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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-11/2 - 4q-9/2 + 8q-7/2 - 13q-5/2 + 16q-3/2 - 18q-1/2 + 17q1/2 - 15q3/2 + 10q5/2 - 6q7/2 + 3q9/2 - q11/2

A2 (sl(3)) Invariant:

- q-16 + 2q-14 - q-12 + q-10 + 3q-8 - 2q-6 + 4q-4 - 2q-2 + 1 + q2 - 2q4 + 4q6 - q8 + q10 + q12 - q14 + q16

HOMFLY-PT Polynomial:

- a-3z-1 - 2a-3z - 3a-3z3 - a-3z5 + 2a-1z-1 + 5a-1z + 6a-1z3 + 4a-1z5 + a-1z7 - 2az-1 - 3az + az3 + 3az5 + az7 + a3z-1 - 2a3z3 - a3z5

Kauffman Polynomial:

- 4a-5z3 + 4a-5z5 - a-5z7 + 4a-4z2 - 14a-4z4 + 12a-4z6 - 3a-4z8 + a-3z-1 - 5a-3z + 14a-3z3 - 20a-3z5 + 15a-3z7 - 4a-3z9 + 5a-2z2 - 16a-2z4 + 14a-2z6 - 2a-2z10 + 2a-1z-1 - 12a-1z + 29a-1z3 - 37a-1z5 + 31a-1z7 - 10a-1z9 + 1 + z2 - 14z4 + 24z6 - 7z8 - 2z10 + 2az-1 - 8az + 3az3 + 6az5 + 4az7 - 6az9 - a2z2 - 4a2z4 + 14a2z6 - 10a2z8 + a3z-1 - a3z - 6a3z3 + 15a3z5 - 11a3z7 - a4z2 + 7a4z4 - 8a4z6 + 2a5z3 - 4a5z5 - a6z4

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 = 12                       1 j = 10                     2   j = 8                   4 1   j = 6                 6 2     j = 4               9 4       j = 2             8 6         j = 0           10 9           j = -2         8 10             j = -4       5 8               j = -6     3 8                 j = -8   1 5                   j = -10   3                     j = -12 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, 47]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 47]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[18, 9, 19, 10], X[8, 17, 9, 18], > X[4, 19, 1, 20], X[12, 6, 13, 5], X[10, 4, 11, 3], X[22, 14, 5, 13], > X[14, 22, 15, 21], X[20, 12, 21, 11], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 7, -5}, {6, -1, 2, -4, 3, -7, 10, -6, 8, -9, 11, -2, 4, -3, > 5, -10, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 4 8 13 16 18 3/2 q - ---- + ---- - ---- + ---- - ------- + 17 Sqrt[q] - 15 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 10 q - 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 -12 -10 3 2 4 2 2 4 6 8 1 - q + --- - q + q + -- - -- + -- - -- + q - 2 q + 4 q - q + 14 8 6 4 2 q q q q q 10 12 14 16 > q + q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 47]][a, z]
Out[8]=	3 3 3 1 2 2 a a 2 z 5 z 3 z 6 z 3 3 3 -(----) + --- - --- + -- - --- + --- - 3 a z - ---- + ---- + a z - 2 a z - 3 a z z z 3 a 3 a a z a a 5 5 7 z 4 z 5 3 5 z 7 > -- + ---- + 3 a z - a z + -- + a z 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 47]][a, z]
Out[9]=	3 2 2 1 2 2 a a 5 z 12 z 3 2 4 z 5 z 1 + ---- + --- + --- + -- - --- - ---- - 8 a z - a z + z + ---- + ---- - 3 a z z z 3 a 4 2 a z a a a 3 3 3 2 2 4 2 4 z 14 z 29 z 3 3 3 5 3 4 > a z - a z - ---- + ----- + ----- + 3 a z - 6 a z + 2 a z - 14 z - 5 3 a a a 4 4 5 5 5 14 z 16 z 2 4 4 4 6 4 4 z 20 z 37 z 5 > ----- - ----- - 4 a z + 7 a z - a z + ---- - ----- - ----- + 6 a z + 4 2 5 3 a a a a a 6 6 7 3 5 5 5 6 12 z 14 z 2 6 4 6 z > 15 a z - 4 a z + 24 z + ----- + ----- + 14 a z - 8 a z - -- + 4 2 5 a a a 7 7 8 9 9 15 z 31 z 7 3 7 8 3 z 2 8 4 z 10 z > ----- + ----- + 4 a z - 11 a z - 7 z - ---- - 10 a z - ---- - ----- - 3 a 4 3 a a a a 10 9 10 2 z > 6 a z - 2 z - ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 3 1 5 3 8 5 8 10 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 8 2 2 2 4 2 4 3 6 3 6 4 > ---- + 9 t + 8 q t + 6 q t + 9 q t + 4 q t + 6 q t + 2 q t + 2 q t 8 4 8 5 10 5 12 6 > 4 q t + q t + 2 q t + q t

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