 4 12 4 14 4 16 4 5 5 > 4 a z + 5 a z - 17 a z - 27 a z - 7 a z + 2 a z - a z + 7 5 9 5 11 5 13 5 15 5 6 6 > 11 a z + 19 a z + 19 a z + 22 a z + 10 a z - 3 a z + 8 6 10 6 12 6 14 6 16 6 7 7 9 7 > 4 a z + 29 a z + 36 a z + 13 a z - a z - 6 a z - 5 a z + 11 7 13 7 15 7 8 8 10 8 12 8 > 3 a z - 2 a z - 4 a z - 6 a z - 14 a z - 14 a z - 14 8 9 9 11 9 13 9 10 10 12 10 > 6 a z - 4 a z - 8 a z - 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 3 1 5 3 8 5 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 28 11 26 10 24 10 24 9 22 9 22 8 20 8 q t q t q t q t q t q t q t 10 8 10 11 11 9 7 11 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 20 7 18 7 18 6 16 6 16 5 14 5 14 4 12 4 q t q t q t q t q t q t q t q t 5 7 3 5 3 > ------ + ------ + ------ + ----- + ---- 12 3 10 3 10 2 8 2 6 q t q t q t q t q t

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