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L11n234 L11n236

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DrawMorseLink

PD Presentation:

X10,1,11,2 X5,20,6,21 X14,3,15,4 X4,15,5,16 X19,22,20,9 X16,7,17,8 X18,12,19,11 X12,18,13,17 X2,9,3,10 X8,13,1,14 X21,6,22,7

Gauss Code:

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

Jones Polynomial:

- q-19/2 + q-15/2 - 2q-13/2 + 2q-11/2 - 3q-9/2 + 3q-7/2 - 3q-5/2 + 2q-3/2 - q-1/2

A2 (sl(3)) Invariant:

2q-30 + q-26 + q-24 + q-22 + 2q-20 + 2q-16 - q-14 + q-2

HOMFLY-PT Polynomial:

- 3a3z - 4a3z3 - a3z5 + 2a5z + 6a5z3 + 5a5z5 + a5z7 - a7z-1 - 5a7z - 5a7z3 - a7z5 + a9z-1 + 2a9z

Kauffman Polynomial:

3a3z - 7a3z3 + 5a3z5 - a3z7 + 5a4z2 - 13a4z4 + 10a4z6 - 2a4z8 + a5z - 7a5z3 + 3a5z5 + 3a5z7 - a5z9 + 5a6z2 - 18a6z4 + 15a6z6 - 3a6z8 - a7z-1 + 5a7z - 7a7z3 + 4a7z7 - a7z9 + a8 - a8z2 - 4a8z4 + 5a8z6 - a8z8 - a9z-1 + 3a9z - 2a9z3 + a9z5 - a10z2 + a10z4 - 4a11z + 5a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                     1 j = -2                   1   j = -4                 2 1   j = -6               2 2     j = -8           1 2 1       j = -10           1 2         j = -12       1 3 2           j = -14         1             j = -16     1 2               j = -18 1                     j = -20 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, 235]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 235]]
Out[4]=	PD[X[10, 1, 11, 2], X[5, 20, 6, 21], X[14, 3, 15, 4], X[4, 15, 5, 16], > X[19, 22, 20, 9], X[16, 7, 17, 8], X[18, 12, 19, 11], X[12, 18, 13, 17], > X[2, 9, 3, 10], X[8, 13, 1, 14], X[21, 6, 22, 7]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 3, -4, -2, 11, 6, -10}, > {9, -1, 7, -8, 10, -3, 4, -6, 8, -7, -5, 2, -11, 5}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) -(15/2) 2 2 3 3 3 2 1 -q + q - ----- + ----- - ---- + ---- - ---- + ---- - ------- 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 -26 -24 -22 2 2 -14 -2 --- + q + q + q + --- + --- - q + q 30 20 16 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 235]][a, z]
Out[8]=	7 9 a a 3 5 7 9 3 3 5 3 7 3 -(--) + -- - 3 a z + 2 a z - 5 a z + 2 a z - 4 a z + 6 a z - 5 a z - z z 3 5 5 5 7 5 5 7 > a z + 5 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 235]][a, z]
Out[9]=	7 9 8 a a 3 5 7 9 11 4 2 6 2 a - -- - -- + 3 a z + a z + 5 a z + 3 a z - 4 a z + 5 a z + 5 a z - z z 8 2 10 2 3 3 5 3 7 3 9 3 11 3 > a z - a z - 7 a z - 7 a z - 7 a z - 2 a z + 5 a z - 4 4 6 4 8 4 10 4 3 5 5 5 9 5 > 13 a z - 18 a z - 4 a z + a z + 5 a z + 3 a z + a z - 11 5 4 6 6 6 8 6 3 7 5 7 7 7 > a z + 10 a z + 15 a z + 5 a z - a z + 3 a z + 4 a z - 4 8 6 8 8 8 5 9 7 9 > 2 a z - 3 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 2 1 1 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 20 8 18 8 16 6 16 5 12 5 14 4 12 4 q q q t q t q t q t q t q t q t 2 1 1 2 2 1 2 t t 2 > ------ + ------ + ----- + ------ + ----- + ---- + ---- + -- + -- + t 12 3 10 3 8 3 10 2 8 2 8 6 4 2 q t q t q t q t q t q t q t q q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n235

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