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L11n335 L11n337

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DrawMorseLink

PD Presentation:

X6172 X16,12,17,11 X8493 X2,18,3,17 X14,6,15,5 X18,7,19,8 X12,16,5,15 X13,20,14,21 X9,13,10,22 X21,11,22,10 X4,19,1,20

Gauss Code:

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

Jones Polynomial:

q-2 - 3q-1 + 6 - 7q + 8q2 - 7q3 + 8q4 - 4q5 + 3q6 - q7

A2 (sl(3)) Invariant:

q-8 + q-6 - 2q-4 + q-2 - q2 + 2q4 + q6 + 5q8 + 5q10 + 5q12 + 6q14 + q16 + 2q18 + q20 - q22

HOMFLY-PT Polynomial:

a-6z-2 - a-6z2 - 2a-4z-2 - 2a-4 + a-4z4 + a-2z-2 + 3a-2 + 4a-2z2 + 2a-2z4 - 2 - 3z2 + a2

Kauffman Polynomial:

4a-7z3 - 4a-7z5 + a-7z7 + a-6z-2 - 2a-6 - 9a-6z2 + 20a-6z4 - 14a-6z6 + 3a-6z8 - 2a-5z-1 + 2a-5z + 6a-5z3 - 3a-5z5 - 5a-5z7 + 2a-5z9 + 2a-4z-2 - 2a-4 - 17a-4z2 + 39a-4z4 - 33a-4z6 + 8a-4z8 - 2a-3z-1 - 2a-3z + 18a-3z3 - 16a-3z5 - a-3z7 + 2a-3z9 + a-2z-2 - a-2 - 3a-2z2 + 15a-2z4 - 17a-2z6 + 5a-2z8 - 6a-1z + 19a-1z3 - 17a-1z5 + 5a-1z7 - 1 + 6z2 - 4z4 + 2z6 - 2az + 3az3 - a2 + a2z2

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 15                   1 j = 13                 2   j = 11               2 1   j = 9             6 2     j = 7           3 4       j = 5         5 4         j = 3     1 3 3           j = 1     4 5             j = -1   1 4               j = -3   2                 j = -5 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, NonAlternating, 336]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 336]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 12, 17, 11], X[8, 4, 9, 3], X[2, 18, 3, 17], > X[14, 6, 15, 5], X[18, 7, 19, 8], X[12, 16, 5, 15], X[13, 20, 14, 21], > X[9, 13, 10, 22], X[21, 11, 22, 10], X[4, 19, 1, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11}, {5, -1, 6, -3, -9, 10, 2, -7}, > {-8, -5, 7, -2, 4, -6, 11, 8, -10, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-2 3 2 3 4 5 6 7 6 + q - - - 7 q + 8 q - 7 q + 8 q - 4 q + 3 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 2 -2 2 4 6 8 10 12 14 16 q + q - -- + q - q + 2 q + q + 5 q + 5 q + 5 q + 6 q + q + 4 q 18 20 22 > 2 q + q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 336]][a, z]
Out[8]=	2 2 4 4 2 3 2 1 2 1 2 z 4 z z 2 z -2 - -- + -- + a + ----- - ----- + ----- - 3 z - -- + ---- + -- + ---- 4 2 6 2 4 2 2 2 6 2 4 2 a a a z a z a z a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 336]][a, z]
Out[9]=	2 2 -2 2 1 2 1 2 2 2 z 2 z -1 - -- - -- - a - a + ----- + ----- + ----- - ---- - ---- + --- - --- - 6 4 6 2 4 2 2 2 5 3 5 3 a a a z a z a z a z a z a a 2 2 2 3 3 3 6 z 2 9 z 17 z 3 z 2 2 4 z 6 z 18 z > --- - 2 a z + 6 z - ---- - ----- - ---- + a z + ---- + ---- + ----- + a 6 4 2 7 5 3 a a a a a a 3 4 4 4 5 5 5 19 z 3 4 20 z 39 z 15 z 4 z 3 z 16 z > ----- + 3 a z - 4 z + ----- + ----- + ----- - ---- - ---- - ----- - a 6 4 2 7 5 3 a a a a a a 5 6 6 6 7 7 7 7 8 17 z 6 14 z 33 z 17 z z 5 z z 5 z 3 z > ----- + 2 z - ----- - ----- - ----- + -- - ---- - -- + ---- + ---- + a 6 4 2 7 5 3 a 6 a a a a a a a 8 8 9 9 8 z 5 z 2 z 2 z > ---- + ---- + ---- + ---- 4 2 5 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	4 3 1 2 1 3 3 2 5 2 - + 4 q + q + ----- + ---- + --- + 5 q t + 3 q t + 3 q t + 5 q t + q 5 2 3 q t q t q t 5 3 7 3 7 4 9 4 9 5 11 5 11 6 > 4 q t + 3 q t + 4 q t + 6 q t + 2 q t + 2 q t + q t + 13 6 15 7 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n336

L11n335 L11n337