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Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X11,17,12,16 X8,9,1,10 X17,9,18,20 X3,12,4,13 X7,14,8,15 X13,6,14,7 X5,18,6,19 X19,4,20,5 X15,2,16,3

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 2q-17/2 - 5q-15/2 + 6q-13/2 - 7q-11/2 + 7q-9/2 - 6q-7/2 + 4q-5/2 - 2q-3/2

A2 (sl(3)) Invariant:

q-30 + q-28 + 3q-24 + q-22 + q-20 + 2q-18 - q-16 + q-14 - 2q-12 + q-8 - q-6 + 2q-4

HOMFLY-PT Polynomial:

- 3a3z - 2a3z3 + a5z + 2a5z3 + a5z5 - a7z-1 - 3a7z - 2a7z3 + a9z-1 + a9z

Kauffman Polynomial:

3a3z - 3a3z3 + 2a4z2 - 2a4z4 - a4z6 + a5z - 3a5z3 + 2a5z5 - 2a5z7 - 3a6z2 + 4a6z4 - 2a6z6 - a6z8 - a7z-1 + 7a7z - 12a7z3 + 11a7z5 - 5a7z7 + a8 - 6a8z2 + 10a8z4 - 3a8z6 - a8z8 - a9z-1 + 7a9z - 9a9z3 + 8a9z5 - 3a9z7 - a10z2 + 4a10z4 - 2a10z6 - 2a11z + 3a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                 2 j = -4               3 1 j = -6             3 1   j = -8           4 3     j = -10         3 3       j = -12       3 4         j = -14     2 3           j = -16     3             j = -18 1 2               j = -20 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, NonAlternating, 55]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 55]]
Out[4]=	PD[X[10, 1, 11, 2], X[11, 17, 12, 16], X[8, 9, 1, 10], X[17, 9, 18, 20], > X[3, 12, 4, 13], X[7, 14, 8, 15], X[13, 6, 14, 7], X[5, 18, 6, 19], > X[19, 4, 20, 5], X[15, 2, 16, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 10, -5, 9, -8, 7, -6, -3}, > {3, -1, -2, 5, -7, 6, -10, 2, -4, 8, -9, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 2 5 6 7 7 6 4 2 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 3 -22 -20 2 -16 -14 2 -8 -6 2 q + q + --- + q + q + --- - q + q - --- + q - q + -- 24 18 12 4 q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 55]][a, z]
Out[8]=	7 9 a a 3 5 7 9 3 3 5 3 7 3 5 5 -(--) + -- - 3 a z + a z - 3 a z + a z - 2 a z + 2 a z - 2 a z + a z z z
In[9]:=	Kauffman[Link[10, NonAlternating, 55]][a, z]
Out[9]=	7 9 8 a a 3 5 7 9 11 4 2 6 2 a - -- - -- + 3 a z + a z + 7 a z + 7 a z - 2 a z + 2 a z - 3 a z - z z 8 2 10 2 3 3 5 3 7 3 9 3 11 3 > 6 a z - a z - 3 a z - 3 a z - 12 a z - 9 a z + 3 a z - 4 4 6 4 8 4 10 4 5 5 7 5 9 5 > 2 a z + 4 a z + 10 a z + 4 a z + 2 a z + 11 a z + 8 a z - 11 5 4 6 6 6 8 6 10 6 5 7 7 7 > a z - a z - 2 a z - 3 a z - 2 a z - 2 a z - 5 a z - 9 7 6 8 8 8 > 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 1 1 2 3 2 3 3 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 20 8 18 8 18 7 16 6 14 6 14 5 12 5 q q t q t q t q t q t q t q t 4 3 3 4 3 3 1 3 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t

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