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L11n122 L11n124

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DrawMorseLink

PD Presentation:

X6172 X14,3,15,4 X9,16,10,17 X20,11,21,12 X8,21,9,22 X18,7,19,8 X12,19,13,20 X15,10,16,11 X22,17,5,18 X2536 X4,13,1,14

Gauss Code:

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

Jones Polynomial:

q-23/2 - 3q-21/2 + 7q-19/2 - 9q-17/2 + 10q-15/2 - 11q-13/2 + 8q-11/2 - 7q-9/2 + 3q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-36 - 2q-34 - 3q-30 + q-26 + 5q-22 + q-20 + 5q-18 + 2q-16 + 2q-12 - 2q-10 + q-8

HOMFLY-PT Polynomial:

- 2a5z3 - a5z5 - 3a7z-1 - 11a7z - 11a7z3 - 3a7z5 + 5a9z-1 + 10a9z + 4a9z3 - 2a11z-1 - a11z

Kauffman Polynomial:

2a5z3 - a5z5 + 5a6z4 - 3a6z6 - 3a7z-1 + 11a7z - 18a7z3 + 17a7z5 - 6a7z7 + 5a8 - 8a8z2 + 2a8z4 + 6a8z6 - 4a8z8 - 5a9z-1 + 15a9z - 25a9z3 + 20a9z5 - 6a9z7 - a9z9 + 5a10 - 4a10z2 - 8a10z4 + 10a10z6 - 5a10z8 - 2a11z-1 + 4a11z - 2a11z3 - a11z5 - a11z9 + 6a12z2 - 6a12z4 + a12z6 - a12z8 + 3a13z3 - 3a13z5 - a14 + 2a14z2 - a14z4

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                   1 j = -6                 3 1 j = -8               4     j = -10             4 3     j = -12           7 4       j = -14         4 5         j = -16       5 6           j = -18     2 4             j = -20   1 5               j = -22   2                 j = -24 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, 123]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 123]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[9, 16, 10, 17], X[20, 11, 21, 12], > X[8, 21, 9, 22], X[18, 7, 19, 8], X[12, 19, 13, 20], X[15, 10, 16, 11], > X[22, 17, 5, 18], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 6, -5, -3, 8, 4, -7, 11, -2, -8, 3, 9, -6, > 7, -4, 5, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 3 7 9 10 11 8 7 3 -(5/2) q - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - q 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-36 2 3 -26 5 -20 5 2 2 2 -8 -q - --- - --- + q + --- + q + --- + --- + --- - --- + q 34 30 22 18 16 12 10 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 123]][a, z]
Out[8]=	7 9 11 -3 a 5 a 2 a 7 9 11 5 3 7 3 ----- + ---- - ----- - 11 a z + 10 a z - a z - 2 a z - 11 a z + z z z 9 3 5 5 7 5 > 4 a z - a z - 3 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 123]][a, z]
Out[9]=	7 9 11 8 10 14 3 a 5 a 2 a 7 9 11 5 a + 5 a - a - ---- - ---- - ----- + 11 a z + 15 a z + 4 a z - z z z 8 2 10 2 12 2 14 2 5 3 7 3 9 3 > 8 a z - 4 a z + 6 a z + 2 a z + 2 a z - 18 a z - 25 a z - 11 3 13 3 6 4 8 4 10 4 12 4 14 4 > 2 a z + 3 a z + 5 a z + 2 a z - 8 a z - 6 a z - a z - 5 5 7 5 9 5 11 5 13 5 6 6 8 6 > a z + 17 a z + 20 a z - a z - 3 a z - 3 a z + 6 a z + 10 6 12 6 7 7 9 7 8 8 10 8 12 8 > 10 a z + a z - 6 a z - 6 a z - 4 a z - 5 a z - a z - 9 9 11 9 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 2 1 5 2 4 5 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q t q t q t q t q t q t q t 6 4 5 7 4 4 3 4 3 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ---- 16 5 14 5 14 4 12 4 12 3 10 3 10 2 8 2 6 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 L11n123

L11n122 L11n124