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L10n28 L10n30

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X3,10,4,11 X7,14,8,15 X15,20,16,5 X9,17,10,16 X19,9,20,8 X13,19,14,18 X17,13,18,12 X2536 X11,4,12,1

Gauss Code:

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

Jones Polynomial:

- 3q-9/2 + 5q-7/2 - 8q-5/2 + 9q-3/2 - 9q-1/2 + 8q1/2 - 6q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

q-18 + q-16 + 4q-14 + q-12 + 2q-10 + q-8 - 3q-6 + q-4 - 3q-2 + 2 + 2q6 - q8 + q10

HOMFLY-PT Polynomial:

- a-1z-1 - 3a-1z - 3a-1z3 - a-1z5 + 3az-1 + 7az + 9az3 + 5az5 + az7 - 4a3z-1 - 7a3z - 4a3z3 - a3z5 + 2a5z-1 + a5z

Kauffman Polynomial:

- a-3z + 2a-3z3 - a-3z5 + a-2 - 3a-2z2 + 6a-2z4 - 3a-2z6 - a-1z-1 + 2a-1z - 2a-1z3 + 7a-1z5 - 4a-1z7 + 3 - 10z2 + 14z4 - 3z6 - 2z8 - 3az-1 + 12az - 21az3 + 21az5 - 9az7 + 3a2 - 11a2z2 + 11a2z4 - 3a2z6 - 2a2z8 - 4a3z-1 + 17a3z - 23a3z3 + 13a3z5 - 5a3z7 + 2a4 - 4a4z2 + 3a4z4 - 3a4z6 - 2a5z-1 + 8a5z - 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               2   j = 4             4 1   j = 2           4 2     j = 0         5 4       j = -2       5 5         j = -4     3 4           j = -6   2 5             j = -8 1 3               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, 29]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 29]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 10, 4, 11], X[7, 14, 8, 15], X[15, 20, 16, 5], > X[9, 17, 10, 16], X[19, 9, 20, 8], X[13, 19, 14, 18], X[17, 13, 18, 12], > X[2, 5, 3, 6], X[11, 4, 12, 1]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, -2, 10}, {9, -1, -3, 6, -5, 2, -10, 8, -7, 3, -4, 5, -8, 7, > -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-3 5 8 9 9 3/2 5/2 7/2 ---- + ---- - ---- + ---- - ------- + 8 Sqrt[q] - 6 q + 3 q - q 9/2 7/2 5/2 3/2 Sqrt[q] q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 4 -12 2 -8 3 -4 3 6 8 10 2 + q + q + --- + q + --- + q - -- + q - -- + 2 q - q + q 14 10 6 2 q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 29]][a, z]
Out[8]=	3 5 3 1 3 a 4 a 2 a 3 z 3 5 3 z 3 -(---) + --- - ---- + ---- - --- + 7 a z - 7 a z + a z - ---- + 9 a z - a z z z z a a 5 3 3 z 5 3 5 7 > 4 a z - -- + 5 a z - a z + a z a
In[9]:=	Kauffman[Link[10, NonAlternating, 29]][a, z]
Out[9]=	3 5 -2 2 4 1 3 a 4 a 2 a z 2 z 3 3 + a + 3 a + 2 a - --- - --- - ---- - ---- - -- + --- + 12 a z + 17 a z + a z z z z 3 a a 2 3 3 5 2 3 z 2 2 4 2 2 z 2 z 3 > 8 a z - 10 z - ---- - 11 a z - 4 a z + ---- - ---- - 21 a z - 2 3 a a a 4 5 5 3 3 5 3 4 6 z 2 4 4 4 z 7 z > 23 a z - 6 a z + 14 z + ---- + 11 a z + 3 a z - -- + ---- + 2 3 a a a 6 7 5 3 5 6 3 z 2 6 4 6 4 z 7 > 21 a z + 13 a z - 3 z - ---- - 3 a z - 3 a z - ---- - 9 a z - 2 a a 3 7 8 2 8 > 5 a z - 2 z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 3 1 3 2 5 3 4 5 5 + -- + ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 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 > 4 q t + 2 q t + 4 q t + q t + 2 q t + q t

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