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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 2q-3/2 - 5q-1/2 + 7q1/2 - 10q3/2 + 10q5/2 - 10q7/2 + 9q9/2 - 6q11/2 + 3q13/2 - q15/2

A2 (sl(3)) Invariant:

q-8 + q-6 + q-4 + 3q-2 + 2q2 + q4 - q6 + 2q8 - 3q10 + q12 - q14 + 2q18 - q20 + q22

HOMFLY-PT Polynomial:

- a-5z-1 - 3a-5z - 3a-5z3 - a-5z5 + 3a-3z-1 + 10a-3z + 10a-3z3 + 5a-3z5 + a-3z7 - 4a-1z-1 - 10a-1z - 8a-1z3 - 2a-1z5 + 2az-1 + 3az + az3

Kauffman Polynomial:

- a-9z3 - 3a-8z4 - a-7z + 4a-7z3 - 6a-7z5 + a-6 - 7a-6z2 + 14a-6z4 - 9a-6z6 - a-5z-1 + 4a-5z - 10a-5z3 + 18a-5z5 - 9a-5z7 + 3a-4 - 14a-4z2 + 15a-4z4 + 4a-4z6 - 5a-4z8 - 3a-3z-1 + 19a-3z - 45a-3z3 + 44a-3z5 - 11a-3z7 - a-3z9 + 3a-2 - 6a-2z2 - 11a-2z4 + 21a-2z6 - 7a-2z8 - 4a-1z-1 + 21a-1z - 39a-1z3 + 25a-1z5 - 3a-1z7 - a-1z9 + 2 + z2 - 9z4 + 8z6 - 2z8 - 2az-1 + 7az - 9az3 + 5az5 - az7

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 16                     1 j = 14                   2   j = 12                 4 1   j = 10               5 2     j = 8             5 4       j = 6           5 5         j = 4         5 5           j = 2       4 7             j = 0     1 3               j = -2   1 4                 j = -4   1                   j = -6 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, 9]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 9]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[18, 14, 19, 13], > X[16, 9, 17, 10], X[8, 17, 9, 18], X[20, 16, 5, 15], X[14, 20, 15, 19], > X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -6, 5, -2, 10, -3, 4, -8, 7, -5, 6, -4, > 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 2 5 3/2 5/2 7/2 9/2 -q + ---- - ------- + 7 Sqrt[q] - 10 q + 10 q - 10 q + 9 q - 3/2 Sqrt[q] q 11/2 13/2 15/2 > 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 -4 3 2 4 6 8 10 12 14 18 q + q + q + -- + 2 q + q - q + 2 q - 3 q + q - q + 2 q - 2 q 20 22 > q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 9]][a, z]
Out[8]=	3 3 3 1 3 4 2 a 3 z 10 z 10 z 3 z 10 z 8 z -(----) + ---- - --- + --- - --- + ---- - ---- + 3 a z - ---- + ----- - ---- + 5 3 a z z 5 3 a 5 3 a a z a z a a a a 5 5 5 7 3 z 5 z 2 z z > a z - -- + ---- - ---- + -- 5 3 a 3 a a a
In[9]:=	Kauffman[Link[10, Alternating, 9]][a, z]
Out[9]=	-6 3 3 1 3 4 2 a z 4 z 19 z 21 z 2 + a + -- + -- - ---- - ---- - --- - --- - -- + --- + ---- + ---- + 7 a z + 4 2 5 3 a z z 7 5 3 a a a a z a z a a a 2 2 2 3 3 3 3 3 2 7 z 14 z 6 z z 4 z 10 z 45 z 39 z 3 > z - ---- - ----- - ---- - -- + ---- - ----- - ----- - ----- - 9 a z - 6 4 2 9 7 5 3 a a a a a a a a 4 4 4 4 5 5 5 5 4 3 z 14 z 15 z 11 z 6 z 18 z 44 z 25 z > 9 z - ---- + ----- + ----- - ----- - ---- + ----- + ----- + ----- + 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 7 7 7 5 6 9 z 4 z 21 z 9 z 11 z 3 z 7 8 > 5 a z + 8 z - ---- + ---- + ----- - ---- - ----- - ---- - a z - 2 z - 6 4 2 5 3 a a a a a a 8 8 9 9 5 z 7 z z z > ---- - ---- - -- - -- 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 -2 4 3 4 q 4 7 q + 5 q + ----- + ----- + ----- + t + ----- + - + ---- + 5 q t + 6 4 4 3 2 3 2 2 t t q t q t q t q t 6 6 2 8 2 8 3 10 3 10 4 12 4 > 5 q t + 5 q t + 5 q t + 4 q t + 5 q t + 2 q t + 4 q t + 12 5 14 5 16 6 > q t + 2 q t + q t

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