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DrawMorseLink

PD Presentation:

X8192 X18,9,19,10 X6718 X20,14,7,13 X12,5,13,6 X10,4,11,3 X4,15,5,16 X16,12,17,11 X14,20,15,19 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

q-11/2 - 4q-9/2 + 8q-7/2 - 13q-5/2 + 15q-3/2 - 17q-1/2 + 15q1/2 - 12q3/2 + 8q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

- q-16 + 2q-14 - 2q-12 + 2q-10 + 4q-8 - q-6 + 5q-4 - 2q-2 + 2 - 2q4 + 3q6 - 2q8 + q10 + q12 - q14

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-4z2 + 2a-4z4 - a-4z6 + 2a-3z - 7a-3z3 + 10a-3z5 - 4a-3z7 - 7a-2z4 + 14a-2z6 - 6a-2z8 + 6a-1z - 25a-1z3 + 29a-1z5 - 5a-1z7 - 3a-1z9 + 2z2 - 19z4 + 34z6 - 15z8 + az-1 + 5az - 29az3 + 39az5 - 12az7 - 3az9 - a2 - 2a2z4 + 11a2z6 - 9a2z8 + a3z-1 + a3z - 9a3z3 + 16a3z5 - 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 j = 10                     1 j = 8                   3   j = 6                 5 1   j = 4               7 3     j = 2             8 5       j = 0           9 7         j = -2         7 9           j = -4       6 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[10, Alternating, 53]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 53]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 9, 19, 10], X[6, 7, 1, 8], X[20, 14, 7, 13], > X[12, 5, 13, 6], X[10, 4, 11, 3], X[4, 15, 5, 16], X[16, 12, 17, 11], > X[14, 20, 15, 19], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 6, -7, 5, -3}, > {3, -1, 2, -6, 8, -5, 4, -9, 7, -8, 10, -2, 9, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 4 8 13 15 17 3/2 q - ---- + ---- - ---- + ---- - ------- + 15 Sqrt[q] - 12 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 > 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 2 2 4 -6 5 2 4 6 8 10 2 - q + --- - --- + --- + -- - q + -- - -- - 2 q + 3 q - 2 q + q + 14 12 10 8 4 2 q q q q q q 12 14 > q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 53]][a, z]
Out[8]=	3 3 3 5 a a z 3 z 3 z 5 z 3 3 3 2 z -(-) + -- + -- - --- + 2 a z - a z + -- - ---- + 6 a z - 2 a z - ---- + z z 3 a 3 a a a a 5 3 5 7 > 4 a z - a z + a z
In[9]:=	Kauffman[Link[10, Alternating, 53]][a, z]
Out[9]=	3 2 3 3 2 a a 2 z 6 z 3 2 z 4 2 7 z 25 z -a + - + -- + --- + --- + 5 a z + a z + 2 z - -- - a z - ---- - ----- - z z 3 a 4 3 a a a a 4 4 3 3 3 5 3 4 2 z 7 z 2 4 4 4 > 29 a z - 9 a z + 2 a z - 19 z + ---- - ---- - 2 a z + 7 a z - 4 2 a a 5 5 6 6 6 4 10 z 29 z 5 3 5 5 5 6 z 14 z > a z + ----- + ----- + 39 a z + 16 a z - 4 a z + 34 z - -- + ----- + 3 a 4 2 a a a 7 7 8 2 6 4 6 4 z 5 z 7 3 7 8 6 z > 11 a z - 8 a z - ---- - ---- - 12 a z - 11 a z - 15 z - ---- - 3 a 2 a a 9 2 8 3 z 9 > 9 a z - ---- - 3 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 3 1 5 3 8 6 8 9 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 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 7 2 2 2 4 2 4 3 6 3 6 4 > ---- + 7 t + 8 q t + 5 q t + 7 q t + 3 q t + 5 q t + q t + 2 q t 8 4 10 5 > 3 q t + q t

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