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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-11/2 - 3q-9/2 + 6q-7/2 - 8q-5/2 + 9q-3/2 - 10q-1/2 + 7q1/2 - 6q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-16 + q-14 - 2q-12 - 2q-6 + 3q-4 + 5 + 2q2 + q4 + 2q6 - q8 + q10

HOMFLY-PT Polynomial:

- 2a-1z-1 - 3a-1z - 3a-1z3 - a-1z5 + 3az-1 + 7az + 9az3 + 5az5 + az7 - a3z-1 - 3a3z - 3a3z3 - a3z5

Kauffman Polynomial:

- a-3z + 2a-3z3 - a-3z5 - 2a-2z2 + 6a-2z4 - 3a-2z6 - 2a-1z-1 + 4a-1z - 5a-1z3 + 8a-1z5 - 4a-1z7 + 3 - 7z2 + 10z4 - 2z6 - 2z8 - 3az-1 + 9az - 18az3 + 20az5 - 9az7 + 3a2 - 10a2z2 + 12a2z4 - 4a2z6 - 2a2z8 - a3z-1 + 4a3z - 8a3z3 + 8a3z5 - 5a3z7 + a4 - 4a4z2 + 7a4z4 - 5a4z6 + 3a5z3 - 3a5z5 + a6z2 - a6z4

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                   1 j = 6                 2   j = 4               4 1   j = 2             4 3     j = 0           6 3       j = -2         4 5         j = -4       4 5           j = -6     2 4             j = -8   1 4               j = -10   2                 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[9, Alternating, 31]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 31]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 9, 17, 10], X[6, 7, 1, 8], X[18, 13, 7, 14], > X[10, 4, 11, 3], X[14, 6, 15, 5], X[4, 12, 5, 11], X[12, 17, 13, 18], > X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -8, 4, -6, 9, -2, 8, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 6 8 9 10 3/2 5/2 q - ---- + ---- - ---- + ---- - ------- + 7 Sqrt[q] - 6 q + 3 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 7/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 2 2 3 2 4 6 8 10 5 - q + q - --- - -- + -- + 2 q + q + 2 q - q + q 12 6 4 q q q
In[8]:=	HOMFLYPT[Link[9, Alternating, 31]][a, z]
Out[8]=	3 3 5 -2 3 a a 3 z 3 3 z 3 3 3 z 5 --- + --- - -- - --- + 7 a z - 3 a z - ---- + 9 a z - 3 a z - -- + 5 a z - a z z z a a a 3 5 7 > a z + a z
In[9]:=	Kauffman[Link[9, Alternating, 31]][a, z]
Out[9]=	3 2 2 4 2 3 a a z 4 z 3 2 2 z 3 + 3 a + a - --- - --- - -- - -- + --- + 9 a z + 4 a z - 7 z - ---- - a z z z 3 a 2 a a 3 3 2 2 4 2 6 2 2 z 5 z 3 3 3 5 3 > 10 a z - 4 a z + a z + ---- - ---- - 18 a z - 8 a z + 3 a z + 3 a a 4 5 5 4 6 z 2 4 4 4 6 4 z 8 z 5 3 5 > 10 z + ---- + 12 a z + 7 a z - a z - -- + ---- + 20 a z + 8 a z - 2 3 a a a 6 7 5 5 6 3 z 2 6 4 6 4 z 7 3 7 > 3 a z - 2 z - ---- - 4 a z - 5 a z - ---- - 9 a z - 5 a z - 2 a a 8 2 8 > 2 z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 2 1 4 2 4 4 5 6 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 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 4 2 2 2 4 2 4 3 6 3 8 4 > ---- + 3 t + 4 q t + 3 q t + 4 q t + q t + 2 q t + q t 2 q t

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