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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,10,13,9 X18,15,19,16 X16,7,17,8 X8,17,9,18 X20,13,5,14 X14,19,15,20 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

q-15/2 - 2q-13/2 + 5q-11/2 - 8q-9/2 + 9q-7/2 - 11q-5/2 + 10q-3/2 - 9q-1/2 + 5q1/2 - 3q3/2 + q5/2

A2 (sl(3)) Invariant:

- q-24 - q-22 - 3q-18 + 2q-14 + 4q-10 + q-8 + 2q-6 + 2q-4 + 4 - q2 + q6 - q8

HOMFLY-PT Polynomial:

a-1z + a-1z3 - az-1 - 3az - 2az3 - az5 - a3z3 - a3z5 + 2a5z-1 + 3a5z + 2a5z3 - a7z-1 - a7z

Kauffman Polynomial:

a-2z2 - a-2z4 - 2a-1z + 4a-1z3 - 3a-1z5 + 3z4 - 4z6 + az-1 - 5az + 5az3 + az5 - 4az7 - a2 - 2a2z2 + 4a2z4 - 3a2z8 - 3a3z3 + 10a3z5 - 5a3z7 - a3z9 + 3a4 - 8a4z2 + 4a4z4 + 7a4z6 - 5a4z8 - 2a5z-1 + 5a5z - 9a5z3 + 12a5z5 - 3a5z7 - a5z9 + 5a6 - 12a6z2 + 8a6z4 + 2a6z6 - 2a6z8 - a7z-1 + 2a7z - 5a7z3 + 6a7z5 - 2a7z7 + 2a8 - 5a8z2 + 4a8z4 - a8z6

Khovanov Homology:

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

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