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L11n411 L11n413

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DrawMorseLink

PD Presentation:

X8192 X5,15,6,14 X10,3,11,4 X13,5,14,4 X2738 X6,9,1,10 X11,18,12,19 X17,12,18,7 X15,20,16,21 X19,22,20,13 X21,16,22,17

Gauss Code:

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

Jones Polynomial:

- q-10 + 2q-9 - 4q-8 + 6q-7 - 6q-6 + 8q-5 - 6q-4 + 6q-3 - 3q-2 + 2q-1

A2 (sl(3)) Invariant:

- q-32 - 2q-30 - q-28 - 3q-26 - q-24 + 3q-22 + 3q-20 + 7q-18 + 6q-16 + 5q-14 + 4q-12 + q-10 + 3q-8 + q-6 + 2q-2

HOMFLY-PT Polynomial:

2a2 + 2a2z2 + 2a4z-2 + 2a4 - a4z4 - 5a6z-2 - 9a6 - 6a6z2 - 2a6z4 + 4a8z-2 + 6a8 + 3a8z2 - a10z-2 - a10

Kauffman Polynomial:

- 2a2 + 3a2z2 + a3z + a3z3 + a3z5 - 2a4z-2 + 6a4 - 4a4z2 + 2a4z6 + 5a5z-1 - 16a5z + 18a5z3 - 12a5z5 + 4a5z7 - 5a6z-2 + 20a6 - 32a6z2 + 26a6z4 - 15a6z6 + 4a6z8 + 9a7z-1 - 33a7z + 44a7z3 - 28a7z5 + 4a7z7 + a7z9 - 4a8z-2 + 17a8 - 32a8z2 + 38a8z4 - 26a8z6 + 6a8z8 + 5a9z-1 - 21a9z + 35a9z3 - 20a9z5 + a9z7 + a9z9 - a10z-2 + 4a10 - 7a10z2 + 12a10z4 - 9a10z6 + 2a10z8 + a11z-1 - 5a11z + 8a11z3 - 5a11z5 + a11z7

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -1                   2 j = -3                 3 2 j = -5               3     j = -7             4 4     j = -9           4 2       j = -11         2 4         j = -13       4 4           j = -15     1 3             j = -17   1 3               j = -19   1                 j = -21 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, 412]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 412]]
Out[4]=	PD[X[8, 1, 9, 2], X[5, 15, 6, 14], X[10, 3, 11, 4], X[13, 5, 14, 4], > X[2, 7, 3, 8], X[6, 9, 1, 10], X[11, 18, 12, 19], X[17, 12, 18, 7], > X[15, 20, 16, 21], X[19, 22, 20, 13], X[21, 16, 22, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -5, 3, 4, -2, -6}, {5, -1, 6, -3, -7, 8}, > {-4, 2, -9, 11, -8, 7, -10, 9, -11, 10}]
In[6]:=	Jones[L][q]
Out[6]=	-10 2 4 6 6 8 6 6 3 2 -q + -- - -- + -- - -- + -- - -- + -- - -- + - 9 8 7 6 5 4 3 2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 2 -28 3 -24 3 3 7 6 5 4 -10 -q - --- - q - --- - q + --- + --- + --- + --- + --- + --- + q + 30 26 22 20 18 16 14 12 q q q q q q q q 3 -6 2 > -- + q + -- 8 2 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 412]][a, z]
Out[8]=	4 6 8 10 2 4 6 8 10 2 a 5 a 4 a a 2 2 2 a + 2 a - 9 a + 6 a - a + ---- - ---- + ---- - --- + 2 a z - 2 2 2 2 z z z z 6 2 8 2 4 4 6 4 > 6 a z + 3 a z - a z - 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 412]][a, z]
Out[9]=	4 6 8 10 5 7 2 4 6 8 10 2 a 5 a 4 a a 5 a 9 a -2 a + 6 a + 20 a + 17 a + 4 a - ---- - ---- - ---- - --- + ---- + ---- + 2 2 2 2 z z z z z z 9 11 5 a a 3 5 7 9 11 2 2 > ---- + --- + a z - 16 a z - 33 a z - 21 a z - 5 a z + 3 a z - z z 4 2 6 2 8 2 10 2 3 3 5 3 7 3 > 4 a z - 32 a z - 32 a z - 7 a z + a z + 18 a z + 44 a z + 9 3 11 3 6 4 8 4 10 4 3 5 5 5 > 35 a z + 8 a z + 26 a z + 38 a z + 12 a z + a z - 12 a z - 7 5 9 5 11 5 4 6 6 6 8 6 10 6 > 28 a z - 20 a z - 5 a z + 2 a z - 15 a z - 26 a z - 9 a z + 5 7 7 7 9 7 11 7 6 8 8 8 10 8 7 9 > 4 a z + 4 a z + a z + a z + 4 a z + 6 a z + 2 a z + a z + 9 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 3 1 3 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 21 9 19 8 17 8 17 7 15 7 15 6 13 6 q q t q t q t q t q t q t q t 4 2 4 4 2 4 4 3 3 > ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- 13 5 11 5 11 4 9 4 9 3 7 3 7 2 5 2 3 q t q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n412

L11n411 L11n413