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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 7q-5/2 - 13q-3/2 + 17q-1/2 - 22q1/2 + 22q3/2 - 20q5/2 + 15q7/2 - 10q9/2 + 5q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-14 - 3q-8 + 4q-6 + q-2 + 7 - q2 + 5q4 - 2q6 + 2q10 - 4q12 + 3q14 - q16 - 2q18 + q20

HOMFLY-PT Polynomial:

- a-5z3 + a-3z-1 + 3a-3z + 4a-3z3 + 2a-3z5 - 4a-1z-1 - 9a-1z - 9a-1z3 - 4a-1z5 - a-1z7 + 4az-1 + 8az + 6az3 + 2az5 - a3z-1 - 2a3z - a3z3

Kauffman Polynomial:

- a-7z5 + 5a-6z4 - 5a-6z6 - 5a-5z3 + 15a-5z5 - 10a-5z7 + a-4 - 5a-4z2 + 4a-4z4 + 10a-4z6 - 10a-4z8 - a-3z-1 + 6a-3z - 23a-3z3 + 35a-3z5 - 11a-3z7 - 5a-3z9 + 4a-2 - 12a-2z2 - 6a-2z4 + 32a-2z6 - 18a-2z8 - a-2z10 - 4a-1z-1 + 17a-1z - 38a-1z3 + 36a-1z5 - 3a-1z7 - 8a-1z9 + 7 - 14z2 - 3z4 + 23z6 - 12z8 - z10 - 4az-1 + 15az - 28az3 + 25az5 - 5az7 - 3az9 + 4a2 - 10a2z2 + 5a2z4 + 5a2z6 - 4a2z8 - a3z-1 + 4a3z - 8a3z3 + 8a3z5 - 3a3z7 + a4 - 3a4z2 + 3a4z4 - a4z6

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 = 14                       1 j = 12                     4   j = 10                   6 1   j = 8                 9 4     j = 6               11 6       j = 4             11 9         j = 2           11 11           j = 0         8 13             j = -2       5 9               j = -4     2 8                 j = -6   1 5                   j = -8   2                     j = -10 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, 78]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 78]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 12, 15, 11], X[22, 15, 5, 16], > X[16, 7, 17, 8], X[8, 21, 9, 22], X[18, 10, 19, 9], X[20, 18, 21, 17], > X[10, 20, 11, 19], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 5, -6, 7, -9, 3, -2, 11, -3, 4, -5, 8, -7, > 9, -8, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 7 13 17 3/2 5/2 q - ---- + ---- - ---- + ------- - 22 Sqrt[q] + 22 q - 20 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 15 q - 10 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 3 4 -2 2 4 6 10 12 14 16 7 - q - -- + -- + q - q + 5 q - 2 q + 2 q - 4 q + 3 q - q - 8 6 q q 18 20 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 78]][a, z]
Out[8]=	3 3 3 3 1 4 4 a a 3 z 9 z 3 z 4 z 9 z ---- - --- + --- - -- + --- - --- + 8 a z - 2 a z - -- + ---- - ---- + 3 a z z z 3 a 5 3 a a z a a a 5 5 7 3 3 3 2 z 4 z 5 z > 6 a z - a z + ---- - ---- + 2 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 78]][a, z]
Out[9]=	3 -4 4 2 4 1 4 4 a a 6 z 17 z 7 + a + -- + 4 a + a - ---- - --- - --- - -- + --- + ---- + 15 a z + 2 3 a z z z 3 a a a z a 2 2 3 3 3 3 2 5 z 12 z 2 2 4 2 5 z 23 z 38 z > 4 a z - 14 z - ---- - ----- - 10 a z - 3 a z - ---- - ----- - ----- - 4 2 5 3 a a a a a 4 4 4 5 3 3 3 4 5 z 4 z 6 z 2 4 4 4 z > 28 a z - 8 a z - 3 z + ---- + ---- - ---- + 5 a z + 3 a z - -- + 6 4 2 7 a a a a 5 5 5 6 6 6 15 z 35 z 36 z 5 3 5 6 5 z 10 z 32 z > ----- + ----- + ----- + 25 a z + 8 a z + 23 z - ---- + ----- + ----- + 5 3 a 6 4 2 a a a a a 7 7 7 8 2 6 4 6 10 z 11 z 3 z 7 3 7 8 10 z > 5 a z - a z - ----- - ----- - ---- - 5 a z - 3 a z - 12 z - ----- - 5 3 a 4 a a a 8 9 9 10 18 z 2 8 5 z 8 z 9 10 z > ----- - 4 a z - ---- - ---- - 3 a z - z - --- 2 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 5 2 8 5 8 13 + 11 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t q t q t q t q t q t q t q t 9 2 4 4 2 6 2 6 3 8 3 > ---- + 11 q t + 11 q t + 9 q t + 11 q t + 6 q t + 9 q t + 2 q t 8 4 10 4 10 5 12 5 14 6 > 4 q t + 6 q t + q t + 4 q t + q t

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