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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 8q-11/2 - 14q-9/2 + 18q-7/2 - 23q-5/2 + 22q-3/2 - 20q-1/2 + 15q1/2 - 9q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-24 - q-22 + q-20 - 3q-18 + 4q-14 - 2q-12 + 6q-10 + 2q-8 + 2q-6 + 4q-4 - 4q-2 + 4 - 4q2 - q4 + 3q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- 2a-1z - 2a-1z3 - a-1z5 + 5az + 8az3 + 4az5 + az7 - 2a3z-1 - 8a3z - 8a3z3 - 3a3z5 + 3a5z-1 + 5a5z + 3a5z3 - a7z-1 - a7z

Kauffman Polynomial:

a-3z3 - a-3z5 - 2a-2z2 + 5a-2z4 - 4a-2z6 + 3a-1z - 8a-1z3 + 12a-1z5 - 8a-1z7 - 2z4 + 10z6 - 9z8 + 7az - 28az3 + 35az5 - 10az7 - 5az9 + 6a2z2 - 28a2z4 + 43a2z6 - 20a2z8 - a2z10 - 2a3z-1 + 10a3z - 33a3z3 + 35a3z5 - a3z7 - 9a3z9 + 3a4 + 4a4z2 - 27a4z4 + 39a4z6 - 16a4z8 - a4z10 - 3a5z-1 + 9a5z - 20a5z3 + 20a5z5 - 2a5z7 - 4a5z9 + 3a6 - 3a6z2 - 3a6z4 + 9a6z6 - 5a6z8 - a7z-1 + 3a7z - 6a7z3 + 7a7z5 - 3a7z7 + a8 - 3a8z2 + 3a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                       1 j = 6                     3   j = 4                   6 1   j = 2                 9 3     j = 0               11 6       j = -2             12 10         j = -4           11 10           j = -6         8 13             j = -8       6 10               j = -10     2 8                 j = -12   1 6                   j = -14   2                     j = -16 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, 220]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 220]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[22, 10, 7, 9], X[10, 14, 11, 13], > X[16, 6, 17, 5], X[20, 15, 21, 16], X[18, 21, 19, 22], X[14, 19, 15, 20], > X[2, 7, 3, 8], X[4, 11, 5, 12], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 5, -11}, > {9, -1, 3, -4, 10, -2, 4, -8, 6, -5, 11, -7, 8, -6, 7, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 8 14 18 23 22 20 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 15 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -20 3 4 2 6 2 2 4 4 2 4 - q - q + q - --- + --- - --- + --- + -- + -- + -- - -- - 4 q - 18 14 12 10 8 6 4 2 q q q q q q q q 4 6 8 10 > q + 3 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 220]][a, z]
Out[8]=	3 5 7 3 -2 a 3 a a 2 z 3 5 7 2 z 3 ----- + ---- - -- - --- + 5 a z - 8 a z + 5 a z - a z - ---- + 8 a z - z z z a a 5 3 3 5 3 z 5 3 5 7 > 8 a z + 3 a z - -- + 4 a z - 3 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 220]][a, z]
Out[9]=	3 5 7 4 6 8 2 a 3 a a 3 z 3 5 7 3 a + 3 a + a - ---- - ---- - -- + --- + 7 a z + 10 a z + 9 a z + 3 a z - z z z a 2 3 3 2 z 2 2 4 2 6 2 8 2 z 8 z 3 > ---- + 6 a z + 4 a z - 3 a z - 3 a z + -- - ---- - 28 a z - 2 3 a a a 4 3 3 5 3 7 3 4 5 z 2 4 4 4 > 33 a z - 20 a z - 6 a z - 2 z + ---- - 28 a z - 27 a z - 2 a 5 5 6 4 8 4 z 12 z 5 3 5 5 5 7 5 > 3 a z + 3 a z - -- + ----- + 35 a z + 35 a z + 20 a z + 7 a z + 3 a a 6 7 6 4 z 2 6 4 6 6 6 8 6 8 z 7 > 10 z - ---- + 43 a z + 39 a z + 9 a z - a z - ---- - 10 a z - 2 a a 3 7 5 7 7 7 8 2 8 4 8 6 8 9 > a z - 2 a z - 3 a z - 9 z - 20 a z - 16 a z - 5 a z - 5 a z - 3 9 5 9 2 10 4 10 > 9 a z - 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 2 1 6 2 8 6 10 11 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 8 13 11 10 12 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 6 t + 9 q t + 3 q t + 6 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 3 q t + q t

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