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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X2536 X16,12,17,11 X10,3,11,4 X4,9,1,10 X14,7,15,8 X8,13,5,14 X18,16,13,15 X12,18,9,17

Gauss Code:

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

Jones Polynomial:

- q-13/2 + q-11/2 - 5q-9/2 + 5q-7/2 - 10q-5/2 + 7q-3/2 - 8q-1/2 + 6q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant:

q-22 + 3q-20 + 4q-18 + 8q-16 + 12q-14 + 10q-12 + 13q-10 + 11q-8 + 8q-6 + 7q-4 + q-2 + 3 - q2 + 2q6 - q8

HOMFLY-PT Polynomial:

a-1z3 - az-3 - 3az-1 - 3az - 2az3 - az5 + 3a3z-3 + 7a3z-1 + 6a3z + 3a3z3 - 3a5z-3 - 5a5z-1 - 3a5z + a7z-3 + a7z-1

Kauffman Polynomial:

- a-2z4 + 4a-1z3 - 4a-1z5 + 8z4 - 6z6 - az-3 + 4az-1 - 6az + 4az3 + 2az5 - 4az7 + 3a2z-2 - 8a2 + 6a2z2 + 7a2z4 - 6a2z6 - a2z8 - 3a3z-3 + 9a3z-1 - 14a3z + 8a3z3 + 4a3z5 - 5a3z7 + 6a4z-2 - 15a4 + 12a4z2 - 2a4z4 - a4z6 - a4z8 - 3a5z-3 + 9a5z-1 - 14a5z + 12a5z3 - 3a5z5 - a5z7 + 3a6z-2 - 8a6 + 6a6z2 - a6z6 - a7z-3 + 4a7z-1 - 6a7z + 4a7z3 - a7z5

Khovanov Homology:

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

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