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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9/2 - 4q-7/2 + 8q-5/2 - 14q-3/2 + 18q-1/2 - 22q1/2 + 22q3/2 - 19q5/2 + 14q7/2 - 9q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-14 + q-12 + q-10 - 2q-8 + 5q-6 + q-2 + 5 - 3q2 + 3q4 - 4q6 + 2q10 - 3q12 + 4q14 - q18 + q20

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-7z3 - a-7z5 - a-6z2 + 5a-6z4 - 4a-6z6 - a-5z-1 + 4a-5z - 8a-5z3 + 13a-5z5 - 8a-5z7 - a-4 + 4a-4z2 - 8a-4z4 + 13a-4z6 - 9a-4z8 - 3a-3z-1 + 18a-3z - 38a-3z3 + 35a-3z5 - 8a-3z7 - 5a-3z9 - 3a-2 + 17a-2z2 - 45a-2z4 + 50a-2z6 - 20a-2z8 - a-2z10 - 4a-1z-1 + 26a-1z - 58a-1z3 + 45a-1z5 - a-1z7 - 9a-1z9 - 2 + 14z2 - 39z4 + 46z6 - 17z8 - z10 - 2az-1 + 15az - 37az3 + 34az5 - 5az7 - 4az9 - a2 + a2z2 - 5a2z4 + 12a2z6 - 6a2z8 + 3a3z - 8a3z3 + 10a3z5 - 4a3z7 - a4z2 + 2a4z4 - 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                     3   j = 10                   6 1   j = 8                 8 3     j = 6               11 6       j = 4             11 8         j = 2           11 11           j = 0         9 13             j = -2       5 9               j = -4     3 9                 j = -6   1 5                   j = -8   3                     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, 55]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 55]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 8, 15, 7], X[22, 16, 5, 15], > X[16, 9, 17, 10], X[8, 21, 9, 22], X[20, 17, 21, 18], X[18, 12, 19, 11], > 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, 3, -6, 5, -9, 8, -2, 11, -3, 4, -5, 7, -8, > 9, -7, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 4 8 14 18 3/2 5/2 q - ---- + ---- - ---- + ------- - 22 Sqrt[q] + 22 q - 19 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 14 q - 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 -10 2 5 -2 2 4 6 10 12 5 - q + q + q - -- + -- + q - 3 q + 3 q - 4 q + 2 q - 3 q + 8 6 q q 14 18 20 > 4 q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 55]][a, z]
Out[8]=	3 3 3 1 3 4 2 a z 6 z 9 z 3 z 5 z 9 z -(----) + ---- - --- + --- - -- + --- - --- + 5 a z - a z - -- + ---- - ---- + 5 3 a z z 5 3 a 5 3 a a z a z a a a a 5 5 7 3 3 3 2 z 4 z 5 z > 5 a z - a z + ---- - ---- + 2 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 55]][a, z]
Out[9]=	-4 3 2 1 3 4 2 a 4 z 18 z 26 z -2 - a - -- - a - ---- - ---- - --- - --- + --- + ---- + ---- + 15 a z + 2 5 3 a z z 5 3 a a a z a z a a 2 2 2 3 3 3 3 2 z 4 z 17 z 2 2 4 2 z 8 z 38 z > 3 a z + 14 z - -- + ---- + ----- + a z - a z + -- - ---- - ----- - 6 4 2 7 5 3 a a a a a a 3 4 4 4 58 z 3 3 3 4 5 z 8 z 45 z 2 4 > ----- - 37 a z - 8 a z - 39 z + ---- - ---- - ----- - 5 a z + a 6 4 2 a a a 5 5 5 5 6 4 4 z 13 z 35 z 45 z 5 3 5 6 4 z > 2 a z - -- + ----- + ----- + ----- + 34 a z + 10 a z + 46 z - ---- + 7 5 3 a 6 a a a a 6 6 7 7 7 13 z 50 z 2 6 4 6 8 z 8 z z 7 3 7 > ----- + ----- + 12 a z - a z - ---- - ---- - -- - 5 a z - 4 a z - 4 2 5 3 a a a a a 8 8 9 9 10 8 9 z 20 z 2 8 5 z 9 z 9 10 z > 17 z - ---- - ----- - 6 a z - ---- - ---- - 4 a z - z - --- 4 2 3 a 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 5 3 9 5 9 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 + 8 q t + 11 q t + 6 q t + 8 q t + 2 q t 8 4 10 4 10 5 12 5 14 6 > 3 q t + 6 q t + q t + 3 q t + q t

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