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L11n276 L11n278

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X9,18,10,19 X22,17,11,18 X20,11,21,12 X16,21,17,22 X15,1,16,4 X19,10,20,5

Gauss Code:

{{1, -4, -3, 10}, {-2, -1, 5, 3, -6, 11}, {8, 2, 4, -5, -10, -9, 7, 6, -11, -8, 9, -7}}

Jones Polynomial:

- q-9 + 2q-8 - 3q-7 + 4q-6 - 3q-5 + 4q-4 - q-3 + 2q-2

A2 (sl(3)) Invariant:

- q-32 - q-30 - 2q-28 - q-26 + 3q-20 + 3q-18 + 6q-16 + 6q-14 + 5q-12 + 5q-10 + 2q-8 + 2q-6

HOMFLY-PT Polynomial:

2a4z-2 + 8a4 + 8a4z2 + 2a4z4 - 5a6z-2 - 15a6 - 14a6z2 - 6a6z4 - a6z6 + 4a8z-2 + 8a8 + 5a8z2 + a8z4 - a10z-2 - a10

Kauffman Polynomial:

- 2a4z-2 + 10a4 - 11a4z2 + 3a4z4 + 5a5z-1 - 16a5z + 13a5z3 - 5a5z5 + a5z7 - 5a6z-2 + 20a6 - 29a6z2 + 19a6z4 - 6a6z6 + a6z8 + 9a7z-1 - 27a7z + 26a7z3 - 10a7z5 + 2a7z7 - 4a8z-2 + 13a8 - 21a8z2 + 18a8z4 - 6a8z6 + a8z8 + 5a9z-1 - 13a9z + 14a9z3 - 5a9z5 + a9z7 - a10z-2 + 2a10 - 3a10z2 + 2a10z4 + a11z-1 - 2a11z + a11z3

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -3               2 j = -5             1 2 j = -7           3     j = -9         1 2     j = -11       3 2       j = -13       1         j = -15   2 3           j = -17   1             j = -19 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 277]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 277]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[2, 14, 3, 13], > X[14, 7, 15, 8], X[9, 18, 10, 19], X[22, 17, 11, 18], X[20, 11, 21, 12], > X[16, 21, 17, 22], X[15, 1, 16, 4], X[19, 10, 20, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 10}, {-2, -1, 5, 3, -6, 11}, > {8, 2, 4, -5, -10, -9, 7, 6, -11, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-9 2 3 4 3 4 -3 2 -q + -- - -- + -- - -- + -- - q + -- 8 7 6 5 4 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 2 -26 3 3 6 6 5 5 2 2 -q - q - --- - q + --- + --- + --- + --- + --- + --- + -- + -- 28 20 18 16 14 12 10 8 6 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 277]][a, z]
Out[8]=	4 6 8 10 4 6 8 10 2 a 5 a 4 a a 4 2 6 2 8 a - 15 a + 8 a - a + ---- - ---- + ---- - --- + 8 a z - 14 a z + 2 2 2 2 z z z z 8 2 4 4 6 4 8 4 6 6 > 5 a z + 2 a z - 6 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 277]][a, z]
Out[9]=	4 6 8 10 5 7 9 4 6 8 10 2 a 5 a 4 a a 5 a 9 a 5 a 10 a + 20 a + 13 a + 2 a - ---- - ---- - ---- - --- + ---- + ---- + ---- + 2 2 2 2 z z z z z z z 11 a 5 7 9 11 4 2 6 2 > --- - 16 a z - 27 a z - 13 a z - 2 a z - 11 a z - 29 a z - z 8 2 10 2 5 3 7 3 9 3 11 3 4 4 > 21 a z - 3 a z + 13 a z + 26 a z + 14 a z + a z + 3 a z + 6 4 8 4 10 4 5 5 7 5 9 5 6 6 > 19 a z + 18 a z + 2 a z - 5 a z - 10 a z - 5 a z - 6 a z - 8 6 5 7 7 7 9 7 6 8 8 8 > 6 a z + a z + 2 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 2 3 1 3 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 19 7 17 6 15 6 15 5 13 4 11 4 11 3 q q q t q t q t q t q t q t q t 1 2 3 1 > ----- + ----- + ----- + ---- 9 3 9 2 7 2 5 q t q t q t q t

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