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L11n348 L11n350

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-3 - 2q-2 + 6q-1 - 8 + 10q - 10q2 + 10q3 - 7q4 + 5q5 - q6

A2 (sl(3)) Invariant:

q-10 + q-8 + 3q-4 + 2 + 3q2 + 5q6 + q8 + 5q10 + 3q12 + q14 + 3q16 - q18

HOMFLY-PT Polynomial:

a-4z-2 + 3a-4 - a-4z4 - 2a-2z-2 - 5a-2 - 2a-2z2 + 2a-2z4 + a-2z6 + z-2 - 4z2 - 2z4 + 2a2 + a2z2

Kauffman Polynomial:

a-7z3 - 2a-6 - a-6z2 + 5a-6z4 + a-5z + a-5z3 + 2a-5z7 - a-4z-2 + 3a-4 - 6a-4z2 + 8a-4z4 - 5a-4z6 + 3a-4z8 + 2a-3z-1 - 8a-3z + 15a-3z3 - 15a-3z5 + 5a-3z7 + a-3z9 - 2a-2z-2 + 9a-2 - 16a-2z2 + 14a-2z4 - 14a-2z6 + 6a-2z8 + 2a-1z-1 - 8a-1z + 17a-1z3 - 20a-1z5 + 5a-1z7 + a-1z9 - z-2 + 3 - 6z2 + 7z4 - 8z6 + 3z8 + az + 2az3 - 5az5 + 2az7 - 2a2 + 5a2z2 - 4a2z4 + a2z6

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13                   1 j = 11                 4   j = 9               5 3   j = 7             5 2     j = 5           5 5       j = 3         5 5         j = 1       4 6           j = -1     2 4             j = -3     4               j = -5 1 2                 j = -7 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, 349]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 349]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[20, 12, 21, 11], X[18, 8, 19, 7], > X[22, 18, 13, 17], X[9, 17, 10, 16], X[15, 11, 16, 10], X[12, 20, 5, 19], > X[8, 22, 9, 21], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -9, -6, 7, 3, -8}, > {11, -2, -7, 6, 5, -4, 8, -3, 9, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-3 2 6 2 3 4 5 6 -8 + q - -- + - + 10 q - 10 q + 10 q - 7 q + 5 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 3 2 6 8 10 12 14 16 18 2 + q + q + -- + 3 q + 5 q + q + 5 q + 3 q + q + 3 q - q 4 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 349]][a, z]
Out[8]=	2 4 4 3 5 2 -2 1 2 2 2 z 2 2 4 z 2 z -- - -- + 2 a + z + ----- - ----- - 4 z - ---- + a z - 2 z - -- + ---- + 4 2 4 2 2 2 2 4 2 a a a z a z a a a 6 z > -- 2 a
In[9]:=	Kauffman[Link[11, NonAlternating, 349]][a, z]
Out[9]=	2 3 9 2 -2 1 2 2 2 z 8 z 8 z 3 - -- + -- + -- - 2 a - z - ----- - ----- + ---- + --- + -- - --- - --- + 6 4 2 4 2 2 2 3 a z 5 3 a a a a a z a z a z a a 2 2 2 3 3 3 3 2 z 6 z 16 z 2 2 z z 15 z 17 z > a z - 6 z - -- - ---- - ----- + 5 a z + -- + -- + ----- + ----- + 6 4 2 7 5 3 a a a a a a a 4 4 4 5 5 3 4 5 z 8 z 14 z 2 4 15 z 20 z 5 > 2 a z + 7 z + ---- + ---- + ----- - 4 a z - ----- - ----- - 5 a z - 6 4 2 3 a a a a a 6 6 7 7 7 8 6 5 z 14 z 2 6 2 z 5 z 5 z 7 8 3 z > 8 z - ---- - ----- + a z + ---- + ---- + ---- + 2 a z + 3 z + ---- + 4 2 5 3 a 4 a a a a a 8 9 9 6 z z z > ---- + -- + -- 2 3 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 2 4 2 4 4 q 3 6 q + 5 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 5 q t + 7 4 5 4 5 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 13 5 > 5 q t + 5 q t + 5 q t + 2 q t + 5 q t + 3 q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n349

L11n348 L11n350