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L11a412 L11a414

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-14 + 3q-13 - 7q-12 + 12q-11 - 16q-10 + 19q-9 - 18q-8 + 17q-7 - 11q-6 + 8q-5 - 3q-4 + q-3

A2 (sl(3)) Invariant:

- q-44 - 2q-42 - q-38 - 2q-36 + 3q-34 - q-32 + 4q-30 + 5q-28 + 3q-26 + 9q-24 + q-22 + 6q-20 + 2q-18 - q-16 + 3q-14 - 2q-12 + q-10

HOMFLY-PT Polynomial:

2a6z2 + 3a6z4 + a6z6 + 2a8z-2 + 11a8 + 19a8z2 + 13a8z4 + 3a8z6 - 5a10z-2 - 19a10 - 20a10z2 - 6a10z4 + 4a12z-2 + 9a12 + 4a12z2 - a14z-2 - a14

Kauffman Polynomial:

2a6z2 - 3a6z4 + a6z6 + 3a7z3 - 7a7z5 + 3a7z7 - 2a8z-2 + 11a8 - 23a8z2 + 26a8z4 - 19a8z6 + 6a8z8 + 5a9z-1 - 19a9z + 28a9z3 - 17a9z5 - 2a9z7 + 4a9z9 - 5a10z-2 + 22a10 - 43a10z2 + 51a10z4 - 39a10z6 + 12a10z8 + a10z10 + 9a11z-1 - 35a11z + 50a11z3 - 29a11z5 - 2a11z7 + 7a11z9 - 4a12z-2 + 13a12 - 16a12z2 + 21a12z4 - 24a12z6 + 11a12z8 + a12z10 + 5a13z-1 - 19a13z + 31a13z3 - 27a13z5 + 8a13z7 + 3a13z9 - a14z-2 + 5a14z2 - 6a14z4 - 2a14z6 + 5a14z8 + a15z-1 - 2a15z + 4a15z3 - 7a15z5 + 5a15z7 - a16 + 3a16z2 - 5a16z4 + 3a16z6 + a17z - 2a17z3 + a17z5

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 = -5                       1 j = -7                     3 1 j = -9                   5     j = -11                 6 3     j = -13               11 5       j = -15             8 7         j = -17           11 10           j = -19         7 10             j = -21       5 9               j = -23     2 7                 j = -25   1 5                   j = -27   2                     j = -29 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, 413]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 413]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[20, 13, 21, 14], X[22, 19, 11, 20], > X[10, 15, 5, 16], X[8, 17, 9, 18], X[16, 7, 17, 8], X[18, 9, 19, 10], > 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, 7, -6, 8, -5}, > {11, -2, 3, -9, 5, -7, 6, -8, 4, -3, 9, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-14 3 7 12 16 19 18 17 11 8 3 -3 -q + --- - --- + --- - --- + -- - -- + -- - -- + -- - -- + q 13 12 11 10 9 8 7 6 5 4 q q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-44 2 -38 2 3 -32 4 5 3 9 -22 6 -q - --- - q - --- + --- - q + --- + --- + --- + --- + q + --- + 42 36 34 30 28 26 24 20 q q q q q q q q 2 -16 3 2 -10 > --- - q + --- - --- + q 18 14 12 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 413]][a, z]
Out[8]=	8 10 12 14 8 10 12 14 2 a 5 a 4 a a 6 2 11 a - 19 a + 9 a - a + ---- - ----- + ----- - --- + 2 a z + 2 2 2 2 z z z z 8 2 10 2 12 2 6 4 8 4 10 4 6 6 > 19 a z - 20 a z + 4 a z + 3 a z + 13 a z - 6 a z + a z + 8 6 > 3 a z
In[9]:=	Kauffman[Link[11, Alternating, 413]][a, z]
Out[9]=	8 10 12 14 9 11 8 10 12 16 2 a 5 a 4 a a 5 a 9 a 11 a + 22 a + 13 a - a - ---- - ----- - ----- - --- + ---- + ----- + 2 2 2 2 z z z z z z 13 15 5 a a 9 11 13 15 17 6 2 > ----- + --- - 19 a z - 35 a z - 19 a z - 2 a z + a z + 2 a z - z z 8 2 10 2 12 2 14 2 16 2 7 3 > 23 a z - 43 a z - 16 a z + 5 a z + 3 a z + 3 a z + 9 3 11 3 13 3 15 3 17 3 6 4 > 28 a z + 50 a z + 31 a z + 4 a z - 2 a z - 3 a z + 8 4 10 4 12 4 14 4 16 4 7 5 > 26 a z + 51 a z + 21 a z - 6 a z - 5 a z - 7 a z - 9 5 11 5 13 5 15 5 17 5 6 6 8 6 > 17 a z - 29 a z - 27 a z - 7 a z + a z + a z - 19 a z - 10 6 12 6 14 6 16 6 7 7 9 7 > 39 a z - 24 a z - 2 a z + 3 a z + 3 a z - 2 a z - 11 7 13 7 15 7 8 8 10 8 12 8 > 2 a z + 8 a z + 5 a z + 6 a z + 12 a z + 11 a z + 14 8 9 9 11 9 13 9 10 10 12 10 > 5 a z + 4 a z + 7 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-7 -5 1 2 1 5 2 7 5 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 29 11 27 10 25 10 25 9 23 9 23 8 21 8 q t q t q t q t q t q t q t 9 7 10 11 10 8 7 11 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 21 7 19 7 19 6 17 6 17 5 15 5 15 4 13 4 q t q t q t q t q t q t q t q t 5 6 3 5 3 > ------ + ------ + ------ + ----- + ---- 13 3 11 3 11 2 9 2 7 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a413

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