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L10n112 L11a1

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X2536 X11,19,12,18 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X20,15,17,16 X16,19,13,20 X17,9,18,12

Gauss Code:

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

Jones Polynomial:

q-3 + q-1 + 5 + 5q2 - q3 + 5q4 - q5 + q6

A2 (sl(3)) Invariant:

q-10 + 2q-8 + 5q-6 + 8q-4 + 16q-2 + 23 + 28q2 + 33q4 + 32q6 + 31q8 + 25q10 + 18q12 + 12q14 + 5q16 + 3q18 + q20

HOMFLY-PT Polynomial:

a-6z-4 + 2a-6z-2 + a-6 - 4a-4z-4 - 9a-4z-2 - 8a-4 - 3a-4z2 + 6a-2z-4 + 15a-2z-2 + 16a-2 + 9a-2z2 + 2a-2z4 - 4z-4 - 11z-2 - 12 - 7z2 - z4 + a2z-4 + 3a2z-2 + 3a2 + a2z2

Kauffman Polynomial:

- a-6z-4 + 5a-6z-2 - 10a-6 + 10a-6z2 - 5a-6z4 + a-6z6 + 4a-5z-3 - 15a-5z-1 + 20a-5z - 10a-5z3 - a-5z5 + a-5z7 - 4a-4z-4 + 14a-4z-2 - 25a-4 + 30a-4z2 - 25a-4z4 + 6a-4z6 + 12a-3z-3 - 41a-3z-1 + 55a-3z - 30a-3z3 + 3a-3z5 + a-3z7 - 6a-2z-4 + 18a-2z-2 - 31a-2 + 40a-2z2 - 25a-2z4 + 5a-2z6 + 12a-1z-3 - 41a-1z-1 + 55a-1z - 30a-1z3 + 5a-1z5 - 4z-4 + 14z-2 - 25 + 30z2 - 11z4 + z6 + 4az-3 - 15az-1 + 20az - 10az3 + az5 - a2z-4 + 5a2z-2 - 10a2 + 10a2z2 - 6a2z4 + a2z6

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                     1 j = 11                       j = 9                 5 1   j = 7               1 5     j = 5             4         j = 3         4   1         j = 1       1 10 4           j = -1         6             j = -3     1                 j = -5 1                     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]=	5
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 113]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 113]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[11, 19, 12, 18], X[3, 11, 4, 10], > X[9, 1, 10, 4], X[7, 15, 8, 14], X[13, 5, 14, 8], X[20, 15, 17, 16], > X[16, 19, 13, 20], X[17, 9, 18, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 10}, {-7, 6, 8, -9}, > {-10, 3, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-3 1 2 3 4 5 6 5 + q + - + 5 q - q + 5 q - q + q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 5 8 16 2 4 6 8 10 23 + q + -- + -- + -- + -- + 28 q + 33 q + 32 q + 31 q + 25 q + 8 6 4 2 q q q q 12 14 16 18 20 > 18 q + 12 q + 5 q + 3 q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 113]][a, z]
Out[8]=	2 -6 8 16 2 4 1 4 6 a 11 2 -12 + a - -- + -- + 3 a - -- + ----- - ----- + ----- + -- - -- + ----- - 4 2 4 6 4 4 4 2 4 4 2 6 2 a a z a z a z a z z z a z 2 2 2 4 9 15 3 a 2 3 z 9 z 2 2 4 2 z > ----- + ----- + ---- - 7 z - ---- + ---- + a z - z + ---- 4 2 2 2 2 4 2 2 a z a z z a a a
In[9]:=	Kauffman[Link[10, NonAlternating, 113]][a, z]
Out[9]=	2 10 25 31 2 4 1 4 6 a 4 12 -25 - -- - -- - -- - 10 a - -- - ----- - ----- - ----- - -- + ----- + ----- + 6 4 2 4 6 4 4 4 2 4 4 5 3 3 3 a a a z a z a z a z z a z a z 2 12 4 a 14 5 14 18 5 a 15 41 41 15 a > ---- + --- + -- + ----- + ----- + ----- + ---- - ---- - ---- - --- - ---- + 3 3 2 6 2 4 2 2 2 2 5 3 a z z a z z z a z a z a z z a z a z 2 2 2 20 z 55 z 55 z 2 10 z 30 z 40 z 2 2 > ---- + ---- + ---- + 20 a z + 30 z + ----- + ----- + ----- + 10 a z - 5 3 a 6 4 2 a a a a a 3 3 3 4 4 4 10 z 30 z 30 z 3 4 5 z 25 z 25 z 2 4 > ----- - ----- - ----- - 10 a z - 11 z - ---- - ----- - ----- - 6 a z - 5 3 a 6 4 2 a a a a a 5 5 5 6 6 6 7 7 z 3 z 5 z 5 6 z 6 z 5 z 2 6 z z > -- + ---- + ---- + a z + z + -- + ---- + ---- + a z + -- + -- 5 3 a 6 4 2 5 3 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	6 3 1 1 1 q 3 2 5 2 7 3 - + 10 q + 4 q + ----- + ----- + ----- + - + 4 q t + q t + 4 q t + q t + q 7 4 5 4 3 2 t q t q t q t 7 4 9 4 9 5 13 6 > 5 q t + 5 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n113

L10n112 L11a1