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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 6q-5/2 - 11q-3/2 + 14q-1/2 - 18q1/2 + 18q3/2 - 16q5/2 + 12q7/2 - 8q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- a-5z3 - a-3z + a-3z5 - a-1z-1 + a-1z + 3a-1z3 + 2a-1z5 + az-1 + az + az3 + az5 - a3z - a3z3

Kauffman Polynomial:

a-7z3 - a-7z5 + 6a-6z4 - 4a-6z6 - 3a-5z3 + 12a-5z5 - 7a-5z7 + 4a-4z2 - 12a-4z4 + 16a-4z6 - 8a-4z8 - 2a-3z + 6a-3z3 - 11a-3z5 + 11a-3z7 - 6a-3z9 + 8a-2z2 - 29a-2z4 + 24a-2z6 - 6a-2z8 - 2a-2z10 - a-1z-1 - 2a-1z + 18a-1z3 - 37a-1z5 + 29a-1z7 - 10a-1z9 + 1 + 6z2 - 17z4 + 14z6 - 2z8 - 2z10 - az-1 + 2az + az3 - 4az5 + 8az7 - 4az9 - 3a2z4 + 9a2z6 - 4a2z8 + 2a3z - 7a3z3 + 9a3z5 - 3a3z7 - 2a4z2 + 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                     3   j = 10                   5 1   j = 8                 7 3     j = 6               9 5       j = 4             9 7         j = 2           9 9           j = 0         7 11             j = -2       4 7               j = -4     2 7                 j = -6   1 4                   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, 54]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 54]]
Out[4]=	PD[X[6, 1, 7, 2], X[20, 7, 21, 8], X[4, 21, 1, 22], X[14, 6, 15, 5], > X[10, 4, 11, 3], X[16, 12, 17, 11], X[12, 16, 13, 15], X[22, 14, 5, 13], > X[18, 9, 19, 10], X[2, 18, 3, 17], X[8, 19, 9, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {4, -1, 2, -11, 9, -5, 6, -7, 8, -4, 7, -6, 10, -9, > 11, -2, 3, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 6 11 14 3/2 5/2 q - ---- + ---- - ---- + ------- - 18 Sqrt[q] + 18 q - 16 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 12 q - 8 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 2 4 -2 2 4 6 10 12 14 5 - q + q - -- + -- + q - q + 3 q - 2 q + 2 q - 3 q + 3 q - 8 6 q q 18 20 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 54]][a, z]
Out[8]=	3 3 5 5 1 a z z 3 z 3 z 3 3 3 z 2 z 5 -(---) + - - -- + - + a z - a z - -- + ---- + a z - a z + -- + ---- + a z a z z 3 a 5 a 3 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 54]][a, z]
Out[9]=	2 2 3 1 a 2 z 2 z 3 2 4 z 8 z 4 2 z 1 - --- - - - --- - --- + 2 a z + 2 a z + 6 z + ---- + ---- - 2 a z + -- - a z z 3 a 4 2 7 a a a a 3 3 3 4 4 4 3 z 6 z 18 z 3 3 3 4 6 z 12 z 29 z > ---- + ---- + ----- + a z - 7 a z - 17 z + ---- - ----- - ----- - 5 3 a 6 4 2 a a a a a 5 5 5 5 2 4 4 4 z 12 z 11 z 37 z 5 3 5 6 > 3 a z + 3 a z - -- + ----- - ----- - ----- - 4 a z + 9 a z + 14 z - 7 5 3 a a a a 6 6 6 7 7 7 4 z 16 z 24 z 2 6 4 6 7 z 11 z 29 z 7 > ---- + ----- + ----- + 9 a z - a z - ---- + ----- + ----- + 8 a z - 6 4 2 5 3 a a a a a a 8 8 9 9 3 7 8 8 z 6 z 2 8 6 z 10 z 9 10 > 3 a z - 2 z - ---- - ---- - 4 a z - ---- - ----- - 4 a z - 2 z - 4 2 3 a a a a 10 2 z > ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 4 2 7 4 7 7 11 + 9 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t q t 2 4 4 2 6 2 6 3 8 3 8 4 > 9 q t + 9 q t + 7 q t + 9 q t + 5 q t + 7 q t + 3 q t + 10 4 10 5 12 5 14 6 > 5 q t + q t + 3 q t + q t

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