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L11n221 L11n223

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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X13,20,14,21 X5,14,6,15 X4,21,5,22 X16,9,17,10 X22,15,9,16 X17,6,18,7 X7,18,8,19 X19,8,20,1

Gauss Code:

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

Jones Polynomial:

- q-27/2 + 2q-25/2 - 2q-23/2 + 2q-21/2 - 2q-19/2 + q-17/2 - q-13/2 - q-9/2

A2 (sl(3)) Invariant:

- q-44 + q-42 - q-34 + q-32 + q-28 + q-26 + q-24 + 2q-22 + 2q-20 + q-18 + q-16

HOMFLY-PT Polynomial:

- 2a9z-1 - 17a9z - 36a9z3 - 28a9z5 - 9a9z7 - a9z9 + 3a11z-1 + 15a11z + 19a11z3 + 8a11z5 + a11z7 - a13z-1 - 2a13z - a13z3

Kauffman Polynomial:

2a9z-1 - 17a9z + 36a9z3 - 28a9z5 + 9a9z7 - a9z9 - 3a10 + 15a10z2 - 19a10z4 + 8a10z6 - a10z8 + 3a11z-1 - 18a11z + 34a11z3 - 27a11z5 + 9a11z7 - a11z9 - 3a12 + 14a12z2 - 19a12z4 + 8a12z6 - a12z8 + a13z-1 - a13z - a13z3 - a14 + 2a14z2 - 2a14z4 + a15z - a15z5 + 3a16z2 - 2a16z4 + a17z - a17z3

Khovanov Homology:

trqj r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -8                     1 j = -10                     1 j = -12                 1     j = -14             1         j = -16           1 1 1       j = -18         2 1           j = -20         1 1           j = -22     2 2               j = -24   1 1                 j = -26   1                   j = -28 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 222]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 222]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[13, 20, 14, 21], > X[5, 14, 6, 15], X[4, 21, 5, 22], X[16, 9, 17, 10], X[22, 15, 9, 16], > X[17, 6, 18, 7], X[7, 18, 8, 19], X[19, 8, 20, 1]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6, -5, 9, -10, 11}, > {7, -1, 2, -3, -4, 5, 8, -7, -9, 10, -11, 4, 6, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) 2 2 2 2 -(17/2) -(13/2) -(9/2) -q + ----- - ----- + ----- - ----- + q - q - q 25/2 23/2 21/2 19/2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-44 -42 -34 -32 -28 -26 -24 2 2 -18 -16 -q + q - q + q + q + q + q + --- + --- + q + q 22 20 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 222]][a, z]
Out[8]=	9 11 13 -2 a 3 a a 9 11 13 9 3 11 3 ----- + ----- - --- - 17 a z + 15 a z - 2 a z - 36 a z + 19 a z - z z z 13 3 9 5 11 5 9 7 11 7 9 9 > a z - 28 a z + 8 a z - 9 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 222]][a, z]
Out[9]=	9 11 13 10 12 14 2 a 3 a a 9 11 13 -3 a - 3 a - a + ---- + ----- + --- - 17 a z - 18 a z - a z + z z z 15 17 10 2 12 2 14 2 16 2 9 3 > a z + a z + 15 a z + 14 a z + 2 a z + 3 a z + 36 a z + 11 3 13 3 17 3 10 4 12 4 14 4 16 4 > 34 a z - a z - a z - 19 a z - 19 a z - 2 a z - 2 a z - 9 5 11 5 15 5 10 6 12 6 9 7 11 7 > 28 a z - 27 a z - a z + 8 a z + 8 a z + 9 a z + 9 a z - 10 8 12 8 9 9 11 9 > a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-10 -8 1 1 1 1 2 2 1 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 28 10 26 9 24 9 24 8 22 8 22 7 20 6 q t q t q t q t q t q t q t 2 1 1 1 1 1 1 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ 18 6 20 5 18 5 16 5 16 4 14 4 16 3 12 2 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 L11n222

L11n221 L11n223