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L11a420 L11a422

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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X16,7,17,8 X20,9,21,10 X8,15,9,16 X10,19,5,20 X18,13,19,14 X22,17,11,18 X14,21,15,22 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-13 + 3q-12 - 8q-11 + 14q-10 - 18q-9 + 23q-8 - 22q-7 + 21q-6 - 15q-5 + 10q-4 - 4q-3 + q-2

A2 (sl(3)) Invariant:

- q-42 - 3q-40 - q-38 - 4q-34 + 5q-32 + 5q-30 + 2q-28 + 10q-26 + 3q-24 + 6q-22 + 2q-20 - q-18 + 5q-16 - 4q-14 + 2q-12 + 3q-10 - 3q-8 + q-6

HOMFLY-PT Polynomial:

a4z4 + a6 + 5a6z2 + 4a6z4 + 2a8z-2 + 7a8 + 9a8z2 + 5a8z4 - 5a10z-2 - 13a10 - 9a10z2 + 4a12z-2 + 5a12 - a14z-2

Kauffman Polynomial:

a4z4 + 4a5z5 - a6 + 5a6z2 - 9a6z4 + 10a6z6 + 7a7z3 - 19a7z5 + 15a7z7 - 2a8z-2 + 8a8 - 15a8z2 + 19a8z4 - 27a8z6 + 16a8z8 + 5a9z-1 - 18a9z + 35a9z3 - 35a9z5 + 9a9z9 - 5a10z-2 + 20a10 - 42a10z2 + 69a10z4 - 71a10z6 + 22a10z8 + 2a10z10 + 9a11z-1 - 33a11z + 48a11z3 - 17a11z5 - 23a11z7 + 13a11z9 - 4a12z-2 + 15a12 - 29a12z2 + 52a12z4 - 44a12z6 + 9a12z8 + 2a12z10 + 5a13z-1 - 19a13z + 26a13z3 - 9a13z5 - 7a13z7 + 4a13z9 - a14z-2 + 3a14 - 7a14z2 + 12a14z4 - 10a14z6 + 3a14z8 + a15z-1 - 4a15z + 6a15z3 - 4a15z5 + a15z7

Khovanov Homology:

trqj r = -11 r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -3                       1 j = -5                     4 1 j = -7                   6     j = -9                 9 4     j = -11               12 6       j = -13             11 10         j = -15           12 11           j = -17         8 13             j = -19       6 10               j = -21     2 8                 j = -23   1 6                   j = -25   2                     j = -27 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, Alternating, 421]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 421]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 7, 17, 8], X[20, 9, 21, 10], > X[8, 15, 9, 16], X[10, 19, 5, 20], X[18, 13, 19, 14], X[22, 17, 11, 18], > X[14, 21, 15, 22], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -5, 4, -6}, > {11, -2, 7, -9, 5, -3, 8, -7, 6, -4, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-13 3 8 14 18 23 22 21 15 10 4 -2 -q + --- - --- + --- - -- + -- - -- + -- - -- + -- - -- + q 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 3 -38 4 5 5 2 10 3 6 2 -18 -q - --- - q - --- + --- + --- + --- + --- + --- + --- + --- - q + 40 34 32 30 28 26 24 22 20 q q q q q q q q q 5 4 2 3 3 -6 > --- - --- + --- + --- - -- + q 16 14 12 10 8 q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 421]][a, z]
Out[8]=	8 10 12 14 6 8 10 12 2 a 5 a 4 a a 6 2 8 2 a + 7 a - 13 a + 5 a + ---- - ----- + ----- - --- + 5 a z + 9 a z - 2 2 2 2 z z z z 10 2 4 4 6 4 8 4 > 9 a z + a z + 4 a z + 5 a z
In[9]:=	Kauffman[Link[11, Alternating, 421]][a, z]
Out[9]=	8 10 12 14 9 6 8 10 12 14 2 a 5 a 4 a a 5 a -a + 8 a + 20 a + 15 a + 3 a - ---- - ----- - ----- - --- + ---- + 2 2 2 2 z z z z z 11 13 15 9 a 5 a a 9 11 13 15 6 2 > ----- + ----- + --- - 18 a z - 33 a z - 19 a z - 4 a z + 5 a z - z z z 8 2 10 2 12 2 14 2 7 3 9 3 > 15 a z - 42 a z - 29 a z - 7 a z + 7 a z + 35 a z + 11 3 13 3 15 3 4 4 6 4 8 4 10 4 > 48 a z + 26 a z + 6 a z + a z - 9 a z + 19 a z + 69 a z + 12 4 14 4 5 5 7 5 9 5 11 5 > 52 a z + 12 a z + 4 a z - 19 a z - 35 a z - 17 a z - 13 5 15 5 6 6 8 6 10 6 12 6 > 9 a z - 4 a z + 10 a z - 27 a z - 71 a z - 44 a z - 14 6 7 7 11 7 13 7 15 7 8 8 > 10 a z + 15 a z - 23 a z - 7 a z + a z + 16 a z + 10 8 12 8 14 8 9 9 11 9 13 9 > 22 a z + 9 a z + 3 a z + 9 a z + 13 a z + 4 a z + 10 10 12 10 > 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-5 -3 1 2 1 6 2 8 6 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 27 11 25 10 23 10 23 9 21 9 21 8 19 8 q t q t q t q t q t q t q t 10 8 13 12 11 11 10 12 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 19 7 17 7 17 6 15 6 15 5 13 5 13 4 11 4 q t q t q t q t q t q t q t q t 6 9 4 6 4 > ------ + ----- + ----- + ----- + ---- 11 3 9 3 9 2 7 2 5 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a421

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