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L11n315 L11n317

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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X7,17,8,16 X20,10,21,9 X18,12,19,11 X22,20,11,19 X15,9,16,8 X10,22,5,21 X17,14,18,15 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

3q - 4q2 + 7q3 - 7q4 + 8q5 - 6q6 + 5q7 - 3q8 + q9

A2 (sl(3)) Invariant:

3q2 + q4 + 3q6 + 5q8 + 2q10 + 4q12 + 2q14 + 3q16 + 3q18 + 2q22 - q24 - q26 + q28

HOMFLY-PT Polynomial:

a-8z2 + a-6z-2 + 2a-6 - a-6z4 - 2a-4z-2 - 6a-4 - 5a-4z2 - 2a-4z4 + a-2z-2 + 4a-2 + 3a-2z2

Kauffman Polynomial:

2a-10z2 - 3a-10z4 + a-10z6 + 7a-9z3 - 10a-9z5 + 3a-9z7 + a-8 - 3a-8z2 + 6a-8z4 - 9a-8z6 + 3a-8z8 + 3a-7z3 - 8a-7z5 + a-7z7 + a-7z9 + a-6z-2 - 3a-6 + 8a-6z4 - 11a-6z6 + 4a-6z8 - 2a-5z-1 + 6a-5z - 8a-5z3 + 5a-5z5 - 2a-5z7 + a-5z9 + 2a-4z-2 - 8a-4 + 10a-4z2 - a-4z4 - a-4z6 + a-4z8 - 2a-3z-1 + 6a-3z - 4a-3z3 + 3a-3z5 + a-2z-2 - 5a-2 + 5a-2z2

Khovanov Homology:

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                 1 j = 17               2   j = 15             3 1   j = 13           4 3     j = 11         4 2       j = 9       3 4         j = 7     4 4           j = 5   1 4             j = 3 2 3               j = 1 3

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, 316]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 316]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[7, 17, 8, 16], X[20, 10, 21, 9], > X[18, 12, 19, 11], X[22, 20, 11, 19], X[15, 9, 16, 8], X[10, 22, 5, 21], > X[17, 14, 18, 15], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 7, 4, -8}, > {5, -2, 11, 9, -7, 3, -9, -5, 6, -4, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 4 5 6 7 8 9 3 q - 4 q + 7 q - 7 q + 8 q - 6 q + 5 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 4 6 8 10 12 14 16 18 22 24 3 q + q + 3 q + 5 q + 2 q + 4 q + 2 q + 3 q + 3 q + 2 q - q - 26 28 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 316]][a, z]
Out[8]=	2 2 2 4 4 2 6 4 1 2 1 z 5 z 3 z z 2 z -- - -- + -- + ----- - ----- + ----- + -- - ---- + ---- - -- - ---- 6 4 2 6 2 4 2 2 2 8 4 2 6 4 a a a a z a z a z a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 316]][a, z]
Out[9]=	2 -8 3 8 5 1 2 1 2 2 6 z 6 z 2 z a - -- - -- - -- + ----- + ----- + ----- - ---- - ---- + --- + --- + ---- - 6 4 2 6 2 4 2 2 2 5 3 5 3 10 a a a a z a z a z a z a z a a a 2 2 2 3 3 3 3 4 4 4 4 3 z 10 z 5 z 7 z 3 z 8 z 4 z 3 z 6 z 8 z z > ---- + ----- + ---- + ---- + ---- - ---- - ---- - ---- + ---- + ---- - -- - 8 4 2 9 7 5 3 10 8 6 4 a a a a a a a a a a a 5 5 5 5 6 6 6 6 7 7 7 10 z 8 z 5 z 3 z z 9 z 11 z z 3 z z 2 z > ----- - ---- + ---- + ---- + --- - ---- - ----- - -- + ---- + -- - ---- + 9 7 5 3 10 8 6 4 9 7 5 a a a a a a a a a a a 8 8 8 9 9 3 z 4 z z z z > ---- + ---- + -- + -- + -- 8 6 4 7 5 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 5 2 7 2 7 3 9 3 9 4 3 q + 2 q + 3 q t + q t + 4 q t + 4 q t + 4 q t + 3 q t + 4 q t + 11 4 11 5 13 5 13 6 15 6 15 7 17 7 > 4 q t + 2 q t + 4 q t + 3 q t + 3 q t + q t + 2 q t + 19 8 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n316

L11n315 L11n317