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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,8,13,7 X13,18,14,19 X16,9,17,10 X8,17,9,18 X19,22,20,5 X15,21,16,20 X21,15,22,14 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 2q-9/2 - 4q-7/2 + 5q-5/2 - 5q-3/2 + 5q-1/2 - 5q1/2 + 3q3/2 - 2q5/2

A2 (sl(3)) Invariant:

q-18 + q-16 + 2q-12 - 2q-4 - 1 + q2 + 2q4 + q6 + 3q8 + q10

HOMFLY-PT Polynomial:

- a-3z-1 + a-1z-1 - a-1z3 + az-1 + 3az + 3az3 + az5 - 2a3z-1 - 4a3z - 2a3z3 + a5z-1 + a5z

Kauffman Polynomial:

a-3z-1 - a-3z - a-2 + a-2z2 - a-2z4 + a-1z-1 + a-1z - 9a-1z3 + 8a-1z5 - 2a-1z7 - 2 + 8z2 - 16z4 + 13z6 - 3z8 - az-1 + 12az - 28az3 + 19az5 - az7 - az9 - 3a2 + 12a2z2 - 26a2z4 + 22a2z6 - 5a2z8 - 2a3z-1 + 15a3z - 27a3z3 + 16a3z5 - a3z9 - a4 + 5a4z2 - 11a4z4 + 9a4z6 - 2a4z8 - a5z-1 + 5a5z - 8a5z3 + 5a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 6                 2 j = 4               1   j = 2             4 2   j = 0           3 3     j = -2         2 3 1     j = -4       3 3         j = -6     1 2           j = -8   1 3             j = -10   1               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[11, NonAlternating, 52]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 52]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[13, 18, 14, 19], > X[16, 9, 17, 10], X[8, 17, 9, 18], X[19, 22, 20, 5], X[15, 21, 16, 20], > X[21, 15, 22, 14], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -6, 5, -2, 11, -3, -4, 9, -8, -5, 6, 4, > -7, 8, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 4 5 5 5 3/2 5/2 -q + ---- - ---- + ---- - ---- + ------- - 5 Sqrt[q] + 3 q - 2 q 9/2 7/2 5/2 3/2 Sqrt[q] q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 2 2 2 4 6 8 10 -1 + q + q + --- - -- + q + 2 q + q + 3 q + q 12 4 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 52]][a, z]
Out[8]=	3 5 3 1 1 a 2 a a 3 5 z 3 3 3 -(----) + --- + - - ---- + -- + 3 a z - 4 a z + a z - -- + 3 a z - 2 a z + 3 a z z z z a a z 5 > a z
In[9]:=	Kauffman[Link[11, NonAlternating, 52]][a, z]
Out[9]=	3 5 -2 2 4 1 1 a 2 a a z z 3 -2 - a - 3 a - a + ---- + --- - - - ---- - -- - -- + - + 12 a z + 15 a z + 3 a z z z z 3 a a z a 2 3 5 2 z 2 2 4 2 9 z 3 3 3 > 5 a z + 8 z + -- + 12 a z + 5 a z - ---- - 28 a z - 27 a z - 2 a a 4 5 5 3 4 z 2 4 4 4 8 z 5 3 5 > 8 a z - 16 z - -- - 26 a z - 11 a z + ---- + 19 a z + 16 a z + 2 a a 7 5 5 6 2 6 4 6 2 z 7 5 7 8 > 5 a z + 13 z + 22 a z + 9 a z - ---- - a z - a z - 3 z - a 2 8 4 8 9 3 9 > 5 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-2 2 1 1 1 3 1 2 3 3 + q + 4 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 q t q t q t q t q t q t q t 3 2 3 3 2 4 6 2 > ----- + ----- + - + ---- + 2 q t + q t + 2 q t 4 2 2 2 t 2 q t q t q t

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