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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-7/2 - 3q-5/2 + 4q-3/2 - 8q-1/2 + 10q1/2 - 12q3/2 + 12q5/2 - 11q7/2 + 9q9/2 - 6q11/2 + 3q13/2 - q15/2

A2 (sl(3)) Invariant:

- q-10 + q-8 + q-6 + 2q-4 + 4q-2 + 1 + 3q2 - q4 - 2q6 - 3q10 + 2q12 + q16 + q18 - q20 + q22

HOMFLY-PT Polynomial:

- a-5z-1 - 2a-5z - 3a-5z3 - a-5z5 + 3a-3z-1 + 5a-3z + 5a-3z3 + 4a-3z5 + a-3z7 - 4a-1z-1 - 3a-1z + 3a-1z3 + 4a-1z5 + a-1z7 + 2az-1 - 3az3 - az5

Kauffman Polynomial:

- a-9z3 - 3a-8z4 - a-7z + 4a-7z3 - 6a-7z5 + a-6 - 5a-6z2 + 13a-6z4 - 9a-6z6 - a-5z-1 + 2a-5z - 9a-5z3 + 20a-5z5 - 10a-5z7 + 3a-4 - 8a-4z2 + a-4z4 + 16a-4z6 - 8a-4z8 - 3a-3z-1 + 9a-3z - 16a-3z3 + 9a-3z5 + 9a-3z7 - 5a-3z9 + 3a-2 - 2a-2z2 - 17a-2z4 + 18a-2z6 - 2a-2z10 - 4a-1z-1 + 7a-1z + 16a-1z3 - 48a-1z5 + 36a-1z7 - 8a-1z9 + 2 + 3z2 - 9z4 - 2z6 + 7z8 - 2z10 - 2az-1 + az + 18az3 - 31az5 + 17az7 - 3az9 + 2a2z2 - 7a2z4 + 5a2z6 - a2z8

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 16                       1 j = 14                     2   j = 12                   4 1   j = 10                 5 2     j = 8               6 4       j = 6             6 5         j = 4           6 6           j = 2         6 8             j = 0       2 4               j = -2     2 6                 j = -4   1 2                   j = -6   2                     j = -8 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, 52]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 52]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[4, 19, 1, 20], X[14, 6, 15, 5], > X[10, 4, 11, 3], X[20, 12, 21, 11], X[22, 14, 5, 13], X[12, 22, 13, 21], > X[16, 9, 17, 10], X[2, 16, 3, 15], X[8, 17, 9, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {4, -1, 2, -11, 9, -5, 6, -8, 7, -4, 10, -9, 11, -2, > 3, -6, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 3 4 8 3/2 5/2 7/2 q - ---- + ---- - ------- + 10 Sqrt[q] - 12 q + 12 q - 11 q + 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 9 q - 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 -6 2 4 2 4 6 10 12 16 18 1 - q + q + q + -- + -- + 3 q - q - 2 q - 3 q + 2 q + q + q - 4 2 q q 20 22 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 52]][a, z]
Out[8]=	3 3 3 1 3 4 2 a 2 z 5 z 3 z 3 z 5 z 3 z 3 -(----) + ---- - --- + --- - --- + --- - --- - ---- + ---- + ---- - 3 a z - 5 3 a z z 5 3 a 5 3 a a z a z a a a a 5 5 5 7 7 z 4 z 4 z 5 z z > -- + ---- + ---- - a z + -- + -- 5 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 52]][a, z]
Out[9]=	-6 3 3 1 3 4 2 a z 2 z 9 z 7 z 2 + a + -- + -- - ---- - ---- - --- - --- - -- + --- + --- + --- + a z + 4 2 5 3 a z z 7 5 3 a a a a z a z a a a 2 2 2 3 3 3 3 3 2 5 z 8 z 2 z 2 2 z 4 z 9 z 16 z 16 z > 3 z - ---- - ---- - ---- + 2 a z - -- + ---- - ---- - ----- + ----- + 6 4 2 9 7 5 3 a a a a a a a a 4 4 4 4 5 5 3 4 3 z 13 z z 17 z 2 4 6 z 20 z > 18 a z - 9 z - ---- + ----- + -- - ----- - 7 a z - ---- + ----- + 8 6 4 2 7 5 a a a a a a 5 5 6 6 6 7 9 z 48 z 5 6 9 z 16 z 18 z 2 6 10 z > ---- - ----- - 31 a z - 2 z - ---- + ----- + ----- + 5 a z - ----- + 3 a 6 4 2 5 a a a a a 7 7 8 9 9 9 z 36 z 7 8 8 z 2 8 5 z 8 z 9 > ---- + ----- + 17 a z + 7 z - ---- - a z - ---- - ---- - 3 a z - 3 a 4 3 a a a a 10 10 2 z > 2 z - ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 2 1 2 2 2 6 4 6 q 8 q + 6 q + ----- + ----- + ----- + ----- + ----- + -- + ----- + - + ---- + 8 5 6 4 4 4 4 3 2 3 2 2 2 t t q t q t q t q t q t t q t 4 6 6 2 8 2 8 3 10 3 10 4 > 6 q t + 6 q t + 5 q t + 6 q t + 4 q t + 5 q t + 2 q t + 12 4 12 5 14 5 16 6 > 4 q t + q t + 2 q t + q t

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