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Knotscape

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DrawMorseLink

PD Presentation:

X12,1,13,2 X7,17,8,16 X3948 X17,2,18,3 X5,14,6,15 X11,6,12,7 X9,18,10,19 X15,11,16,20 X10,13,1,14 X19,5,20,4

Gauss Code:

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

Jones Polynomial:

- 3q-9/2 + 6q-7/2 - 9q-5/2 + 10q-3/2 - 11q-1/2 + 9q1/2 - 7q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

q-18 + 3q-14 + 2q-10 + 2q-8 - 2q-6 + 3q-4 - 2q-2 + 3 - q4 + q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- 2a-1z3 - a-1z5 + 4az3 + 4az5 + az7 - a3z-1 - 3a3z - 3a3z3 - a3z5 + a5z-1 + a5z

Kauffman Polynomial:

a-3z3 - a-3z5 - 2a-2z2 + 7a-2z4 - 4a-2z6 + a-1z - 5a-1z3 + 12a-1z5 - 6a-1z7 - 5z2 + 13z4 - 2z6 - 3z8 + 4az - 18az3 + 26az5 - 12az7 - 3a2z2 + 6a2z4 - a2z6 - 3a2z8 - a3z-1 + 9a3z - 18a3z3 + 13a3z5 - 6a3z7 + a4 - 3a4z6 - a5z-1 + 6a5z - 6a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                 1 j = 6               3   j = 4             4 1   j = 2           5 3     j = 0         6 4       j = -2       5 6         j = -4     4 5           j = -6   2 5             j = -8 1 4               j = -10 3

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, NonAlternating, 63]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 63]]
Out[4]=	PD[X[12, 1, 13, 2], X[7, 17, 8, 16], X[3, 9, 4, 8], X[17, 2, 18, 3], > X[5, 14, 6, 15], X[11, 6, 12, 7], X[9, 18, 10, 19], X[15, 11, 16, 20], > X[10, 13, 1, 14], X[19, 5, 20, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, 10, -5, 6, -2, 3, -7, -9}, > {-6, -1, 9, 5, -8, 2, -4, 7, -10, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-3 6 9 10 11 3/2 5/2 7/2 ---- + ---- - ---- + ---- - ------- + 9 Sqrt[q] - 7 q + 4 q - q 9/2 7/2 5/2 3/2 Sqrt[q] q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 3 2 2 2 3 2 4 6 8 10 3 + q + --- + --- + -- - -- + -- - -- - q + q - 2 q + q 14 10 8 6 4 2 q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 63]][a, z]
Out[8]=	3 5 3 5 a a 3 5 2 z 3 3 3 z 5 3 5 -(--) + -- - 3 a z + a z - ---- + 4 a z - 3 a z - -- + 4 a z - a z + z z a a 7 > a z
In[9]:=	Kauffman[Link[10, NonAlternating, 63]][a, z]
Out[9]=	3 5 2 3 4 a a z 3 5 2 2 z 2 2 z a - -- - -- + - + 4 a z + 9 a z + 6 a z - 5 z - ---- - 3 a z + -- - z z a 2 3 a a 3 4 5 5 5 z 3 3 3 5 3 4 7 z 2 4 z 12 z > ---- - 18 a z - 18 a z - 6 a z + 13 z + ---- + 6 a z - -- + ----- + a 2 3 a a a 6 7 5 3 5 6 4 z 2 6 4 6 6 z 7 > 26 a z + 13 a z - 2 z - ---- - a z - 3 a z - ---- - 12 a z - 2 a a 3 7 8 2 8 > 6 a z - 3 z - 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 3 1 4 2 5 4 5 5 6 + -- + ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 4 t + 2 10 4 8 4 8 3 6 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t q t 2 2 2 4 2 4 3 6 3 8 4 > 5 q t + 3 q t + 4 q t + q t + 3 q t + q t

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