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Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X14,5,15,6 X12,3,13,4 X16,8,17,7 X18,15,9,16 X4,13,5,14 X6,18,7,17 X2,9,3,10 X8,11,1,12

Gauss Code:

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

Jones Polynomial:

q-17/2 - 2q-15/2 + 3q-13/2 - 6q-11/2 + 6q-9/2 - 6q-7/2 + 5q-5/2 - 4q-3/2 + 2q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-26 + 3q-18 + q-16 + 2q-14 + q-12 + 2q-8 - q-6 + q-4 + q2

HOMFLY-PT Polynomial:

- 2az - az3 - a3z-1 - a3z + 2a3z3 + a3z5 + a5z-1 + 3a5z + 3a5z3 + a5z5 - 2a7z - a7z3

Kauffman Polynomial:

- 2az + 3az3 - az5 - 2a2z2 + 5a2z4 - 2a2z6 - a3z-1 + 3a3z - 2a3z3 + 4a3z5 - 2a3z7 + a4 - a4z2 + 2a4z4 - a4z8 - a5z-1 + 6a5z - 12a5z3 + 10a5z5 - 4a5z7 - a6z2 - a6z8 - a7z - 3a7z3 + 3a7z5 - 2a7z7 + 2a8z4 - 2a8z6 - 2a9z + 4a9z3 - 2a9z5 + 2a10z2 - a10z4

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                   1 j = 0                 1   j = -2               3 1   j = -4             3 2     j = -6           3 2       j = -8         3 3         j = -10       3 3           j = -12     1 4             j = -14   1 2               j = -16   1                 j = -18 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[9, Alternating, 34]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 34]]
Out[4]=	PD[X[10, 1, 11, 2], X[14, 5, 15, 6], X[12, 3, 13, 4], X[16, 8, 17, 7], > X[18, 15, 9, 16], X[4, 13, 5, 14], X[6, 18, 7, 17], X[2, 9, 3, 10], > X[8, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 3, -6, 2, -7, 4, -9}, {8, -1, 9, -3, 6, -2, 5, -4, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 2 3 6 6 6 5 4 2 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - Sqrt[q] 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 3 -16 2 -12 2 -6 -4 2 -q + --- + q + --- + q + -- - q + q + q 18 14 8 q q q
In[8]:=	HOMFLYPT[Link[9, Alternating, 34]][a, z]
Out[8]=	3 5 a a 3 5 7 3 3 3 5 3 -(--) + -- - 2 a z - a z + 3 a z - 2 a z - a z + 2 a z + 3 a z - z z 7 3 3 5 5 5 > a z + a z + a z
In[9]:=	Kauffman[Link[9, Alternating, 34]][a, z]
Out[9]=	3 5 4 a a 3 5 7 9 2 2 4 2 a - -- - -- - 2 a z + 3 a z + 6 a z - a z - 2 a z - 2 a z - a z - z z 6 2 10 2 3 3 3 5 3 7 3 9 3 > a z + 2 a z + 3 a z - 2 a z - 12 a z - 3 a z + 4 a z + 2 4 4 4 8 4 10 4 5 3 5 5 5 > 5 a z + 2 a z + 2 a z - a z - a z + 4 a z + 10 a z + 7 5 9 5 2 6 8 6 3 7 5 7 7 7 > 3 a z - 2 a z - 2 a z - 2 a z - 2 a z - 4 a z - 2 a z - 4 8 6 8 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 1 1 2 1 4 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 3 3 3 3 2 3 t 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + t + -- + q t 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q

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