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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X4,19,1,20 X14,12,15,11 X8493 X12,5,13,6 X20,13,5,14 X16,10,17,9 X10,16,11,15 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

q-11/2 - 3q-9/2 + 5q-7/2 - 9q-5/2 + 11q-3/2 - 12q-1/2 + 11q1/2 - 9q3/2 + 6q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

- q-18 + 2q-14 - q-12 + 2q-10 + 3q-8 - q-6 + 2q-4 - 2q-2 - q4 + 4q6 + q10 + 2q12 - q14

HOMFLY-PT Polynomial:

- a-3z-1 + a-3z3 + 2a-1z-1 + a-1z - a-1z3 - a-1z5 - 2az-1 - 2az - az3 - az5 + a3z-1 + 2a3z + 2a3z3 - a5z

Kauffman Polynomial:

2a-4z4 - a-4z6 + a-3z-1 - a-3z - 7a-3z3 + 12a-3z5 - 4a-3z7 + 2a-2z2 - 11a-2z4 + 15a-2z6 - 5a-2z8 + 2a-1z-1 - 4a-1z - 11a-1z3 + 17a-1z5 - 2a-1z7 - 2a-1z9 + 1 + 4z2 - 19z4 + 24z6 - 9z8 + 2az-1 - 6az - az3 + 8az5 - 2az7 - 2az9 + a2z2 - 2a2z4 + 4a2z6 - 4a2z8 + a3z-1 - 5a3z + 7a3z3 - 4a3z7 + 3a4z4 - 4a4z6 - 2a5z + 4a5z3 - 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 r = 5 j = 10                     1 j = 8                   3   j = 6                 3 1   j = 4               6 3     j = 2             5 3       j = 0           7 6         j = -2         6 7           j = -4       3 5             j = -6     2 6               j = -8   1 3                 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[10, Alternating, 7]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 7]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[4, 19, 1, 20], X[14, 12, 15, 11], > X[8, 4, 9, 3], X[12, 5, 13, 6], X[20, 13, 5, 14], X[16, 10, 17, 9], > X[10, 16, 11, 15], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {6, -1, 2, -5, 8, -9, 4, -6, 7, -4, 9, -8, 10, -2, > 3, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 5 9 11 12 3/2 5/2 q - ---- + ---- - ---- + ---- - ------- + 11 Sqrt[q] - 9 q + 6 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 7/2 9/2 > 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 -12 2 3 -6 2 2 4 6 10 12 14 -q + --- - q + --- + -- - q + -- - -- - q + 4 q + q + 2 q - q 14 10 8 4 2 q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 7]][a, z]
Out[8]=	3 3 3 1 2 2 a a z 3 5 z z 3 -(----) + --- - --- + -- + - - 2 a z + 2 a z - a z + -- - -- - a z + 3 a z z z a 3 a a z a 5 3 3 z 5 > 2 a z - -- - a z a
In[9]:=	Kauffman[Link[10, Alternating, 7]][a, z]
Out[9]=	3 2 1 2 2 a a z 4 z 3 5 2 2 z 1 + ---- + --- + --- + -- - -- - --- - 6 a z - 5 a z - 2 a z + 4 z + ---- + 3 a z z z 3 a 2 a z a a 3 3 4 2 2 6 2 7 z 11 z 3 3 3 5 3 4 2 z > a z + a z - ---- - ----- - a z + 7 a z + 4 a z - 19 z + ---- - 3 a 4 a a 4 5 5 11 z 2 4 4 4 6 4 12 z 17 z 5 5 5 > ----- - 2 a z + 3 a z - a z + ----- + ----- + 8 a z - 3 a z + 2 3 a a a 6 6 7 7 6 z 15 z 2 6 4 6 4 z 2 z 7 3 7 > 24 z - -- + ----- + 4 a z - 4 a z - ---- - ---- - 2 a z - 4 a z - 4 2 3 a a a a 8 9 8 5 z 2 8 2 z 9 > 9 z - ---- - 4 a z - ---- - 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	7 1 2 1 3 2 6 3 5 7 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 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 6 2 2 2 4 2 4 3 6 3 6 4 > ---- + 6 t + 5 q t + 3 q t + 6 q t + 3 q t + 3 q t + q t + 2 q t 8 4 10 5 > 3 q t + q t

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