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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-17/2 + 3q-15/2 - 7q-13/2 + 11q-11/2 - 15q-9/2 + 18q-7/2 - 19q-5/2 + 16q-3/2 - 13q-1/2 + 8q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant:

q-28 + 2q-26 - q-24 + 2q-20 - 4q-18 + q-16 + q-14 - 2q-12 + 4q-10 + 4q-6 + q-4 - 2q-2 + 4 - 3q2 + 2q6 - q8

HOMFLY-PT Polynomial:

a-1z3 - az5 - 2a3z-1 - 6a3z - 4a3z3 - 2a3z5 + 4a5z-1 + 8a5z + 5a5z3 - 3a7z-1 - 4a7z + a9z-1

Kauffman Polynomial:

- a-2z4 + 2a-1z3 - 4a-1z5 - z2 + 7z4 - 8z6 + 2az - 8az3 + 15az5 - 11az7 - a2 + 2a2z2 - 4a2z4 + 13a2z6 - 10a2z8 - 2a3z-1 + 15a3z - 38a3z3 + 39a3z5 - 7a3z7 - 5a3z9 - 2a4 + 15a4z2 - 45a4z4 + 52a4z6 - 17a4z8 - a4z10 - 4a5z-1 + 25a5z - 50a5z3 + 30a5z5 + 9a5z7 - 8a5z9 - 3a6 + 18a6z2 - 46a6z4 + 42a6z6 - 10a6z8 - a6z10 - 3a7z-1 + 16a7z - 28a7z3 + 14a7z5 + 4a7z7 - 3a7z9 - a8 + 6a8z2 - 13a8z4 + 11a8z6 - 3a8z8 - a9z-1 + 4a9z - 6a9z3 + 4a9z5 - a9z7

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 = 6                       1 j = 4                     3   j = 2                   5 1   j = 0                 8 3     j = -2               9 6       j = -4             10 7         j = -6           8 9           j = -8         7 10             j = -10       5 9               j = -12     2 6                 j = -14   1 5                   j = -16   2                     j = -18 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, 74]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 74]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 8, 17, 7], X[22, 13, 5, 14], > X[20, 18, 21, 17], X[18, 9, 19, 10], X[8, 19, 9, 20], X[14, 21, 15, 22], > X[10, 16, 11, 15], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -7, 6, -9, 11, -2, 4, -8, 9, -3, 5, -6, > 7, -5, 8, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 7 11 15 18 19 16 13 -q + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 5/2 > 8 Sqrt[q] - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 2 -24 2 4 -16 -14 2 4 4 -4 2 4 + q + --- - q + --- - --- + q + q - --- + --- + -- + q - -- - 26 20 18 12 10 6 2 q q q q q q q 2 6 8 > 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 74]][a, z]
Out[8]=	3 5 7 9 3 -2 a 4 a 3 a a 3 5 7 z 3 3 5 3 ----- + ---- - ---- + -- - 6 a z + 8 a z - 4 a z + -- - 4 a z + 5 a z - z z z z a 5 3 5 > a z - 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 74]][a, z]
Out[9]=	3 5 7 9 2 4 6 8 2 a 4 a 3 a a 3 5 -a - 2 a - 3 a - a - ---- - ---- - ---- - -- + 2 a z + 15 a z + 25 a z + z z z z 3 7 9 2 2 2 4 2 6 2 8 2 2 z > 16 a z + 4 a z - z + 2 a z + 15 a z + 18 a z + 6 a z + ---- - a 4 3 3 3 5 3 7 3 9 3 4 z 2 4 > 8 a z - 38 a z - 50 a z - 28 a z - 6 a z + 7 z - -- - 4 a z - 2 a 5 4 4 6 4 8 4 4 z 5 3 5 5 5 > 45 a z - 46 a z - 13 a z - ---- + 15 a z + 39 a z + 30 a z + a 7 5 9 5 6 2 6 4 6 6 6 8 6 > 14 a z + 4 a z - 8 z + 13 a z + 52 a z + 42 a z + 11 a z - 7 3 7 5 7 7 7 9 7 2 8 4 8 > 11 a z - 7 a z + 9 a z + 4 a z - a z - 10 a z - 17 a z - 6 8 8 8 3 9 5 9 7 9 4 10 6 10 > 10 a z - 3 a z - 5 a z - 8 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 2 1 5 2 6 5 8 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 18 8 16 7 14 7 14 6 12 6 12 5 10 5 q q t q t q t q t q t q t q t 9 7 10 8 9 10 7 9 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 3 t + 10 4 8 4 8 3 6 3 6 2 4 2 4 2 q t q t q t q t q t q t q t q t 2 2 2 4 2 6 3 > 5 q t + q t + 3 q t + q t

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