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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X18,9,19,10 X6718 X13,20,14,7 X12,5,13,6 X10,4,11,3 X4,15,5,16 X16,12,17,11 X19,14,20,15 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-13/2 - 3q-11/2 + 4q-9/2 - 5q-7/2 + 5q-5/2 - 5q-3/2 + 3q-1/2 - 3q1/2 + q3/2

A2 (sl(3)) Invariant:

- q-24 + q-22 + q-18 - q-14 + q-12 - q-10 + 3q-8 + q-6 + 2q-4 + 2q-2 + 1 + q2 - q4

HOMFLY-PT Polynomial:

- az-1 + 3az3 + az5 + a3z-1 - 3a3z - 7a3z3 - 5a3z5 - a3z7 + 3a5z + 4a5z3 + a5z5 - a7z

Kauffman Polynomial:

- z2 + 3z4 - z6 + az-1 + az - 11az3 + 12az5 - 3az7 - a2 - 2a2z4 + 6a2z6 - 2a2z8 + a3z-1 + 5a3z - 23a3z3 + 23a3z5 - 6a3z7 + 2a4z2 - 5a4z4 + 6a4z6 - 2a4z8 + 6a5z - 15a5z3 + 11a5z5 - 3a5z7 - a6z6 + 2a7z - 3a7z3 - a8z2

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                 1 j = 2               2   j = 0             1 1   j = -2           4 2     j = -4         2 2       j = -6       3 3         j = -8     2 3           j = -10   1 2             j = -12   2               j = -14 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, 50]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 50]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 9, 19, 10], X[6, 7, 1, 8], X[13, 20, 14, 7], > X[12, 5, 13, 6], X[10, 4, 11, 3], X[4, 15, 5, 16], X[16, 12, 17, 11], > X[19, 14, 20, 15], 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]=	-(13/2) 3 4 5 5 5 3 3/2 q - ----- + ---- - ---- + ---- - ---- + ------- - 3 Sqrt[q] + q 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -18 -14 -12 -10 3 -6 2 2 2 4 1 - q + q + q - q + q - q + -- + q + -- + -- + q - q 8 4 2 q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 50]][a, z]
Out[8]=	3 a a 3 5 7 3 3 3 5 3 5 -(-) + -- - 3 a z + 3 a z - a z + 3 a z - 7 a z + 4 a z + a z - z z 3 5 5 5 3 7 > 5 a z + a z - a z
In[9]:=	Kauffman[Link[10, NonAlternating, 50]][a, z]
Out[9]=	3 2 a a 3 5 7 2 4 2 8 2 -a + - + -- + a z + 5 a z + 6 a z + 2 a z - z + 2 a z - a z - z z 3 3 3 5 3 7 3 4 2 4 4 4 > 11 a z - 23 a z - 15 a z - 3 a z + 3 z - 2 a z - 5 a z + 5 3 5 5 5 6 2 6 4 6 6 6 7 > 12 a z + 23 a z + 11 a z - z + 6 a z + 6 a z - a z - 3 a z - 3 7 5 7 2 8 4 8 > 6 a z - 3 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 1 2 1 2 2 3 3 3 -- + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 4 2 14 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q q q t q t q t q t q t q t q t q t 2 2 t 2 2 2 4 3 > ---- + t + --- + t + 2 q t + q t 4 2 q t q

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