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L11n306 L11n308

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X3,13,4,12 X7,17,8,16 X9,21,10,20 X15,9,16,8 X19,5,20,10 X13,19,14,18 X17,11,18,22 X21,15,22,14 X2536 X11,1,12,4

Gauss Code:

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

Jones Polynomial:

2q - 4q2 + 8q3 - 8q4 + 11q5 - 9q6 + 8q7 - 6q8 + 3q9 - q10

A2 (sl(3)) Invariant:

2q2 - q4 + 4q8 + 2q10 + 6q12 + 6q14 + 5q16 + 6q18 + q20 + 3q22 - q24 - 3q26 - 2q30 - q32

HOMFLY-PT Polynomial:

- a-10z-2 - a-10 + 4a-8z-2 + 7a-8 + 4a-8z2 - 5a-6z-2 - 12a-6 - 9a-6z2 - 3a-6z4 + 2a-4z-2 + 5a-4 + 2a-4z2 - a-4z4 + a-2 + 2a-2z2

Kauffman Polynomial:

a-11z-1 - 4a-11z + 6a-11z3 - 4a-11z5 + a-11z7 - a-10z-2 + 3a-10 - 6a-10z2 + 14a-10z4 - 12a-10z6 + 3a-10z8 + 5a-9z-1 - 19a-9z + 36a-9z3 - 21a-9z5 - a-9z7 + 2a-9z9 - 4a-8z-2 + 15a-8 - 32a-8z2 + 48a-8z4 - 39a-8z6 + 10a-8z8 + 9a-7z-1 - 33a-7z + 52a-7z3 - 38a-7z5 + 5a-7z7 + 2a-7z9 - 5a-6z-2 + 20a-6 - 37a-6z2 + 35a-6z4 - 24a-6z6 + 7a-6z8 + 5a-5z-1 - 18a-5z + 25a-5z3 - 20a-5z5 + 7a-5z7 - 2a-4z-2 + 8a-4 - 8a-4z2 + a-4z4 + 3a-4z6 + 3a-3z3 + a-3z5 - a-2 + 3a-2z2

Khovanov Homology:

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 21                   1 j = 19                 2   j = 17               4 1   j = 15             4 2     j = 13           6 5       j = 11         5 3         j = 9       3 6           j = 7     5 5             j = 5     4               j = 3 2 4                 j = 1 2

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, 307]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 307]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 13, 4, 12], X[7, 17, 8, 16], X[9, 21, 10, 20], > X[15, 9, 16, 8], X[19, 5, 20, 10], X[13, 19, 14, 18], X[17, 11, 18, 22], > X[21, 15, 22, 14], X[2, 5, 3, 6], X[11, 1, 12, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -3, 5, -4, 6}, > {-11, 2, -7, 9, -5, 3, -8, 7, -6, 4, -9, 8}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 4 5 6 7 8 9 10 2 q - 4 q + 8 q - 8 q + 11 q - 9 q + 8 q - 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 4 8 10 12 14 16 18 20 22 24 2 q - q + 4 q + 2 q + 6 q + 6 q + 5 q + 6 q + q + 3 q - q - 26 30 32 > 3 q - 2 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 307]][a, z]
Out[8]=	2 2 -10 7 12 5 -2 1 4 5 2 4 z 9 z -a + -- - -- + -- + a - ------ + ----- - ----- + ----- + ---- - ---- + 8 6 4 10 2 8 2 6 2 4 2 8 6 a a a a z a z a z a z a a 2 2 4 4 2 z 2 z 3 z z > ---- + ---- - ---- - -- 4 2 6 4 a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 307]][a, z]
Out[9]=	3 15 20 8 -2 1 4 5 2 1 5 --- + -- + -- + -- - a - ------ - ----- - ----- - ----- + ----- + ---- + 10 8 6 4 10 2 8 2 6 2 4 2 11 9 a a a a a z a z a z a z a z a z 2 2 2 2 9 5 4 z 19 z 33 z 18 z 6 z 32 z 37 z 8 z > ---- + ---- - --- - ---- - ---- - ---- - ---- - ----- - ----- - ---- + 7 5 11 9 7 5 10 8 6 4 a z a z a a a a a a a a 2 3 3 3 3 3 4 4 4 4 3 z 6 z 36 z 52 z 25 z 3 z 14 z 48 z 35 z z > ---- + ---- + ----- + ----- + ----- + ---- + ----- + ----- + ----- + -- - 2 11 9 7 5 3 10 8 6 4 a a a a a a a a a a 5 5 5 5 5 6 6 6 6 7 4 z 21 z 38 z 20 z z 12 z 39 z 24 z 3 z z > ---- - ----- - ----- - ----- + -- - ----- - ----- - ----- + ---- + --- - 11 9 7 5 3 10 8 6 4 11 a a a a a a a a a a 7 7 7 8 8 8 9 9 z 5 z 7 z 3 z 10 z 7 z 2 z 2 z > -- + ---- + ---- + ---- + ----- + ---- + ---- + ---- 9 7 5 10 8 6 9 7 a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 2 7 2 7 3 9 3 9 4 2 q + 2 q + 4 q t + 4 q t + 5 q t + 5 q t + 3 q t + 6 q t + 11 4 11 5 13 5 13 6 15 6 15 7 > 5 q t + 3 q t + 6 q t + 5 q t + 4 q t + 2 q t + 17 7 17 8 19 8 21 9 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n307

L11n306 L11n308